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STANDARD ALGEBRA

REVISED

BY

WILLIAM J. MILNE, Ph.D., LL.D.

PRESIDENT OF NEW YORK STATE COLLEGE FOR TEACHERS ALBANY, N.Y.

AMERICAN BOOK COMPANY

NEW YORX CINCINNATI CHICAGO

Copyright, 1908, 1914, by WILLIAM J. MILNE

Entered at Stationers' Hall, London.

standard algebra

E. P. I '

] ^ It

PREFACE

Scope. — This work has been written to meet the demands of colleges and universities for general admission and the require- ments of the course for elementary and intermediate algebra outlined by the Kegents of the State of New York. Every kind of question asked in recent examinations has been covered.

Method. — The author adheres to the inductive method of presentation, but uses declarative statements and observations instead of questions. These are followed by illustrative prob- lems and explanations which bring out the important points that should be. emphasized. The treatment is rounded out by abundant practice. The work is based on the student's knowledge of arithmetic. New ideas of number are intro- duced whenever the development of the science demands it.

Exercises. — The number of exercises is extremely large, and the variety is great. The concrete and abstract work are well proportioned, so that both skill in algebraic processes and ability to solve problems are properly sustained.

Problems. — Some of the traditional problems have been retained because they are often given in examinations and because they are useful in developing intellectual power; but the work contains in addition a large number of fresh and interesting problems drawn from commercial life, from physics and geometry, and from various topics of modern interest. While the formulae of physics and of geometry are used to familiarize the pupils with solutions for other letters than x, y, and z, no attempt has been made to present the subject-matter of these topics.

Algebraic Representation. — Throughout the early part of the book there are sets of exercises designed to teach algebraic language. In these the student is required to translate into algebraic notation expressions stated in words, and also to state in words expressions that are written with algebraic symbols.

Numerical Substitution. — A large amount of work is given in evaluating expressions. This is important not only in impart- ing a better idea of algebraic language, but it is used through- out the book in testing results. Accuracy is thus secured by

3

4 PREFACE

numerous checks and tests and by the requirement that roots of equations be verified. The student in this way becomes self-reliant, and reference to answers becomes unnecessary.

Graphs. — An interesting aid in the general solution of equations is given by the presentation of graphic solutions. They are not to be substituted for the ordinary methods of solution ; consequently, they are put after, rather than before, the particular kinds of equations to which they refer. They will be found to interest the student in a phase of algebra which has relation to his more advanced work in mathematics.

Factoring. — Present-day requirements which omit from high- est common factor and lowest common multiple the method by successive division make it imperative that the student shall be well prepared in the subject of factoring. The author has, therefore, fully treated all the usual cases as well as the factor theorem, giving a great amount of practice. The summary of cases presented at the close of the chapter on factoring will give the student unusual power in this important subject. Factoring by completing the square receives attention in the chapter on quadratics.

The solution of equations by factoring is treated early in the book, and wherever it is feasible to adopt that method.

Reviews. — Helpful and frequent reviews constitute a valua- ble feature of the Standard Algebra. They call for a knowl- edge of principles, processes, definitions, and for the solution of abstract exercises and of problems.

It is hoped that the main features of the book as specified above will commend it to those who are looking for a text that is thoroughly up to date in its matter, clear and intelligible in its presentation, thorough in its method of treatment, and certain to give the student not only a scholarly grasp of the science but delight in its mastery.

The revision of the Standard Algebra has improved the treatment at many points and has greatly increased the num- ber of easy, well-graded exercises and problems throughout the book. It has furthermore provided a chapter of Supple- mentary Exercises designed to afford additional practice work on important subjects throughout the book.

WILLIAM J. MILNE.

CONTENTS

PACE

Introduction 1

Definitions a\d Notation 13

Positive and Negative Numbers 23

Addition 30

Subtraction ► 34

Review 49

Multiplication 51

Division 72

Review 88

Factoring 92

Review of Factoring Ill

Highest Common Factor ... .... 119

Lowest Common Multiple 123

Fractions 126

Review 149

Simple Equations 152

Simultaneous Simple Equations 176

Graphic Solutions — Simple Equations 198

Involution 207

Evolution 214

Theory of Exponents 232

Radicals . .242

5

6 CONTENTS

PAGK

Imagixary Numbers 268

Review. ^ 272

Quadratic Equations 279

Graphic Solutions — Quadratic Equations .... 326

Properties of Quadratic Equations 339

General Review 350

Inequalities 361

Ratio and Proportion . 369

Variation 385

Progressions 394

Interpretation of Results 411

The Binomial Theorem . . 416

Logarithms 423

Complex Numbers 443

Supplementary Exercises 447

Index 490

STAIiTDARD ALGEBRA

^>Hc

INTRODUCTION

1. The basis of algebra is found in arithmetic. Both arith- metic and algebra treat of number, and the student will find in algebra many things that were familiar to him in arithmetic. In fact, there is no clear line of demarcation between arith- metic and algebra. The fundamental principles of each are identical, but in algebra their application is broader than it is in arithmetic.

The very attempt to make these principles universal leads to new kinds of number, and while the signs, symbols, and definitions that are given in arithmetic appear in algebra with their arithmetical meanings, yet in some instances they take on additional meanings.

To illustrate, arithmetic teaches the meaning of 5 — 3 and so does algebra, but it will be seen that algebra is more general than arithmetic in that it gives a meaning also to 3—5, which in arithmetic is meaningless. In this connection the student will see how addition does not always mean an increase, nor subtraction a decrease. Arithmetic teaches the meaning of 9^ ; that is, 9^ = 9 X 9. Later the student will learn that algebra gives a meaning to 9^ ; that is, 9^ = 3, one of the two equal factors of 9.

In short, algebra affords a more general discussion of number and its laws than is found in arithmetic.

INTRODUCTIOISr

ALGEBRAIC SOLUTIONS

2. The numbers in this chapter do not differ in character from the numbers with which the student is already familiar.

The following solutions and problems, however, serve to illustrate how the solution of an arithmetical problem may often be made easier and clearer by the algebraic method, in which the numbers sought are represented by letters, than by the ordinary arithmetical method.

Letters that are used for numbers are called literal numbers.

3. Illustrative Problem. — A man had 400 acres of corn and oats. If there were 3 times as many acres of corn as of oats, how many acres were there of each ?

Arithmetical Solution A certain number = the number of acres of oats. Then, 3 times that number = the number of acres of corn, and 4 times that number =: the number of acres of both ;

therefore, 4 times that number = 400.

Hence, the number = 100, the number of acres of oats,

and 3 times the number = 300, the number of acres of corn.

Algebraic Solution

Let X = the number of acres rf oats.

Then, • Sx = the number of acres of corn,

and 4 a; = the number of acres of both ;

therefore, 4 a; = 400.

Hence, ic = 100, the number of acres of oats,

and 3x = 300, the number of acres of corn.

Observe that in the algebraic solution x is used to stand for •' a certain number" or "that number," and thus the work is abbreviated.

4. An expression of the equality of two numbers or quan- tities is called an equation.

5 a: = 30 is au e(juation.

INTRODUCTION 9

5. A question that can be answered only after a course of reasoning is called a problem.

6. The process of finding the result sought is called the solution of the problem.

Problems

7. Solve, both arithmetically and algebraically, the follow- ing problems :

1. Mary is twice as old as Jane. If the sum of their ages is 15 years, what is the age of each ?

2. Two boys had together 48 marbles. If one had 3 times as many as the other, how many marbles did each have ?

3. A bicycle and suit cost $ 54. How much did each cost, if the bicycle cost twice as much as the suit ?

4. Two boys dug 160 clams. If one dug 3 times as many as the other, how many did each dig ?

5. Two boys bought a boat for $ 45. One boy furnished 4 times as much money as the other. How much money did each furnish ?

6. Find a number the double of which equals 52.

' 7. Six times a certain number equals 78. What is the number ?

8. Five times Eoy's age is 80 years. How old is Eoy ?

9. Find a number which added to its double equals 12.

10. Three times a certain number is twice 24. Find the number.

11. A house and lot cost $ 3000. If the house cost 4 times as much as the lot, what was the cost of each ?

12. One number is 5 times another and their sum is 84. Find the numbers.

13. Separate 90 into two parts, one of which is 5 times the other.

10 INTRODUCTION

14. A certain number added to 3 times itself equals 96. What is the number ?

15. The water and steam in a boiler occupied 120 cubic feet of space, and the water occupied twice as much space as the steam. How many cubic feet of space did each occupy ?

16. The sum of three numbers is 14. The second is twice the first and the third is 4 times the first. Find the numbers.

17. Separate 21 into three parts, such that the first is twice the second, and the second is twice the third.

18. Separate 36 into three parts, such that the first is twice the second, and the third is twice the sum of the first two.

19. In a fire B lost twice as much as A, and C lost 3 times as much as A. If their combined loss was $ 6000, how much did each lose ?

20. A boy bought a bat, a ball, and a glove for $ 2.25. If the bat cost twice as much as the ball, and the glove cost 3 times as much as the bat, what was the cost of each ?

21. Three newsboys sold 60 papers. If the first boy sold twice as many as the second, and the third sold 3 times as many as the second, how many did each sell ?

22. In 510 bushels of grain there was 4 times as much corn as wheat and 3 times as much barley as corn. How many bushels of each kind were there ?

23. In a business enterprise the joint capital of A, B, and C was $ 8400. If A's capital was twice B's, and B's was twice C's, what was the capital of each ?

24. A plumber and two helpers together earned $ 7.50 per day. How much did each earn per day, if the plumber earned 4 times as much as each helper ?

25. How much did John earn daily, if with 5 days^ earn- ings he bought a rifle and with 20 days' earnings, a bicycle, both together costing $ 50 ? How much did the rifle cost ? the bicycle ?

INTRODUCTION 11

26. The sides of any square (Fig. 1) are equal in length. How long is one side of a square, if the perimeter (distance around it) is 36 inches ?

Fig. 1 Fig. 2 Fig. 3

27. The length of each of the sides, a and 6, of the triangle (Fig. 2) is twice the length of the side c. If the perimeter is 40 inches, what is the length of each side ?

28. The opposite sides of any rectangle (Fig. 3) are equal. If a rectangle is twice as long as it is wide and its perimeter is 48 inches, how wide is it ? how long ?

29. How long is one side of a square, if the perimeter added to the length of one side is 15 inches ?

30. A and B began business with a capital of $ 7500. If A furnished half as much capital as B, how much did each furnish ?

SuGGESTjON. — Let X = the number of dollars A furnished.

31. James bought a pony and a saddle for % 60. If the saddle cost \ as much as the pony, how much did each cost?

32. The sum of two numbers is 35 and one number is \ of the other. Find the numbers.

33. Separate 12 into two parts, one of which is ^ of the other.

34. Separate 78 into two parts, one of which is \ of the other.

35. Two boys bought a football for 98 ^. If one paid \ as much as the other, how much did each pay ?

36. George earned $ 1.05 in two days. He earned half as much the first day as he did the second day. How much did he earn each day ?

12 INTRODUCTION

37. Ann paid $ 3.00 for three books. The first cost 4- as much as the second and J as much as the third. Find the cost of each.

38. A basket-ball team won 16 games, or f of the number of games it played. Find the number of games it played.

	Solution
Let	X — the number of games it played.
Then,	tx=16,
and	\x = ^.
Therefore,	X = 24, the number of games it played,

39. If it takes me 15 minutes to walk f of a mile, how long does it take me to walk a mile ?

40. Charles solved 14 problems or ^ of the problems in his lesson. How many problems were there in his lesson ?

41. The distance by rail between two cities is 35 miles. This is "I of the distance by boat. Find the distance by boat.

42. What is the number of feet in the width of a street, if f of the width, or 48 feet, lies between the curbstones ?

43. If I of the number of persons who went on an excursion to Niagara Falls were teachers, and 240 teachers went, what was the whole number of persons who went on the excursion ?

44. If ^ of a number is added to the number, the sum is 12. What is the number ?

Suggestion. x-\-lx = ^x=12.

45. If I of a number is added to the number, the sum is 30. Find the number.

46. If ^ of a number is added to twice the number, the sum is 35. What is the number ?

47. If \ of the number of cents Lucy has is added to the num- ber she has, the sum is 819 cents. How many cents has she ?

48. Ada is J as old as her brother. If the sum of their ages is 28 years, how old is each ?

49. I owe A and B $ 45. If I owe A | as much as I owe B, how much do I owe each ?

DEFINITIONS AND NOTATION

8. A unit or an aggregate of units is called a whole number, or an integer ; one of the equal parts of a unit or an aggregate of equal parts of a unit is called a fractional number.

Such numbers are called arithmetical, or absolute, numbers.

9. Arithmetical numbers have fixed and known values, and are represented by symbols called numerals; as the Arabic figures, 1, 2, 3, etc., and the Roman letters, I, V, X, etc.

10. You have seen that it is convenient, in solving problems, to use letters for the numbers whose values are sought. So also, in stating rules, letters are used to represent not only the numbers whose values are to be found, but also the num- bers that must be given whenever the rule is applied.

For example, the volume of any rectangular prism is equal to the area of the base multiplied by the height. By using V for volume, A for area of base, and h for height, this rule is stated in symbols, thus :

When A = 60 3i.nd h = 5, F=60x5 =300;

when ^ = 36 and /i = 10, F= 36 x 10 = 360 ; etc.

An equation that states a rule in brief form is called a formula.

In each particular problem to which the above formula applies, A and h represent fixed, known values, but in conse- quence of being used for all problems of this class, A and h represent numbers to which any arithmetical values whatever may be assigned,

13

14 DEFINITIONS AND NOTATION

11. A literal number to which any value may be assigned at pleasure is called a general number.

12. A general number or a number whose value is known is called a known number.

The general numbers, A and A, in the formula F= ^ x ^ (§ 10), are known numbers ; so also are the numerals 3 and 5.

13. A number whose value is to be found is called an unknown number.

In 3ic = 21, ic is an unknown number; in the formula for volume, V = Ay.h^ F is an unknown number ; but when this formula is changed to the formula for height, h = V ^ A^ V and A are known numbers and h is an unknown number.

ALGEBRAIC SIGNS

14. The sign of addition is +, read 'plus.''

It indicates that the number following it is to be added to the number preceding it.

a + 6, read ' a plus 6,' means that b is to be added to a.

15. The sign of subtraction is — , read 'minus.'

It indicates that the number following it is to be subtracted from the number preceding it.

a — 6, read ' a minus 6,' means that h is to be subtracted from a.

16. The sign of multiplication is x or the dot (•), read 'mul- tiplied by.'

It indicates that the number preceding it is to be multiplied by the number following it.

a X 6, or a . &, means that a is to be multiplied by b.

The sign of multiplication is usually omitted in algebra, except between figures.

Instead of o x 6, or a-b, usually ab is used. But 3x6 cannot be writ- ten 35, because 36 means 30 4- 6.

17. The sign of division is -j-, read ' divided by.'

It indicates that the number preceding it is to be divided by the number following it.

a -i-b means that a is to be divided by b.

DEFINITIONS AND NOTATION 15

Division may be indicated also by writing the dividend above the divisor with a line between them. Such indicated divisions are called fractions.

-, sometimes read ' a over &,' means that a is to be divided by b. h

18. The sign of equality is =, read * is equal to' or ^equals.'

19. Order of operations. — It has become a matter of agree- ment, or a custom, among mathematicians to employ signs of operation, when written in a sequence, as follows :

When only -\- and — occur in a sequence or only x and -*-, the operations are performed in order from left to right.

Thus, 3+4-2 + 3= 7-2+3 = 5 + 3= 8;

also, 3"x 4-^-2x3 = 12 -T-2x3 = 6x3 = 18.

a + 6 — c + d means that h is to be added to a, then from this result c is to be subtracted, and to the result just obtained d is to be added.

When X , -i-fOr both, occur in connection with +-, — , or both^ the indicated midtiplications and divisions are performed first unless otherwise indicated.

Thus, 7 + 10 -6-3x4 = 7 + 10 -2x4 = 7 + 10 -8 = 9.

There are apparent exceptions to the established order.

For example, in m -r- a& the multiplication is considered as already performed ; consequently, m -^ ah means that m is to be divided by ah, not that m is to be divided by a and the result multiplied by h ; but m -^ a y.h (the multiplication being indicated by x ) means that m is to be 'divided by a and the result multiplied by h.

20. The signs of aggregation are : the parentheses, ( ) ; the vincidum, ; the brackets, [ ] ; the brakes, \ j ; and the vertical bar, I .

They are used to group numbers, each group being regarded as a single number.

Thus, each of the forms (a + &)c, a ■}■ h • c, la -\- 6]c, {a + b}c, and a c signifies that the sum of a and h is to be multiplied by c. +6

All operations within groups should be performed first. When numbers are included by any of the signs of aggregation, they are commonly said to be in parenthesis, in a parenthesis, or in parentheses.

16 DEFINITIONS AND NOTATION

21. Tlie sign of continuation is • • •, read Uind so on/ or 'and so on to.^

2, 4, 6, 8, . • ., 50 is read '2, 4, 6, 8, and so on to 50.'

22. The sign of deduction is .-., read ' therefore ' or ' hence.'

EXERCISES

23. Read and tell the meaning of :

1. m + n. 6. 2 •3 — 4 w. 11. a + mH-n.

2. x — y, 7. 3p+5g. 12. a + (m — 7i).

3. a-H6. 8. 7(^ + 2;). 13. a — m — n.

4. ^ + ^. 9. ^. 14. i-i + 1

&s b-\-s a m n

5. ab — rs. 10. 0^ + 2/^3. 15. (a + m)(6 — n).

Indicate results :

16. Add 2 times c to 5 times d.

17. Subtract 2 times 4 from m times n.

18. Multiply the sum of x and 2/ by z,

19. Divide v — w; by r times s.

20. Find the product of 2 ic -|- 7 and 3 1/ — 2.

21. Express the product of a and a-\-b divided by the prod- uct of b and a—b.

22. A boy had a apples and his brother gave him b more. How many apples had he then ?

23. Edith is 14 years old. How old was she 4 years ago ? a years ago ? How old will she be in 3 years ? in b years ?

24. At X cents each, how much will 5 oranges cost ?

25. If z caps cost 10 dollars, how much will 1 cap cost ?

26. At y cents each, how many pencils can be bought for x cents ?

27. A boy who has p marbles loses q marbles, and afterward buys r marbles. How many marbles does he then have ?

DEFINITIONS AND NOTATION 17

28. What two whole numbers are nearest to 9 ? to x, if x is a whole number ? to a, if a is a whole number ?

29. If y is an even number, what are the two nearest even numbers ?

30. A woman exchanged x dozen eggs for 8 pounds of sugar at a cents a pound and 5 pounds of coffee at h cents a pound. How much were the eggs worth a dozen ?

FACTORS, POWERS, AND ROOTS

24. Each of two or more numbers whose product is a given number is called a factor of the given number.

Since 12 = 2 x 6, or 4 x 3, each of these numbers is a factor of 12. Since 3 a6 = 3 x a x 5, or 3 a x 6, or 3 x a6, or 3 6 x a, each of these numbers, 3, a, 6, 3 a, 3 6, and a&, is a factor of 3 ah.

25. When one of the two factors into which a number can be resolved is a known number, it is usually written first and called the coefficient of the other factor.

In 6 a;?/, 5 is the coefficient of xy ; in ax, if a is a known number, it is the coefficient of x.

In a broader sense, either one of the two factors into which a number can be resolved may be considered the coefficient^ or co-factor, of the other.

In 6 ax^ ax may be considered the coefficient of 5, 5 a of x, cc of 5 a, etc.

Coefficients are numerical, literal, or mixed, according as they are composed of figures, letters, or both figures and letters.

When no numerical coefficient is expressed, the coefficient is considered to be 1.

26. When a number is used a certain number of times as a factor, the product is called a power of the number.

When a is used twice as a factor, the product is the second power of a, or the square of a; when a is used three times as a factor, the product is the third power of a, or the cube of a ; four times, the fourth power of a ; n times, that is, any number of times, the nth power of a.

milne's stand, ai.g. — 2

18 DEFINITIONS AND NOTATION

27. The product indicated by axaxaxaxa may be indicated more briefly by a^. Likewise, if a is to be used n times as a factor, the product may be indicated by a".

A figure or a letter placed a little above and to the right of a number is called an index, or an exponent, of the power thus indicated.

The exponent indicates how many times the number is to be used as a factor.

6^ indicates that 5 is to be used twice as a factor ; a^ indicates that a is to be used 3 times as a factor.

a^ is read ' a square,' or ' a second power ' ; a^ is read ' a cube,' or ' a third power'; a* is read 'a fourth,' 'a fourth power,' or 'a exponent 4 ' ; a'* is read ' a nth,' ' a nth power,' or ' a exponent %.'

When no exponent is written, the exponent is regarded as 1. 5 is regarded as the first power of 5, and a^ is usually written a.

The terms coefficient and exponent should be distinguished. 5 a means a-j-a + a + a + a, but a^ means axaxaxaxa.

28. When the factors of a number are all equal, one of the factors is called a root of the number.

5 is a root of 25 ; a is a root of a* ; 4 x is a root of 64 x^.

One of the two equal factors of a number is its second, or square, root; one of the three equal factors of a number is its third, or cube, root ; one of the four equal factors, the fourth root ; one of the n equal factors, the nth root.

29. The symbol which denotes that a root of a number is sought is -y/j written before the number.

It is called the root sign, or the radical sign. A figure or a letter written in the opening of the radical sign indicates what root of the number is sought. It is called the index of the root. When no index is written, the second, or square, root is meant.

v^ indicates that the third, or cube, root of 8 is sought.

Vox indicates the square root of ax, and Va—b, the square root of a — &.

DEFINITIONS AND NOTATION 19

ALGEBRAIC EXPRESSIONS

30. A number represented by algebraic symbols is called an algebraic expression.

31. An algebraic expression whose parts are not separated by + or — is called a term; as 2a^, — 6xyz, and — .

In the expression 2a;2 — bxyz + — there are three terms.

z

The expression m{a-^h) is a term, the parts being m and (a + 6).

32. Terms that contain the same letters with the same expo- nents are called similar terms.

3x2 and 12 oj^ are similar terms ; also 3(a + 6)2 and 12 (a + 6)2 ; also a« and 6x, when a and h are regarded as the coefficients of x.

33. Terms that contain different letters, or the same letters with different exponents, are called dissimilar terms.

5 a and 3 hy are dissimilar terms ; also 3 aV) and 3 aft^.

34. An algebraic expression of one term only is called a monomial, or a simple expression.

xy and 3 ah are monomials.

35. An algebraic expression of more than one term is called a polynomial, or a compound expression.

3 a + 2 6, xy + y;s + 0X, and a^ + 62 _ c2 + 2 a6 are polynomials.

36. A polynomial of two terms is called a binomial. 3 a + 2 6 and x? — y^ are binomials.

37. A polynomial of three terms is called a trinomial.

a + 6 + c and 3 x — 2 y — s are trinomials.

38. An expression, any term of which is a fraction, is called a fractional expression.

2x2 a2

x2 a

— — 3x+-isa fractional expression.

»2 />.

20 DEFINITIONS AND NOTATION

39. An expression that contains no fraction is called an integral expression.

5 a2 _ 2 a and 6 x are integral expressions.

Expressions like x^ + f «^ + i aj + 1 are sometimes regarded as integral, since the literal numbers are not in fractional form.

Numerical Substitution

40. When a particular number takes the place of a letter or general number, the process is called substitution.

EXERCISES

41. 1. When a = 2 and 5=3, find the numerical value of, that is, evaluate 3 ah \ a^.

Solutions. 3 a6 = 3 • 2 • 3 = 18 ; also, a* = 2 • 2 • 2 • 2 = 16. When a = 5, 5 = 3, c = 10, m = 4, evaluate :

2. 10 a. 6. 5m\ 10. am\ 14. ^ab^

3. 2ab. 7. 2an}. 11. {aby. 15. ^bm.

4. 3 cm. 8. 3 6ml 12. a^b^ 16. j^ abc.

5. 6 be. 9. 4:a^b. 13. aJ'c. 17. Sb'^cml

18. When x= 6, y =S,z=:2, 7n = 0, n = 4, find the value of

Vxyi^ ; of 3 m^n.

Solutions

y/xyz^= Ve . 3 . 2 . 2 . 2 = vlii = 12. 3 m% = 3 . 02 . 4 = 3 . 0 . 4 = 0.

Note. — It will be seen in § 641 that when any factor of a product is 0, the product is 0 ; therefore, any power of 0 is 0 ; also any root of 0 is 0.

When a = 4, 6 = 2, r = 0, s = 5, evaluate :

19. V2a6. 22. V7 r^s. 25. 3aVbh\ 28. Gf'V^b'.

20. 7 6V. 23. 3s^6«. 26. f a^fts. 29. 2«63sV.

21. Vas^. 24. VSab. 27. .8 s Vo^ft^. 30. ^9v^«.

DEFINITIONS AND NOTATION 21

31. ?^^. 32. ^. 33. ^^. 34. ^i^^.

35. When a? = 3 and y = 2f evaluate x^ — y^] (^ + 2/)^;

Solutions x2~y2 =3.3-2.2 = 9-4 = 5.

(x + y)2 = (3 4- 2)2= 5 . 5 = 25.

x2 + 2 a;y + 2/2 = 3.3 + 2. 3-2 + 2. 2 = 9 + 12 + 4= 25.

When a = 5, 6 = 3, m = 4, n = 1, evaluate :

36. a^ + b^ 39. (7i - 1)^ 42. m«-^

37. (a + by. 40. n^ — 1. 43. (6m)«-^

38. ^ + ^^. 41. m+ ^^ . 44. ^!-:^'. m — 2n m —2 n a — b

45. a6 — bn + m^^ ^ 3 mn\

46. (a6-6n + m52)-^3mn2.

47. Show that 2x-\-3x = 5x when a; = 2; when a; = 3. Giving X any value you choose, find whether 2x-\-3x = 5x.

48. Show that m(a ■i-b)—ma-{- mb when m = 5, a= 4:, and 6=3. Find whether the same relation holds true for other values of m, a, and b.

49. Show that (a +- by= a^-^2ab + ¥ when a = 3 and 6 = 2. Find whether this is true for other values of a and b.

50. When a = 0, b = S,c=5,d = 3, find the value of c^d ■i-abc-bd'^-^ad\

Solution

32d + a6c-&d2-fa# = 5.5.3 + 0.8.5-8.3.3-|.0.3.3.3.3 = 75 + 0-72-0=3.

Whena; = 6, y = S, z = 0, r = 2, s = 10, find the value of: •

51. rx -\- yz -\- rs — xz. 53. 12 !^ -}- r^y^ -\- 5 xy -7- 3 s.

52. sa;2 - r^s +. a;2/2! - a;?/2. 54. 4: sy'^ ^ ^ x^r^ — {^ a:^y^z.

22 DEFINITIONS AND NOTATION

42. Solution of problems by substitution in formulae.

(a) If the base of a rectangle is 6 feet and its altitude is 3 feet, the area is 6 times 3 square feet.

In general, the area (A) of any rectangle is equal to the product of its base (b) and altitude or height (h). This rule of mensuration is stated by the algebraic formula,

A = bh.

By substitution in this formula find the area of :

1. A rectangle whose base is 12 feet and altitude, 5 feet.

2. A rectangle whose base is 20 feet and altitude, 10 feet.

(b) If an automobile goes 30 feet per second, in 25 seconds it will go 25 times 30 feet.

In general, the space (s) passed over by anything moving with uniform velocity (v) in any given time (t) is equal to the product of the velocity and the time.

This physical law is expressed briefly by the algebraic formula,

« = vt

By substitution in this formula find the space passed over :

3. By a train in 30 sec, uniform velocity 48 ft. per sec.

4. By a launch in 11 hr., uniform velocity 8.2 mi. per hr.

5. By a torpedo in 75 sec, uniform velocity 44 ft. per sec

(c) The formula for the space (s) through which a freely falling body acted upon by gravity (g) will fall in t seconds, starting from rest, is

Use 32. 16 or 9.8 for g according as s is to be obtained in feet or in meters.

6. How many feet will a body fall from rest in 5 seconds ?

7. A stone dropped from the top of an overhanging cliff reached the bottom in 4 seconds. How many feet high was the cliff ? how many meters high ?

8. A ball thrown vertically into the air returned in 6 seconds. How many meters high was it thrown ? (Time of fall = 3 sec.)

POSITIVE AND NEGATIVE NUMBERS

43. For convenience, arithmetical numbers may be arranged in an ascending scale :

0, 1, 2, 3, 4, 6, ...

I I I I I I

The operations of addition and subtraction are thus reduced to counting along a scale of numbers. 2 is added to 3 by be- ginning at 3 in the scale and counting 2 units in the ascending, or additive, direction ; and consequently, 2 is subtracted from 3 by beginning at 3 and counting 2 units in the descending, or subtractive, direction. In the same way 3 is subtracted from 3. But if we attempt to subtract 4 from 3, we discover that the operation of subtraction is restricted in arithmetic, inas- much as a greater number cannot be subtracted from a less. If this restriction held in algebra, it would be impossible to subtract one literal number from another without taking into account their arithmetical values. Therefore, this restriction must be removed in order to proceed with the general dis- cussion of numbers.

To subtract 4 from 3 we begin at 3 and count 4 units in the descending direction, arriving at 1 on the opposite, or subtrac- tive, side of 0. It now becomes necessary to extend the scale 1 unit in the subtractive direction from 0.

To subtract 5 from 3 we begin at 3 and count 5 units in the descending direction, arriving at 2 on the opposite, or sub- tractive, side of 0. The scale is again extended; and in a similar way the scale may be extended indefinitely in the sub- tractive direction.

24 POSITIVE AND NEGATIVE NUMBERS

Numbers on opposite sides of 0 may be distinguished by means of the small signs "*" and ~, called signs of quality, or direction signs, + being prefixed to those numbers which stand in the additive direction from 0 and - to those which stand in the subtractive direction from 0.

The former are called positive numbers, the latter negative numbers.

Zero occupies a position at the middle of our number system. There are as many negative numbers below it as there are posi- tive numbers above it. Zero is neither positive nor negative.

Including zero, the scale of algebraic numbers is written :

..., -5, -4, -3, -2, -1, 0, +1, +2, -^3, +4, -*-5, ...

I I \ I \ I \ I I I \

44. By repeating +1 as a unit any positive integer may be obtained, and by repeating "1 as a unit any negative integer may be obtained. Hence, positive integers are measured by the positive unit, +1, and negative integers by the negative unit, ~1.

Fractions are measured by positive or negative fractional units. Thus, the unit of +(|) is +{\) ; the unit of -(|) is -(|).

45. If +1 and ~1, or +2 and ~2, or any two numbers nu- merically equal but opposite in quality are taken together, they cancel each other. For counting any number of units from 0 in either direction and then counting an equal number of units from the result in the opposite direction, we arrive at 0.

Hence, if a positive and a negative number are united into one number, any number of the units or parts of units of which one is composed cancels an equal number of units or parts of units of the other,

46. Quantities opposed to each other in such a way that, if united, any number of units of one cancels an equal number of the other may be distinguished as positive and negative.

Thus, if money gained is positive, money lost is negative ; if a rise in temperature is positive, a fall in temperature is negative ; if west longi- tude is positive, east longitude is negative ; etc.

POSITIVE AND NEGATIVE NUMBERS 25

47. Positive and negative numbers, whether integers or fractions, are called algebraic numbers.

Arithmetical numbers are positive numbers.

48. The value of a number without regard to its sign is called its absolute value.

Thus, the absolute value of both + 4 and — 4 is 4.

ADDITION AND SUBTRACTION

49. The aggregate value of two or more algebraic num- bers is called their algebraic sum; the numbers are called addends.

The process of finding a simple expression for the algebraic sum of two or more numbers is called addition.

50. In addition, two numbers are given, and their algebraic sum is required; in subtraction, the algebraic sum, called the minuend, and one of the numbers, called the subtrahend, are given, and the other number, called the remainder, or difference, is required. Subtraction is, therefore, the inverse of addition.

The difference is the algebraic number that added to the sub- trahend gives the minuend.

51. Negative numbers give the foregoing definitions a wider range of meaning than they had in arithmetic. In algebra addition does not always imply an increase, nor subtraction a decrease.

	< \	Sum of Two or More Numbers
52. Add:		EXERCISES
1. +5 2.	-5	3. +5 4. +5 5. +5 6.	-5	7. -5
+2	I?	z^ :± 11	+_S	+2

Suggestions. — The sum of 2 positive units and 6 positive units is 7 positive units ; of 2 negative units and 5 negative units, 7 negative units ; (§ 45) of 5 negative units and 5 positive units, 0 ; of 4 negative units and 6 positive units, 1 positive unit ; of -9 and +6, -4 ; etc.

5	16.	+8	17. +6	18. -9	19. +10
-3		-9	+5	+3	-40
■8		+1	-5	+2	+25

26 POSITIVE AND NEGATIVE NUMBERS

To add two algebraic numbers (see exercises 1-7) : Rule. — If they have like signs, add the absolute values and prefix the common sign ; if they have unlike signs, ^nc? the differ- ence of the absolute values and prefix the sign of the numerically greater.

By successive applications of the above rule any number of numbers may be added.

Add: 8. +8 9. +1 10. -8 11. +2 12. -6 13. "20

+2 21 '^A +8 -4 +5

14. +7 15. -3

+2 _ _ _ _

20. +10 + -4 + +6 + -7 + +9. 23. +6 + "9 + +5 + +3 + "4.

21. -12 + +8 + +2 + -6++2. 24. 0 + "7 + +4 + "3 + -4.

22. -40 + -6 + +8 + +7 + +6. 25. "2 + +3 -f- +6 + +8 -f "8.

53. Abbreviated notation for addition.

Reference to the scale of algebraic numbers shows that adding positive units to any number is equivalent to count- ing them in the positive direction from that number, and adding negative units to any number is equivalent to counting them in the negative direction from that number. Hence, in addition, the signs + and — denoting quality have primarily the same meanings as the signs + and — denoting arith- metical addition and subtraction. For example,

+1 means 0 -|- 1 and ~1 means 0 — 1;

also +5 means 0 + 5 and ~5 means 0 — 5 ; etc.

•

Hence, in finding the sum of any given numbers, only one set of signs, + and — , is necessary, and they may be regarded either as signs of quality or as signs of operation, though com- monly it is preferable to regard them as signs of operation.

Thus, +5 + +3 + -C may be written +5 + 3-6, or 5 + 3-6.

POSITIVE AND NEGATIVE NUMBERS

27

When the first term of an expression is positive, the sign is usually omitted, as in the preceding illustration.

If there is need of distinguishing between the signs of quality + and — and the signs of operation + and — , the num- bers and their signs of quality may be inclosed in parentheses.

Thus, if a = 5, 6 = - 3, and c = - 2, then a + &4-c = 5+(-3) + (_2); a-&-c = 5-(-3)-(-2); aftc = 5(- 3)(- 2) ; etc.

54. A term preceded by + , expressed or understood, is called a positive term, and a term preceded by — , a negative term.

Thus, in the polynomial 3 + 2 & — 5 c the first and second terras are positive and the third term is negative.

EXERCISES

[Additional exercises are given on page 447.]

55. Write with one set of signs and find the sum :

1. +7 + +8. 3. -3 + -7. 5. -6 + -3 + +16.

2. +6-\--5. 4. +2 + -4. 6. +8 + +9 + "15.

Find the sum :
7. 10-7 + 4-9-6.	10. 3 + 5-16 + 24-11.
8. 21 + 3-6-5 + 8.	11. 7_|_44.12_30 + 15.
9. l7_2-3-4-6.	12. 8-2 + 40-18 + 13.

25

45

3

13. In a football game the ball was advanced 5 yards T +5 ^ from the Juniors' 25-yard line ^j I ^^ toward the Seniors' goal, then |i 1*^ +i3

6 yards, then — 8 yards (i.e. it "l i ^

went back 8 yards), and so on, gb ^ 4k

as shown in the diagram.

What was the position of the ball after 3 plays? after 4 plays? after 5 plays ? after 6 plays ?

14. Plot the following and find the last position of the ball : On 15-yard line; gained 4 yards; gained 5 yards; lost 2 yards;

gained 30 yards ; lost 6 yards ; lost 2 yards ; gained 12 yards.

28 POSITIVE AND NEGATIVE NUMBERS

Difference of Two Numbers

EXERCISES

[Additional exercises are given on page 447.]

56. On account of the extension of the scale of numbers below zero (§ 43), subtraction is always possible in algebra.

When the subtrahend is positive, algebraic subtraction is like arithmetical subtraction, and consists in counting backward along the scale of numbers.

Subtract the lower number from the upper one ; that is, tell what number added to the subtrahend will give the minuend :

8. -3

When the subtrahend is negative, it is no longer possible to subtract as in arithmetic by counting backward. 15. Subtract ~2 from 8.

Explanation. — If 0 were subtracted from 8, the

result would be 8, the minuend itself.

8 The subtrahend, however, is not 0, but is a number 2

-o units below 0 in the scale of numbers. Hence, the

Q rt —To difference is not 8, but is 8 + 2, or the minuend plus the

6	2.	6	3.	6	4.	6	5.	6	6.	6	7.	6
3		4		5		6		7		8		9
■3	9.	-3	10.	-3	11.	-4	12.	-5	13.	-6	14.	-7
0		1		2		3		4		5		6

subtrahend with its sign changed. Subtract the lower number from the upper one :

16.	4	17.	4	18.	4	19.	4	20.	5	21.	7	22.	9
	0		-1		-2		-3		-6		-7		-5
23.	-5	24.	-5	25.	-5	26.	-5	27.	-1	28.	-4	29.	-6
	0		-1		-2		-6		-3		-7		-5

From the preceding exercises, observe that : Principle. — Subtracting any number is equivalent to adding it with its sign changed.

POSITIVE AND NEGATIVE NUMBERS

29

Subtract the lower number from the upper one

30.	10	31.	12	32.	20	33.	16	34.	40	35.	32
	-2		-_5		-6		-4		-8		-7
36.	0	37.	-3	38.	~<	39.	-10	40.	- 5	41.	-12
	-2		-6		-4		- 5		-10		-20
42.	4	43.	4	44.	-4	45.	-9	46.	3	47.	-7
	4		-4		4		^		^9		8
48.	5	49.	-3	50.	-5	51.	10	52.	-10	53.	13
	-6		7		-4		"7		7		:?
54.	20	55.	44	56.	28	57.	10	58.	-10	59.	-5
	3		-4		-6		10		-10		12

60. Subtract 12 from ~1.

61. Subtract 4 from 14.

62. Subtract 11 from "8.

63. Subtract "7 from 0.

64. From ~6 subtract 4.

65. From 0 subtract "3.

66. From "3 subtract 0.

67. From '6 subtract -4.

A weather map for January 16 gave the following minimum and maximum temperatures (Fahrenheit) :

	Chicago	DULUTIl	Helena	MONTKEAL	New Orleans	New York
Minimum Maximum	24" 30°	-go 2°	-12^ -4°	-12° 18°	64° 76°	20° 42°

68. The range of temperature in Chicago was 6°. Find the range of temperature in each of the other cities.

69. The freezing point is 32° F. How far below the freez- ing point did the temperature fall in Montreal ?

70. How much colder was it in Duluth than in Chicago ? in Montreal than in New York ?

Exercises in multiplication and division of positive and negative num- bers are given in §§ 8-5, 124.

ADDITION

57. Arithmetically, 2 + 3 = 3-1-2.

In general, a-\-b = b-\-a. That is,

Numbers may be added in any order.

This is the law of order, or the commutative law, for addition.

68. Arithmetically, 2 -f- 3 -h 5 = (2 -h 3) + 5 = 2 -h (3 + 5) = (2 H- 5) -f 3. In general, a -h 6 -f- c = (a -f 6) + c = a + (6 -f c) = (a + c) -h 5. That is,

The sum of three or more numbers is the same in whatever manner the numbers are grouped.

This is the law of grouping, or the associative law, for addition.

59. To add monomials.

EXERCISES

[Additional exercises are given on page 448.] 1. Add 4 a and 3 a. PROCESS Explanation. — Just as 3 a's and 4 o's are 7 a's, so 3 a

4a -|-4a = 7a; that is, wlien the mouomials are similar the sum

3 d may be obtained by adding the numerical coefficients and

rT^ annexing to their sum the common literal part.

Add: 2. 2a; 3. a 4. — a 5. —4c 6. Zd

3a; 5a 4a -3c -6r2

7.	4n	8.	2s	9.	— z	10.	e>t	11.	-5x
	2n		-8s		32		-4:t		9x
12.	A:V	13.	- y	14.	126	15.	4a;2	16.	-SOu
	2v		4.7		- 26		-Ss^		50 u
	Iv		-9^		- 66		-ex'		-70m
				30

ADDITION 31

17. Add 4 a, f a, —3 a, and i a.

PROCESS

Explanation. — By §§ 57, 58, the numbers may be added in any order or grouped in any manner. For convenience, then, we may first add those with integral coefficients, then those with fractional coefficients, and afterward add these sums, as in the process.

18. Add 5 a?, — f ic, 2 x, and —\x.

19. Add 1^ ah, — 2 ah, 5\ah, and — 3 ah.

20. Add 4 xyz, — f xyz, f xyz, and — 2| xyz,

21. Add - (?de, 21 c'de, | cHe, and - 5 c^de.

Simplify :

22. 2y-7y-^y-y-[-10y-e^y + ^y.

23. 5a-3a + 8a-10a — 5a — lla+24a.

24. 3 6y - 5 6?/ - 10 6i/ - 14 62/ + 48 % + 46?/ - 12 6y.

25. 8 a^ft + 6 a^?) - 11 a% -2a^h+9a%-7 a^h -h lOa^ft.

26. {a — x) -\- 5{a — x) -\-7 (a — x) — 3{a — x) — 2(a — x).

27. 2(a;-l)-13(a;-l)H-5(aj-l)-f 10(a;-l) +6(a;-l).

Since only similar terms can be united into a single term, dissimilar terms are considered to have been added when they have been written in succession with their proper signs.

28. Add 6 a, —5h, —2 a, 3 b, 2 c, and — a.

Solution. — Sum = 6a-2a-a-56 + 36 + 2c=3a-26 + 2c.

Add:

29. 2xy, A ah, 3 xy, and ah.

30. mn, —3cd, — 6 mn, and 4 cd.

31. a, —6, 2c, —2a, 3 6, —7 a, and —4c.

32. 2 a, 2 6, 2 c, 2 cZ, — a, — 3 6, — c, 3 6, — 4 a, and -3d.

32 ADDITION

60. To add polynomials.

EXERCISES

[Additional exercises are given on page 448.] 1. Add 3a-36 + 5c, -3a + 2 6, ande-46+2a.

PROCESS Explanation. — For convenience, similar terms

„ Q 7^ K "^^y ^® written in the same column (§ 57).

~ ' The algebraic sum of the first column is 2 a, of

3 a + 2 6 the second - 5 6, of the third +G c; and these

2 a — 4 6 + c numbers written in succession express in its sim-

2a — 56 + 6c plest form the sum sought.

Rule. — Arrange similar terms to stand in the same column. Find the algebraic sum of each column, and write the results in succession with their proper signs.

Add: ^. x-\-2y 3. r — 3s 4. a — h-^-c

X- Sy —r-\-2s 2a-{-b

5. 3a- c, 2 c. 10. 7 X- 8 2/, 4 2/ -9 a;.

6. x-{-3y, —Sx. 11. 4:W-\-Sv, 2v — 3w.

7. 4a + 4&, — 6 6. 12. 2 m + n, 3 n — 6 m.

8. Sx-\-y, x-2y. 13. dx^-^-y^ y^-3x\

9. a-^ 5c, a -2b. 14. 2a^b-{-c% 2c^-a^b.

Simplify the following polynomials:

15. c — 2d-\-c + d — c. 19. m + n-{-d — 2n-\-d.

16. x — y-\-2x — y — x. 20. 12r + 6s — 8r — 9s.

17. 2a + b-c-b-c. 21. 2 a^ + 4 «- 10 «; + a.

18. x-\-z — 2x — z + x. 22. ab — 4: a^b + ab + 5 a^b.

23. 7 x — lly-{-4:Z—7 z + llx — Ay -{-7y — llz—4:X-\-y+z.

24. a + 36 + 5c — 6a + cZ + 46-2c — 26+5a — d+a — 6.

25 . 2 a;?/ ^ 5 2/^ + a;^2/^ - 7 a;?/ + 3 2/^ - 4 a^2^2 _f_ 5 aj^ _j_ 4 2/2 _|_ ^y ^

26. 4ar^-3a;2/+52/24-10a;2/-172/2-ll.T2-5a;2/+12a;2-2a;2/.

ADDITION 33

Add:

27. 4:r-5t, 2r-s-\-3t, 2s-3r, and s + 2^.

28. 5 X — 3 y — 2 z, 4 ?/ — 2 a;, 3z — 2x, and 4 x + 2 2;.

29. 6an-S a^n'' + 4, a2^^2 -4 an- 2, and a^M^ _ 6.

30. 7 « - 3 6 -I- 5 c, 2 6 H- d — 3 c, and 5c — 6a + 2d.

31. .^' + ?/^-2;, ic — ?/-|-5;, y — z — x, z — x — y, and it' — 2.

32. 2 « — 3 6, 2 & — 3 c, 5 c — 4 a, 10 a — 5 6, and 7 6 - 3 c.

33. a — ^b+c, f a H- ^ 6, |- a — ^ c, and f 6 — J c.

34. (2a + 6)v4-(a-f-c).v, — 3(2a + 6)y, and (a -\- c)y,

35. 6 Va; — by/xy — 3 Vy, 6 Vic^ — Vx, and 3Vic + V?y.

36. 2{a-{-h)-\-{G-d), 2(c-d)-3(a + 6), and (tt + &)-(c-c?).

37. i .X' 4- 2 2/ — 2, 2x — \y-\-\z, and iz — |a;— iy.

38. Va + Va6, 2 Va — V6 + 2 Va6, and 2V6 - 3 Va6 — Va.

39. a; + 3(a + l)-?/, - (a + 1)- 2a; + 4?/, and 3a;-4(a + 1).

40. (c + 2)a; + (a + d) y, {a + c2)y - (c + 2) x, and (c + 2)a; -

{a + d)y.

41. 4 a^ - 2 a;2 - 7 a; + 1, a.-' 4- 3 a;2 4- 5 a; - 6, 4 a;2 - 8 a-' H- 2 -6 a?, 2a^-2a;2 + 8a;-|-4, and 2 aj^ - 3a;2 - 2 a; + 1.

42. 2c-7d + 6?i, llm-3c-5^i, 7n-2d-8c, 8d-3m + 10 c, 4 fZ — 3 H — 8 7/1, m — 6 n, and 2 m — 3d.

43. a^ 4- 5 a^ + 5a + 1, a^ - 2 a^ + a^ - 2, a^ - 3a2 - 4 a^- a^ and 2a^4-a^-2a3-f 2a2-3a4-l.

44. 5a;6-a.'3H-7a;-9, 4ar5-3a;«+6a;2 + 12, a;^ - 5 a?^-a;-7, 4-a;2-a^, 4a;4-10a;2-h3a;«4-4, and a;« 4-a^-3 a.-^- 4 a; -5.

45. 3(a + &) + 6(5+c), 5(c«H- 6)- 10(6 4- c), 2 (a 4- 6)4- (64- c), 3(6 + c) - (a 4- 6), 2(6 +c)- 10(a + 6), and 3(a + 6) - 3(6 4- c).

46. (a 4- b)x— (c— d)2/, (« 4- b)x-\-(c—d)y, 2 c—d)y — (a-\-b)x, (c — d)y — (a 4- 6)a;, (a 4- b)x — (c — d)?/, and (a 4- 6)a; — (c — d)iy.

milne's stani>. ai.g. — 3

SUBTRACTION

EXERCISES

[Additional exercises are given on page 449.] 61. 1. From 10 x subtract 15 x.

PROCESS Subtract: Add: 10a; 10a; 15 a; - 15 a; — ^x — 5 X	Explanation. — Since (§56, Prin.) subtract- ing any number is equivalent to adding it with its sign changed, 15 x may be subtracted from 10 x by changing the sign of 16 x and adding - 15 cc to 10 X. The change of sign should be made men- tally.
	2.	3.	4.	5.	6.
From	12 a	9 am	Sx'y^	24: mn^	11 (a + 6)
Take	5a	21am	18 xY	12 mn^	21 (a 4- 6)
	7.	8.	9.	10.	11.
From	-6d -	-4a6	— 7 xyz	-lli/V	- 5(a-6)
Take	2d	5a6	Sxyz	14 ?/V	9(a-6)
	12.	13.	14.	15.	16.
From	86	5 6c	12 mV	6a262	18 (6 - c)
Take	-46 - 17.	-8 6c 18.	— 4 m^n^	- 12 a262 20.	- 2(6-c)
	19.	21.
From	-7c -	- 4mn	- 15 b^c^	- Ix^f	-19{c-^d)
Take	-2c - 22.	-Smn	- 9 6V 24.	- 14 xy 25.	-13(c + d)
	23.	26.
From	5a	6xy	— 9 mn	- 13 V^	- S(a + b)
Take	-2a -	-3xy	— 4mw	- bVx	-10(a + 6)
	54

SUBTRACTION

27. From Sx—Sy subtract 5x—7y.

35

PROCESS

Sx-Sy 5x — 7y 3x-\-4:y

Explanation. — Since (§ 5G, Prin.) subtracting any number is equivalent to adding it with its sign changed, subtracting 6 x from 8 a; is equivalent to adding — 5 x to 8 oj, and subtracting —7y from — 3 y is equivalent to adding -\-7 y to —Sy.

Rule. — Consider the sign of each term of the subtrahend to be changed, and add the result to the minuend.

	28.	29.	30.	31.
From	9a4-76	5 a -f 10 6	l^x-2y	3m-3n
Take	2a-3b	7a+ 46	6x + Ay	2 m — 5n
	32.	38.	34.	35.
From	15 m— n	lx-2y	_4a; + 4?/	Sp-\-3q
Take	12m + 2n	4:X — Ay	7x-2y	10p-\-2q

36. From 8p 4- 3 2 subtract 10 p — z.

37. From 15m — n subtract 5m -\- 3n.

38. From 9r — 3st subtract 5 s^ — 11 r.

39. From 3 ax— 5 by subtract 4 aa; — 6 by.

40. From 7 yz — Scd subtract 4 cd — 3yz.

41. From 8 abc -f- 19 mx subtract 20 abc + 7 mx.

42. From — 7 a;?/ + 3 rV subtract 14 rV _ 12 xy.

	43.	44.		45.
From	4m — 3n4-2p	8 a - 10 6 + c		3x-\-2y-z
Take	2m — 5n— p 46.	6a- 5b-c		5 X — 4:y — z
	47.	48.
From	a— 6-hc	8a26-5ac2-l-9	a^c	r-s-fi
Take	2a4.&-c	3 a^ft -f 2 ac2 - 9 ^	a^c	r + s-t

36 SUBTRACTION

49. From 5x — 3y-\-z take 2x — y-]-Sz.

50. From 3 a'b -\-b'-a^ take 4 d'b - 8 a^ + 2 b\

51. From ISa^ + 5 ft^- 4 c^ take Sa^ + Ofe^ + 10 c^.

52. From 15a? — 3y -\-2z subtract Sce + S^^ — 9^;.

53. From m^ — mn -\- n^ subtract 2 m^ — 3 mn + 2 n^,

54. From ^oi? — 2xy-f' subtract 2o?-\-2oyy — Zy'^.

55. From 2 ax— by — oxy subtract 2 by — 2 ax — 3 xy.

56. From 4 a6 + c subtract a"^ — b'^ -\- 2 ab — 2 c.

57. From a' + (c' subtract a^ — 3 a^x -{- 3 aoi^ — a^.

58. From a" + 1 subtract 1 — a + «^ — «^ + a^-

59. From a -\- b -\- c subtract the sum of a— b — c, b — c — a, and c — a — b.

60. Subtract the sum of m^n — 2 mn^ and 2 m^^i — m^ — 7i^ + 2m7i'^ from m^ — n^.

61. Subtract the sum of 2c — 9a — 36 and 36 — 5a — 5c from 6 — 3 c -|- a.

62. From the sum of 1 -h a; and 1 — x"^ subtract l — x-}-x'^ — x^.

63. From |«3_ 2 a.2_^3a;_7 subtract |a;3-fa^-f- fa;- 10.

64. From 5(a + b) — 3{x -\- y) + 4(m + n) subtract 4(a + b) H- 2(x + 2/) + (m 4- n).

65. Take 4 a;'" + 2 x'^y" +02/" from Ix"'-^- 2a;'»2/" + 9 y'\

66. Take a"' + 3 ft^*- + 7 c^'* from 3 a"^ + 5 b^' -\- 9 c'^\

67. Take 6 m* + 11 m'n^ + 5 n' from 10 m" + 11 mV + 8 n^

68. From the sum of 3 aj"* + 4 2/" + 2^'' and 2 s;^* + 2 aj*" — 3 ?/" subtract 4x"' — 2y''-\- z^".

If a; = a2 + 62, y = 2ab, z = a^-b^ ojid v = a^ -2 ab -\-b%

69. a;+2/ + 2; — v = ? 71. x — y + z — v = ?

70. x—y — z-{-v=? 72. y — x — v-\-z=?

SUBTRACTION 37

PARENTHESES 62. Removal of parentheses preceded by + or -.

EXERCISES

[Additional exercises are given on page 460.]

1. Remove parentheses and simplify 3 a + (6 + c — 2 a).

Solution. — The given expression indicates that (6 + c — 2a) is to be added to S a. This may be done by writing the terms of (6 + c — 2 a) after 3 a in succession, each with its proper sign, and uniting terms.

.-. 3a + (& + c-2a)=3a + 6 + c-2a = a + & -\- c.

2. Remove parentheses and simplify 4: a— (2a — 2b).

Solution. — The given expression indicates that ( + 2 a — 2 6) is to be subtracted from 4 a. Proceeding as in subtraction, that is, changing the sign of each term of the subtrahend and adding, we have

4a-(2a-2 6)=4a-2a + 2& = 2a + 2 6.

Principles. — 1. A parenthesis preceded by a plus sign may be removed from an expression without changing the signs of the terms in parenthesis.

2. A parenthesis preceded by a minus sign may be removed from an expression, if the signs of all the terms in parenthesis are changed.

Simplify each of the following :

3. a + (6 — c). 10. a — b-(c — d).

4. a — Q) — c). 11. a — b — {—c-\-a).

5. X— (y — z). 12. a — m—{n — m).

6. x-\-{—y + z). 13. na—2b—{a — 2b).

7. m — n — {—a). 14. a — (6 — c + a) — (c — 6).

8. m — {n-2a). 15. 2 xy -{-3y'^ — {x^ -\-xy — y"^).

9. ^x—{2x + y). 16. m-\-{Z7n — n) — (2n — m)-\-n.

38 SUBTRACTION

When an expression contains parentheses within paren- theses, they may be removed in succession, beginning with either the outermost or the innermost, preferably the latter.

17. Simplify 6a;-[3 a- J4 & +(8 6- 2 a)- 3 ^j + 4 a;].

Solution 6 a; - [3 a -{4 & + (8 6 - 2 a)- 3 6}+ 4 a] Prin. 1, = 6 cc -[3 a -{4 6 + 8 & - 2 a - 3 6}+ 4 X]

Uniting terms, = 6 a; -[3 a — {9 6 — 2 a}+ 4 a:] Prin. 2, =6a;-[3a-96 + 2a + 4a;]

Uniting terms, =6x — [5 a — 9& + 4x] Prin. 2, =6x — 5a + 9& — 4a;

Uniting terms, =2x — 6a + 9b.

Simplify each of the following :

18. 4:a-{-b — \x-\-4:a-{-b — 2y — {x-\-y)\.

19. ab — \ab-\-ac — a — (2a — ac)-\-(2a — 2ac)l.

20. a + 2/-S5 + 4a-(62/ + 3)S-(7 2/-4a-l).

21. 4m-[p-l-37i — (m + w) + 3— (6i? — 3n — 5m)].

22. a + 2 6 4- (14 a - 5 &) - 5 6 a + 6 ft - (5 a - 4 « + 4 6) J .

23. 12 a- [4 -3 6 -(6 & + 3 c)+ 6-8 -(5 a- 2 6 -6) j.

24. a -{- b ~ \ — [a -\-b — c — x'] — \3 a — c -{- X — a — b']-\- 4^ a\.

25. a^-[x''-(l-x);\-\l-^x^-(l-x)-^a^\.

26. 4-f52/-34-2a;-2-aj-(52/-a;-3)j.

27. a6 — S5 + a; - (6 + c — a6 4- ^) S + aj — (6 — c - 7).

28 . — \S ax — [5 xy — S z^-{- z — (4: xy + 16 z -^ 7 ax']-\- 3 z)\.

29. 1 — ic-jl— a — [l-o; — 1— a; — » — l]-a;-i-lj.

30. l-.x-\l-[x-l+(x-l)-(l-x)-x']-\-l-xl,

31. a-(6-c)-[a-J6-c-(6 + c-a) + (a-6) + (c-a)j].

I

SUBTRACTION 39

63. Grouping terms by means of parentheses.

It follows from § 62 that :

Principles. — 1. Any number of terms of an expression may be inclosed in a parenthesis preceded by a plus sign without chang- ing the signs of the terms to be inclosed.

2. Any number of terms of an expression may be inclosed in a parenthesis preceded by a minus sign, if the signs of the terms to be inclosed are changed,

EXERCISES

64. 1. In a^ + 2 a6 4- b^, group the terms involving b.

Solution

a2 + 2 a6 + 62 = a2 + (2 a6 + 62).

2. In a^ — a:^ — 2xy — y^, group as a subtrahend the terms

involving x and y.

Solution

3. In aa;^ 4- a6 + 2 a^ + 2 6, group the terms involving a?, and

also the terms involving b, as addends.

4. In a^ + 3 a^b + 3 a62 + b^^ group the first and fourth terms, and also the second and third terms, as addends.

In each of the following expressions group the last three terms as a subtrahend:

5. a2-62_26c-c2. 7. a"" -{- 2 ab + b^ - c" -{- 2 cd - dJ".

6. a'-b^-\-2bc-c\ 8. a^-2ab-^b^-c^-2cd-'d\

9. In a^ + 2 a6 H- 62 _ 4 a — 4 6 4- 4, group the first three terms as an addend and the fourth and fifth as a subtrahend.

Errors like the following are common. Point them out.

10. x^-a^-{-x-l = (a^-l)-(a^-\-x).

11. x'-y^-{-2yz-z^ = x^-(y^-\-2yz-z^).

40 SUBTRACTION

65. Collecting literal coefficients.

EXERCISES

[Additional exercises are given on page 450.] Add:

1. ax 2. bm 3. — ex 4. (t + r)x

bx —cm —dx (t-{-2r)x

(a-\-b)x (b — c)m — (c + d)x (2^ + 3 r)x

5. ax 6. ey 7. — mp 8. (a + b)x

nx —dy —np (2 a + b)x

Subtract the lower expression from the upper one :

9. mx 10. dy-\-az 11. ax— by 12. {n — 2T)y

nx ey — bz 2 x — cy (2 w — r)y

13. a'^x + aby 14. mx — ??i!/ 15. ax— 5y 16. (2 a -|- 6)^ b'^x -f a% ^^ ~ '^y 5 x-\- ay (a — 6)v

17. Collect the coefficients of x and of yin ax — ay — bx— by.

Solution. — The total coefficient of x is (« — &); the total coefiBcient of 2/ is (— a — 6), or —(a + b).

.'. ax— ay— bx — by =(a — b)x—(a + b)y.

Collect in alphabetical order the coefficients of x and of y in each of the following, giving each parenthesis the sign of the first coefficient to be inclosed therein :

18. ax — by — bx — cy-\-dx — ey. 22. bx — cy — 2ay -\-by.

19. 5a^-\-Say—2dx-\-ny—5x—y. 23. rx — ay — sx-\-2 cy.

20. cx—2bx-\-lay-{-^ax—lx—ty. 24. ^tx-\-ax — by -{-ay.

21. bx-\-cy — 2 ax -{-by — ex — dy. 25. mx — ay — ax — ny.

Group the same powers of x in each of the following :

26. am^ + bx^ — cx-\'ex^ — dx^ —fx.

27. a^ + 3 a;2 + 3 aj — aaj2 — 3 aa^ + bx.

28. x"^ — abx — j^ — bx^ — rx — m7ix^ -f dx.

29. ax* — x^ — ax^-{-x^-{-ax^x — abx^ + ar'.

SUBTRACTION 41

TRANSPOSITION IN EQUATIONS

66. In an equation, the number on the left of the sign of equality is called the first member of the equation, and the number on the right is called the second member.

In the equation a; = 6 + 2, ic is the first member and 6 + 2 is the second.

67. Observe how (2) is obtained from (1) in each of the following :

1.	-2 + 5 = 3 (1)	3.	6= 6(1)
Adding Sums,	2 =2 5 = 5 (2)	Multiplying by Products,	2= 2 12 = 12(2)
2.	4 + 3 = 7 (1)	4.	8= 8(1)
Subtracting Eemainders,	4 =4 3 = 3 (2)	Dividing by Quotients,	4= 4 2= 2(2)

The following principles, illustrated above, are useful in solving equations. They are so simple as to be self-evident. Such self-evident principles are called axioms.

68. Axioms. — 1. If equals are added to equals, the sums are equal.

2. If equal's are subtracted from equals, the remainders are equal.

3. If equals are multiplied by equals, the products are equal.

4. If equals are divided by equals, the quotients are equal.

In the application of Ax. 4, it is not allowable to divide by zero (§ 547).

EXERCISES

69. 1. Solve ic— 6=4 by adding 6 to both members (Ax. 1).

2. Solve the equation a; + 3 = 11 by employing Ax. 2.

3. Solve ^ a;= 10 by employing Ax. 3.

4. Solve 7 a; = 21. Explain how Ax. 4 applies.

5. Solve I a; = 16 in two steps, first finding the value of }, x by Ax. 4, then the value of x by Ax. 3.

42 SUBTRACTION

Solve, and give the axiom applying to each step:

6. 2x = 6. 17. a; 4-2 = 10. 28. fm=9.

18. w— 5 = 11.

19. ^+1 = 12.

20. s-7 = 10.

21. 9 + s=12.

22. 5 + 2/ = 15.

23. 10 +a; = 12.

24. 11 +a; = 15.

25. 20 -fa; = 30.

26. 7?/ — 5 = 2.

27. 22! -1-3 = 9.

70. 1. Adding 7 to both members of the equation

a;-7 = 3,

we obtain, by Ax. 1, a; = 3 -f 7.

— 7 has been removed from the first member, but reappears in the second member with the opposite sign.

2. Subtracting 5 from both members of the equation

a; -f 5 = 9, we obtain, by Ax. 2, a; = 9 — 5.

When plus 5 is removed, or transposed, from the first mem- ber to the second, its sign is changed.

3. Explain the transposition of terms in each of the following :

7.	5a; = 5.
8.	42/ = 8.
9.	32/ = 9.
10.	i.=5.
11.	iz=2.
12.	\v=S.
13.	SV=24:.
14.	9r = 54.
15.	ir=1.5.
16.	la; = 2.5.

29.	-|n = 8.
30.	|a; = 10.
31.	faj = 21.
32.	1^=20.
33.	|.=15.
34.	5m-l = 9.
35.	4;i-f 3 = 7.
36.	6r-7 = 5.
37.	i.6-f-3 = 8.
38.	ix + 2 = 6.

2a;-l = 5; 2a; = 5 + l.

3a;-f2 = ll; 3a; = ll-2.

4a; = 14-3a;; 4a;-f-3a;=14.

71. Principle. — Any term may be transposed from one member of an equation to the other y provided its sign is changed.

SUBTRACTION 43

EXERCISES

72. 1. Solve the equation 2 a; + 20 = 80 — 4 a?.

PROCESS

2a;+20 = 80-4a; 2a; + 4a; = 80-20 6a; = 60 a; = 10

Explanation. — The first step in solving an equation is to collect the unknown terms into one member (usually the first member) and the known terms into the other member.

Adding 4 x to both members, or transposing — 4x from the second member to the first and changing its sign, places all unknown terms in the first member.

Subtracting 20 from both members, or transposing + 20 from the first member to the second and changing its sign, places all known terms in the second member.

Uniting similar terms and dividing both members by 6, the coefficient of X, we find the value of x to be 10.

Verification. — The result should always be verified by substituting it for the unknown number in the given equation. If the members of the given equation reduce to the same number, the result obtained is correct.

Substituting 10 for x, makes the first member 20 + 20, or 40, and the second member 80 — 40, or 40. Hence, 10 is the true value of x.

2. Solve the equation 7 — 5 a; = 7 — 30.

FIRST PROCESS SECOND PROCESS

_5a;=-30 30 = 5a;

5 a; = 30 6 = a;ora; = 6

x = 6 Test. 7-5-6 = 7-30

Suggestions. — 1. By Ax. 2 the same number may be subtracted, or canceled, from both members.

2. By Ax. 2 the signs of all the terms of an equation may be changed (first process) ; for each member may be subtracted from the correspond- ing member of the equation 0 = 0.

3. To avoid multiplying by — 1, the second process may be adopted.

44 SUBTRACTION

Solve and verify :

3. 3 = 5 -X. 12. 8 + 7 a = 5 a + 20.

4. 9-5x=-l. 13. 2 + 13/i = 50-9.

5. 10 + 'W = 18-'U. 14. 50 -w = 20 + 71.

6. 2 2 + 2 = 32-2. 15. 3a;- 23 = 0.'- 17.

7. 7a; + 2 = .^• + 14. 16. 4 a; +12 = 2a.' + 36.

8. 3p + 2 = j9+30. 17. 2a; + ia; = 30-ia;.

9. 571 — 5 = 2ri + 4. 18. 3a; — ^a; = 30 + ia;.

10. 6y -2 = Sy-{-7. 19. 5^(; - 10 = f w + 16.

11. 8a;-7 = 3 + 6a;. 20. 4r- 18 = 20 + |^?\

Simplify as much as possible before transposing terms, solve, and verify :

21. 10 a; + 30 -4 a; -(9 a; -33 -12 a;) =90 +12 -4 a;.

22. 16 a; + 12 - 75 + 2 a; - 12 - 70 = 8 a; - (50 + 25).

23. lls-60 + 5s + 17-(2s-41)~3s = 2s + 97.

24. 10 2 - 35 - (12 2 - 16) + 32 = 4 2 - (35 - 10 2) + 32.

25. 36 + 5 a; - 22 - (7 a; - 16) = 5 a; + 17 - (2 a; + 22).

26. 12a;-(6a;-17a;-15-a;) = 15- (2-17a;+6a;-4-12a;).

27. 14 yi - 35 = 9 - (11 71 - 4 - 16 + 10 71 - ?i) + 136 - 16 n.

Algebraic Representation

73. 1. Express the sum of 2 and 3 ; of x,—y, and — z.

2. What number is 10 less than 25 ? 71 less than 25 ?

3. Express the number that exceeds 4 by 9 ; ahy b.

4. How much does 20 exceed 10 + 4 ? z exceed 10 + ?/ ?

5. What number must be added to 6 so that the sum shall be 10 ? to 771 so that the sum shall be 4 ?

6. Mary read 10 pages in a book, stopping at the top of page a. On what page did she begin to read?

SUBTRACTION 45

7. A man made three purchases of a, b, and 2 dollars, respectively, and tendered a 10-dollar bill. Express the number of dollars in change due him.

8. A has X dollars and B, y dollars. If A gives B m dollars, how much will each then have ?

9. If 40 is separated into two parts, one of which is x, rep- resent the other part. i

10. What two whole numbers are nearest to ic, if a; is a whole number ? to a + 6, if a -h 6 is a whole number ?

11. If ic is an even number, what are the two even numbers nearest to a; ? the two odd numbers nearest to a? ?

12. If 71 + 2 is an odd number, what are the two odd num- bers nearest to n + 2 ? the two even numbers ?

13. There is a family of three children, each of whom is 2 years older than the next younger. When the youngest is x years old, what are the ages of the others ? When the oldest is y years old, what are the ages of the others ?

14. A boy who had x dollars spent y cents of his money. How much money had he left?

15. The number 25 may be written 20 + 5. Write the num- ber whose first digit is x and second digit y.

16. The number 376 may be written 800 + 70 + 6. Write the number whose first digit is x, second digit y, and third digit z.

SOLUTION OF PROBLEMS

74. If 3 a? = a certain number and cc -f 10 = the same number, ithen, 3a7 = aj-f-10.

This illustrates another axiom to be added to the list in § 68. I It will be found useful in the solution of problems.

!

Axiom 5. — Numbers that are equal to the same number, or to equal numbers, are equal to each other.

46 SUBTRACTION

75. Illustrative Problem. — Of the steam vessels built on the Great Lakes one year, 21, or 5 less than ^ of all, were of steel. How many steam vessels were built on the Lakes that year ?

Solution. — Let x = the number of steam vessels built.

Then, 1 05 — 5 = the number of steel vessels.

But 21 = the number of steel vessels.

.-.Ax. 6, ^ic — 5 = 21.

Transposing, | OJ = 21 + 5 = 26.

Hence, Ax. 3, x = 78, the number of steam vessels built.

76. A problem gives certain conditions, or relations, that exist between known numbers and one or more unknown numbers. The statement in algebraic language of these con- ditions is called the equation of the problem.

The equation of the problem just solved is | a; — 6 = 21.

77. General Directions for Solving Problems. — 1. Represent one of the unknoivn numbers by some letter, as x.

2. From the conditions of the problem find an expression for each of the other unknown numbers.

3. Find from the conditions two expressions that are equal and write the equation of the problem,

4. Solve the equation.

Problems

[Additional problems are given on page 460.]

78. 1. What number increased by 10 is equal to 19 ?

2. What number diminished by 30 is equal to 20 ?

3. What number decreased by 6 gives a remainder of 17 ?

4. What number exceeds \ of itself by 10 ?

5. What number diminished by 45 is equal to — 15 ?

6. What number is 3 more than J of itself ?

7. Find three consecutive numbers whose sum is 42.

8. Find three consecutive odd numbers whose sum is 57.

SUBTRACTION 47

W 9. Find three consecutive even numbers whose sum is 84. I 10. Separate 64 into two parts whose difference is 12. I 11. If 5 times a certain number is decreased by 12, the re- mainder is 13. What is the number ?

12. Eighty decreased by 7 times a number is 17. Find the number.

13. If I subtract 12 from 16 times a number, the result is 84. Find the number.

14. If from 7 times a number I take 5 times the number, the result is 18. What is the number ?

15. One number is 8 times another; their difference is 35. What are the numbers ?

16. The sum of a number and .04 of itself is 46.8. What is the number ?

17. What number decreased by .35 of itself equals 52 ?

18. Find two numbers whose sum is 60 and whose difference is 36.

19. The sum of two numbers is 82. The larger exceeds the smaller by 16. Find the numbers.

20. John and Frank have $ 72. John has $ 12 more than Frank. How many dollars has each ?

21. Four times a certain number plus 3 times the number minus 6 times the number equals 7. What is the number ?

22. If 5 times a certain number is subtracted from 58, the result is 16 plus the number. Find the number.

23. Twelve times a certain number is decreased by 4. The result is 6 more than 10 times the number. Find the number.

24. Three times a certain number decreased by 4 exceeds the number by 20. Find the number.

25. Three times a certain number is as much less than 72 as 4 times the number exceeds 12. What is the number ?

26. Twice a certain number exceeds J of the number as much as 6 times the number exceeds 65. What is the number ?

48 SUBTRACTION

27. A rectangle having a perimeter of 46 feet is 5 feet longer than it is wide. Find its dimensions.

28. A lawn is 7 rods longer. than it is wide. If the distance around it is 62 rods, what are its dimensions ?

29. The distance around a desk top is 170 inches. If the desk top is 15 inches longer than it is wide, how wide is it ?

30. How old is a man whose age 16 years hence will be 4 years less than twice his present age ?

31. A boy is 8 years younger than his sister. In 4 years the sum of their ages will be 26 years. How old is each ?

32. A prime dark sea-otter skin cost $ 400 more than a brown one. If the first cost 3 times as much as the second, how much did each cost ?

33. The total height of a certain brick chimney in St. Louis is 172 feet. Its height above ground is 2 feet more than 16 times its depth below. How high is the part above ground ?

34. A wagon loaded with coal weighed 4200 pounds. The coal weighed 1800 pounds more than the wagon. How much did the wagon weigh ? the coal ?

35. A mining company sold copper ore at $ 5.28 per ton. The profit per ton was $ .22 less than the cost. What was the profit on each ton ?

36. The length of the steamship Lusitania is 790 feet, or 2 feet less than 9 times its width. What is the width ?

37. The length of a tunnel was 22|- times its width. If the length had been 50 feet less, it would have been 20 times the width. Find its length ; its width.

38. The Canadian Falls, in the Niagara River, are 158 feet high. This is 8 feet more than \^ of the height of the Ameri- can Falls. Find the height of the American Falls.

39. The St. Lawrence River at a point wHere it is spanned by a bridge is 1800 feet wide. This is 180 feet less than f of the length of the bridge. How long is the bridge ?

REVIEW 49

REVIEW

79. 1. Define square of a number ; square root of a number. Show the difference between these by an illustrative example.

2. Distinguish between power and exponent; between exponent and coefficient.

3. By what law do we know that the sum of 2 «, — 3 ?/, 4 «, and 5 ?/ is the same as the sum of 2 ic, 4 a;, 5 y, and — Sy?

4. From the sum oi z + xy and z — x^y subtract the sum of ./;// — z, 9? — a?y, and y^ — ct^.

5. From the sum of 3x^ — 2xy -\- y^ and 2xy — 5y subtract 2 01^ — 6 xy -\- 4: y^ less a^ — xy -\- y^.

6. What number added to — a^ -{- b"^ — 2 ab gives 0 for the sum ?

7. Give the general name that belongs to the two following expressions, and the specific name of each :

x^ — xy -{- y^ and a^ — o^y.

8. Remove parentheses and then regroup the terms in order, first as binomials ; second as trinomials :

2 a - [ 2 a^ 4- 2 6 - c - J - 2 d - (2 2/2 - ^) 4- m S ] + ?i3 .

9. Inclose the last four terms ofa — y— d + a; — 2in brack- ets and the last two terms in parentheses.

10. Collect similar terms within parentheses:

av? — cy -\-ax — 2 aoi? -\-2cy^ — ax — cy^ + aa^ + cy.

11. How does M -^ yz differ in meaning from k xl-^y X z?

12. Define positive numbers ; negative numbers. Illustrate.

^^ 13. The temperature at White River Junction one winter day was — 40° F. The temperature at Washington that day

I^as 40° F. What was the difference in temperature ? I 14. What is the absolute value of 8 ? of — 5 ? of any number ? r milnk's stand, alg. — 4

50 REVIEW

When a = 1, 6 = 2, c = 3, d = 4, and e = 5, find the value of :

15. a-(e + b)-{c-{-d)-(e-d + b-\-c).

16. Sa¥-2b(^- {d'^e'^ ~ ac") + 8 he\

17. ^ac + hVd^a + 2{b-a)(e-d)-{-hGe.

18. V2 ed6 + 4 e - V9 a-^c^ - 2 &d - (abe - abode).

19. Show that a number may be transposed from one mem- ber of an equation to the other, if its sign is changed.

Solve, giving reasons for each step :

20. 5a; + 5 = 7a;-3.

21. 2x-(4: + x)-5x-{-20 = 4:X-\-(4:-5x).

22. Sx-5-6x-i-l-(9x-5-5x)=3x-9.

23. 10a;-3-(4-2aj) + (3a;-4a;4-5-2a;) = 2-3a; + 4aj —(2x + x)i-7.

24. Prove that 5x-\-4: = 6x — l, when x = 5.

25. Prove that 17 — a; = a? — 19, when x = 18.

26. Show that x(x — y -\-xy) = cc^ — xy-\-x'^y, for as many numerical values as you may substitute for x and y.

Supply the missing coefficients in the following equations :

27. 3a-*b-\-6a + 5b-*xy = *a-i-b-2xy.

28. a^-{-2xy-{-3y^-[2x'-^*y^'] = *a^-{-*xy.

29. 6 m2 + 9 mn- 3 n2 - [3 m"^ -^* mn'] + n^^* m} — mn-* n\

30. From a^'^ft" + a"»62n _ a^n ^ake — 2 a^'"^" + a^n.

31. Ux = r^-\-rs-s% y=:27^-{-4:rs-\-2s% and ;3=r2-3rs-s2, find the value of x -^ y — z.^

32. Add .5 m^ + 2.5 m^w -f n + 3, — .6 wiw^ + .5 m^ + .5 w — 3, and — m' — 2.5 m^n — 1.5 7i + 1.

33. Add .5a;^-3ii/3+f 2^, _ 3^24_. 4^4.^ 5^^^ _3^3_^^3.4^ and 9Ja;4-l^;2^ + 2.25 2/3^

MULTIPLICATION

80. As in arithmetic, the number multiplied is called the multiplicand ; the number by which the multiplicand is multi- plied, the multiplier ; and the result, the product.

When the multiplier is a positive integer, multiplication may be defined as the process of taking the multiplicand additively as many times as there are ones in the multiplier.

Thus, since 3 = 1 + 1 + 1, 6-3 = 5 + 5 + 5 = 16.

Since fractional and negative multipliers cannot be obtained by simple repetitions of 1, the definition of multiplication must now be stated in more general terms.

Although fractional and negative multipliers cannot be ob- tained directly from the primary unit 1, they may be obtained from units derived from 1, by taking a part of 1, or by chang- ing the sign of 1, or by both operations.

Thus, 3^ + i + i + Jand-f = (-i) + (-i) + (-i).

In multiplying 5 by 3, we first observe how 3 is derived from the primary unit 1 ; then in this process we replace 1 by 5.

Therefore, in multiplying 5 by J, in order to extend the definition of multiplication in harmony with the existing one, having observed that the multiplier f is derived from the primary unit by taking 3 of the 4 equal parts of it, we should take 3 of the 4 equal parts of the multiplicand 5. Thus, since 3_|_^i4.i^ 5.3 = 5 + 5_,_5^]^5. Similarly,

-3 = (-l) + (-l) + (-l); ...5.(-3) = (-5) + (-5) + (-5) = -16.

■"

Multiplication is the process of finding a number that is ob- tained from the multiplicand just as the multiplier is obtained om unity,

51

52 MULTIPLICATION

81. Arithmetically, 2x3=3x2.

In general, ah = ha. That is,

Tlie factors of a product may he taken in any order. This is the law of order, or the commutative law, for multipli- cation.

82. 2 X 3 X 5 = (2 X 3) X 5 = 2 X (3 X 5) = (2 X 5) X 3. In general, ahc = (ah)c = a (he) = (ac) h. That is.

The factors of a product may he grouped in any manner. This is the law of grouping, or the associative law, for multiplication.

83. Sign of the product.

(1) Suppose that the multiplier is a positive number, as -f- 2. Since + 2 may be obtained from + 1 by taking + 1 addi-

tively 2 times, a process that involves no change of sign, by the definition of multiplication any number may be multiplied by + 2 by taking the number, with its own sign, additively 2 times.

+ 4.2 = ( + 4)+( + 4)=+8; - 4 . 2 = (- 4) + ( - 4) = -8.

The product, therefore, has the same sign as the multiplicand.

(2) Suppose that the multiplier is a negative number, as — 2. Since — 2 may be obtained from + 1 by changing the sign of

+ 1 and taking the result additively 2 times, any number -may be multiplied by —2 by changing the sign of the number and taking the result additively 2 times.

+ 4.-2 = (-4) + (-4) =-8; -4.-2= (+4) + ( + 4)= +8.

The product has the sign opposite to that of the multiplicand.

(3) The conclusions of (1) and (2) may be written as follows : From (1), 4- a multiplied by -\-h= + ah,

and — a multiplied by +h= —ah-,

from (2), + a multiplied by — ?> = — ah,

and — a multiplied by —h=-\- ah. Whence,

84. Law of Signs for Multiplication. — The sign of the product of two factors is + when the factors hav^ like signs, m\d — when they have unlike signs.

MULTIPLICATION 53

EXERCISES [Additional exercises are given on page 447.]

85. Multiply each of the following by -f- 2 ; then by — 2 :

1. 3. 2. -6. 3. 4. 4. -8. 5. a. 6. -b.

7. -5. 8. 7. 9. 2. 10. 9. 11. - x. 12. y.

Multiply: 13. a 14.-6 15. -x 16. - y 17. n

By i _^ Il§ :zl -12

86. Product of two monomials.

The product of two numbers must contain all the factors, numerical and literal, of both numbers. These may be taken in any order or associated in any manner (§§ 81, 82).

Usually the numerical coefficients are grouped to form the coefficient of the product ; then the literal factors are written, any like factors that may exist being grouped by exponents.

Thus^ § 27, 4 m^n ■ 8 m^n^ = 4 • m ■ m • n • 3 ■ m • ni ■ m ' n - n § 82, =(4 • 3)(w ■ m ' m • m ' m)(n ' n ■ n)

§ 27, " =12 m6«3.

87. Law of Coefficients for Multiplication. — The coefficient of the product is equal to the product of the coefficients of the muU tipUcand and the multiplier.

88. Law of Exponents, or Index Law, for Multiplication. — TTie exponent of a number in the product is equal to the sum of its exponents in the midtipUcand and the multiplier.

The proof for positive integral exponents follows :

Let m and n be any positive integers, and let a be any number.

By notation, § 27,

to m factors, to n factors ;

• to m factors) (a • a • a ••• to w factors) to (m -f ii) factors

	a^	— a •	a •	a
and	a"	= a •	a •	a
.-. «»" •	a""	= (a	' a	• a
by assoc. law		= a •	a •	a
by notation,		= «»»+».

54 MULTIPLICATION

EXERCISES

89. 1. Multiply - 4 ax^ by 2 aV.

Explanation. — Since the signs of the monomials are PROCESS unlike, the sign of the product is minus (Law of Signs).

— 4 ax^ 4-2 = 8 (Law of Coefficients).

2 gV a- a^ = a^ • a^ = a^+^ = a^ (Law of Exponents).

— 8 aV a;2 • x* = x2+4 =x^ (Law of Exponents).

Hence, the product is — 8 a*x^.

Multiply : [Additional exercises are given on page 464. ]

2. 10 a^ 3.-5 m^n^ 4.-4 abc 5. 3 a^bd^

5 a' 3 mn 2 a^b

6. xY 7. 5 jp^V 8. —Sab

xy' — 2 rq'^x — 1

10. — 2x 11. —6ah'^x 12. — 3a6

— 4 o^/>?A 2 6a

15. Aa'^bY 16. -1 3 a^6^.y -1

19. 10 mV 20. —p^q

— 3 n^m^ ap2^^

23. — 2 crft^^ 24. — aj"2/~ 7 a^''?)''* 07?/

	2x^
14.	~3n' 6 b'
18.	4a2 -1
22.	Sa^
26.	5y 3 r-^
30.	-5x- X
34.	— a""

27. a^ftVv"-'^ 28. — a;^-"

31. — iB2/2 32. a6V

35. — 4 a'b' 36. y''-'"

— 3 a'"~26» ym-n-^l

38. d*-'» 39. 8 r*-^s* 40. 2;'-+''-3

9.	-^a^o? -2ax^
13.	-2a'x' — 4 aic*
17.	- 5 m^d^ - 2mi°ccP
21.	- 2 a^m^ri^ 8 b'7iV
25. 29.	4aj"-i -2a;«+i ^n-l^n-2^
33.	2^-n^-a 2n-3^U-b
37.	— x^'-Y'^ -xy
41.	m^n'bY mhi'^b^'y'''''

i

MULTIPLICATION 53

90. Product of several monomials.

By the law of signs, — a • — b =-\- ab]

— a ' — b' — c=+ab- — c = — abc ; — a' — b-— C' — d= — abc . — d = -f abed; etc.

TJie product of an even number of negative factors is positive ; of an odd number of negative factors, negative.

Positive factors do not affect the sign of the product (§ 83).

EXERCISES

91. Find the products indicated : [Other exercises on page 454.]

1. (_1)(-1)(_1). 4. {-2xy){-^xy){3:>?y){-xf).

2. (_2)(-a6)(-3a2). 5. {-4rbc)b{-Zc^)c(-b){-bc).

3. {-a-'x){^bx){-5a'). 6. (- 23)(- 2^)(5. 22)(- 5^ -2).

7. Find the product of 2", 2""^ and 2"+^ Test the correct- ness of your answer when n represents 3.

92. To multiply a polynomial by a monomial.

43 40 + 3 4^ + 3 6 a -2b

_2 ^ 2 ^

86 80 + 6 8^ + 6 15a -66

In general, a{x -\- y -{■ z)= ax -[- ay -\- az. That is,

93. Tlie product of a polynomial by a monomial is equal to the algebraic sum of the partial products obtained by midtiplying each term of the polynomial by the monomial.

This is the distributive law for multiplication.

. EXERCISES

94. Multiply : [Additional exercises are given on page 454.]

1. 2> x^ — 2 xy hy b xy^. 4. p^(f- — 2p(f' by — pq.

2. 3a3-6a26by -26. 5. 4 a^ - o ft^c - c^ by aftc^.

3. im?n? — 3 mn'^ by 2 mn. 6. — 2 ac + 4 aa; by — 5 acx.

56 ' MULTIPLICATION

Multiply as indicated :

7. a'bc(Sa'^-4.a'b). 10. - rs(2r^ - 3rs -\- s^).

8. 2 xy(5a^ -lOxy). 11. 3 ^'^^(^^3 _^ 4 ^ _ 2 ^3^3^^

9. 5 m^(6 Kin? — 2 m^n). 12. xyz{— xy + yz -\- 2 xz).

13. abc{aW - 2 a^& - 2 W-& - a' - 4.b' - c' - 5 abc).

14. - bc{b' + c^ - 6^ - c3 4- 5V - 4 &2c + 8 &c2 - 2 5c).

15. ??i«n3(m^ — 5 m^n^ — 16 m^n^^ + 24 mn^^- - 7i^).

16. X'^-^y'^^\0?y'^~^ — 5 a^""?/"'-^ + 10 aj^-n^^m-l _ 5 ^-2ny'2-my

95. To multiply a polynomial by a polynomial.

EXERCISES [Additional exercises are given on page 454.] 1. Multiply a; + 5 by a? + 2; test the result.

PROCESS TEST

X -\- 5 =6 when x = 1

x + 2 =3

X times (cc + 5) = ic^ + 5 a; 2 times (a; + 5) = 2a;+ 10

(a; + 2) times (a; + 5)= a;^ + 7 a; + 10 =18

Test. — The product must equal the multiplicand multiplied by the multiplier, regardless of what value x may represent. To test the result, therefore, we may assign to x any value we choose and observe whether, for that value, product obtained = multiplicand x muUiplier. When X = 1, multiplicand = 6, multiplier = 3, and x^ + 7rK + 10 = 18 j since 6 X 3 = 18, it may be assumed that a:^ + 7 a; + 10 is the correct product.

E/ULE. — Multiply the multiplicand by each term of the multi- plier and find the algebraic sum of the partial products.

2. Multiply a; + 4 by a; + 6 ; test the result when a; = 1.

3. Multiply a; — 1 by a; — 2 ; test the result when x = 5.

4. Multiply 2a;-|-3by4a; — 1; test the result when a; = 1.

5. Multiply ar^ + a? + lbya;— 1; test the result when x = 2.

MULTIPLICATION 57

6. Multiply 2a — b + c by 3a-hb; test the result when a = 1, 6 = 1, and c = 1.

In like manner the multiplication of any two literal expressions may be tested arithmetically by assigning any values we please to the letters.

While it is usually most convenient to substitute + 1 for each letter, since this may be done readily by adding the numerical coefficients, the student should bear in mind that this really tests the coefficients and not necessarily the exponents, for any power of 1 is 1.

Multiply, and test each result :

7. 2a; + 3bya;4-2. 15. 4^ -66 by 2y-|- &.

8. 4 a; + 1 by 3 a; + 4. 16. 2 6 -f- 5 c by 5 6 — 2 c.

9. 2a + 4by4a — 3. 17. 5 a; — 3 a by 3 a; — 4 a.

10. 6 6 + 1 by 2 6 - 4. 18. a6 - 15 by a6 + 10.

11. 5 71 — 1 by 4 n 4- 5. 19. 3 a + cd by 4 a + cd.

12. h-\-2Tchj 3h — k. 20. ax + by by ax — by.

13. 6c + 2 d by 2c + <^. 21. a^ - ay -\-y^ by a + y.

14. Sl-\-5thj 2l-^6t. 22. a;^ + 6a; - 6^ by 6 - a;.

* An indicated product is said to be expanded when the mul- tiplication is performed.

Expand, and test each result :

23. (x + y)(x-\-y). 32. (af» + y) (af + ?r).

24. (a - 6) (a - 6). 33. (a:« - 2^) (a;" — 2^").

25. (c3 + (f3) (c3 + cP). 34. (af* + 2/»)(a;»-2r).

26. (a;2 4-2/2)(a;2-2/2). 35. (2 6c- 3d) (2 6c - 3d).

27. (3a + 6)(3a + 6). 36. (3 aa; 4- 2 6^/) (3 aa; + 2 62/).

28. (3a-|-6)(3a-6). 37. (3 aa; + 2 6?/) (3 aa; - 2 62^).

29. {2x-y)(2x-y). 38. (4 cd - 2 a6) (4 cd - 2 a6).

30. (2a2-f6)(2a2-6). 39. (a + 6+ c) (a + 6 - c).

31. (4a;-2/2)(4a;-?/2). 40. (x - y + z){x-\-y -z).

68 MULTIPLICATION

Multiply, and test each result :

41. a^ -h a^ + 4 a^-a^-i- a by a -hi.

42. 4/-10 + 22/ by 22/2-32/4-5.

43. 31a^-27a;2 + 9ic-3by 3a;-hl. l

44. 5a;-5a;2-|.l0by 12-30a;-|-2a;2.

45. 4aj-3a;2 + 2a.'3by3a;-10aj2 + 10.

46. 2a2_362_^5 by 3a2-4a5-562.

47. a-\-b-\-c-[-dhya — b — c + d.

^S. 4.a:^-3x''y-]-5xy^-6f by 5x-}-6y.

49. 3 ^2 -f- 3 m^ -H mw, by m^ — 2 mn'^ -f m'*?*.

60. a^ -h ft^-j-c^ — a6 - ac — 6c by a -h 6 -h c.

51. m* — m^n^ -|- m'*/i^ — m^yi^ -j- n^^ by ^2 _|_ ^3^

Expand :

52. (a2 + &2)2, 64, (i^_|^)(3^_2^).

53. (a^ + ft'^/. 65. (i62._i5^i)(^5_|).

54. (5a; + y)2. 66. (^x''- xy + iy^)(ix + ^y).

55. (a& + cc?)2. 67. (a -}- 6) (a - 6) (a -f 6) (a - 6).

56. (2x + 2yy. 68. (a - &) (a + 6) (a^ + 6^) (a^ + 6^).

57. (a-}-6 + c)2. 69. (2n3_^^2_(.3^_^)(3^^4),

58. (3 a2 -I- 5 63)2. 70. (ax^'* -h a/'^) (aa;^'* - a^/^'^). 1

59. (4c + d)(4c-(Z). 71. (a"» - 6") (a"» -h 6") K"* + 6^").

60. (|a;-t-2/)(ia; + y). 72. (aaf^'-{-y»-'')(Saaf'-^-\-2y^-^), ^

61. (3x3-i2/)(3a^-|?/). 73. (x"" - nx'^-'^y -\- ^ nx^'-y) (x -\- y). '

62. (i.a4-i6)(ia— ^6). 74. (ic^^-f a;"?/" + 2/'^'*) (aj^"— «;".?/"+ /")•

63. (2a;4-^2/)(fa^+i2/)- 75. (a«"-ha4'*62<^-|-a2'»6^''-f 6«'')(a2"-62c)^

96. When a polynomial is arranged so that in passing from left to right the several powers of some letter are successively higher or lower, the polynomial is said to be arranged according to the ascending or descending powers, respectively, of that letter.

The polynomial x^ — ix^y + 6 x'^y^ — ixy^ + y* is arranged according to the ascending powers of y and the descending powers of x.

MULTIPLICATION 59

97. When polynomials are arranged according to the ascend- ing or the descending powers of some letter, processes may often be abridged by using the detached coefficients.

EXERCISES

98. 1. Expand (2a;*-3i»« + 3a; + l)(3a;-f 2).

FULL PROCESS DETACHED COEFFICIENTS

2x*-3a^+3x-\-l 2 -3 +0 +3 +1

3 a; +2 • 3 +2

6a^-9x* -\-9x^-\-3x 6 -9 +0 +9 +3

4a;*-6a^ +6a;4-2 4 ~6 +0 +6 +2

6a^_5a;*-6a^H-9a^ + 9a;-f2 6a^-5 x*-6 a^-{-9a^ + 9 x-{-2

Since the second power of x is wanting in the first factor, the term, if it were supplied, would be Ox^. Therefore the detached coefficient of the term is 0. The detached coefficients of missing terms should be sup- plied to prevent confusion in placing the coefficients in the partial products and to avoid errors in writing the letters in the result. •

Arrange properly and expand, using detached coefficients:

2. (x-{-x^-{-l-{-x^)(x-l).

3. (a^-\-10-7x-4:3i^(x-2).

4. (14:-9x-6x^-{-x^)(x + l).

5. (a3 + 4a2-lla-30)(a-l).

6. (4a2-2a3-8a + a*-3)(2 + a).

7. (2m-3-f 2m»-4m2)(2m-3).

8. (x + ar'-5)(x^-S-2x),

9. {b''-{-5b-4:){-4.-\-2b^-3b).

10. (4n3 + 6-2n*+16n-8n2-f-ri*)(n4-2).

11. (T-6x-{-5x'-4:a^-\-3x*-2a^ + x^){x^-{-2x + l).

12. (1 -\- x-\-4. x' -\- 10 a^ + A6 a^ + 22 x'){2 a^ -^1 -3 x).

60 MULTIPLICATION

99. Multiply a» + 2 a^ft + 2 oft^ + h^ by a^ + a6 + h\

PROCESS TEST

a2+ ah +62 ^ 3

a*6 + 2tt862_|_2a258^ ^54

a362 + 2a263 + 2a5^ + 6» a'' + 3 a^6 4- 5 a^ft^ _,_ 5 ^2^3 + 3 ^^^,4 ^ js ^ -^g

When each letter of an expression is given the value 1, the expression is equal to the sum of its numerical coefficients. The test on the right of the process, then, tests the signs and coefficients in the product, but not the exponents.

To test the exponents in the product, observe that each term of the multiplicand contains three literal factors, as a<xa, aab, etc., or is of the third degree; also that each term of the mul- tiplier is of the second degree. Therefore, each term of the product should be of the fifth degree.

When all the terms of an expression are of the same degree, the expression is called a homogeneous expression.

The multiplicand in the process is a homogeneous expression of the third degree ; the multiplier is a homogeneous expression of the second degree ; and the product is a homogeneous expression of the fifth degree.

As a further test observe that the multiplicand involves a and b in exactly the same way, b corresponding to a, b^ to a^, and b^ to a^, so that if b and a were interchanged the multi- plicand would not be changed, except in the order of terms. Such an expression is said to be symmetrical in a and b. Since both multiplicand and multiplier are symmetrical in a and b, the product should be symmetrical in a and b.

100. TJie iwoduct of two homogeneous expressions is a homo- geneous expression.

If two expressions are symmetrical in the same letters, their \ product is symmetrical in those letters.

MULTIPLICATION 61

101. Expand, and test each result:

1. (a + 6 + c)(a4-& + c).

3. (a3 + 3a26 4.3a&2 + 68)(a + &).

4. (a;^ -a^z/-f-a;22^- a^ + 2/0(^ + 2/)-

6. (a& 4- 6c 4- cc2 + bd){ab -\-bc — cd — bd).

7. (a^-.x'2/ + / + a'+2/4-l)(aJ + 2/ + l).

8. (a^ + Sa'b + 3ab^ + b^)(a'-\-2ab + b^,

9. (a2_a6-ac + 62_6c4-c2)(a + 64-c).

Addition, Subtraction, Multiplication

EXERCISES

102. 1. Simplify a2^a(6-a)-&(26-3a).

Solution. — The expression indicates the algebraic sum of a^, a(b — a)^ and — b(2b-Sa). Expanding, a(b — d)=ah - a^, and - &(2 6 - 3 a) = — 2 62 ^ 3 a6. Therefore, writing the terms in order with their proper signs, a2 + 05(6 _ «) - 6(2 6 - 3 a) = a2 + a6 - a2 - 2 62 + 3 «6 = 4a6-2 62.

Simplify :

2. a^ + a;(2/ — a;). 5. ds? — y^ — {x — yf-

3. c^ — c(c — d). 6. c(a-6)-c(a4-6).

4. 5-2(a;-3). 7. a^- 63-3a6(a-6).

8. -2{^y -xf)-^{xf -^y).

9. (3a-2)(2a-3)-6(a-2)(a-l).

10. 8a.'^-(4a;2-2a;2/ + rX2« + y)-

11. (3 m - l)(m -h 2) - 3 m{m + 3) + 2(m + 1).

12. (a - 6)2- 2(a2 - 6^)- 2 a(- a - 6)- 4 6-,

62 MULTIPLICATION

13. 4(aaj — bx-\-cx — dx) — S(ax -{-bx — cx — dx).

14. (x + l){x -f 2) - 2(x - l)(x - 2) + 4(a; + 2){x + 3).

15. (ar^ H- 2 xy + f){x' - 2 xy -{- y') - (:^ + y%x^ + 2/').

16. 5^ +(a2 - a6 -}- 62)(a2 + b'')-(a' - 63)(a + 2 b).

17. 2/^ - [2 a;^ - a;2/(aj - ?/) - y'} + 2(aj - 2/)(^' + ^y-\- 2/').

103. Numerical substitution.

EXERCISES

1. When a = — 2, 6 = 3, c = 4, find the value of

a2_(a-c)(5 + c) + 2&.

Solution. a^-.(a- c)(6 + c) + 2 6 =(- 2)2 -(-2-4)(3+4)+2 • 3

= 4-(-6)(7)+6 i

= 4-(-42)+6 }

= 4 + 42 + 6 = 52.

When a; = 3, 2/ = — 4, 2; = 0, »i = 6, n = 2, find the value of :

9. (n — m) + 3 xy. 1

10. (m + ny-(y+zy.

11. (m + 2/)^ + a;2;— n^.

12. (a; + 2/)(m — 7i) + 32;.

13. {m + xY — (n—yy-y\

14. aJ2/2^ — 7i(x — mf -r (nxy.

15. S{x-{-z) — %'mn-{-5y. 4

16. |(2/-2 70-K^-2 2/)(3 2/-4 7i). 1

17. (a; + 2/)^-a;2/(a;-2/) + (a;4-2/)(a5^-2/'^)-

18. 3 m{x — y — ny —{y — n — x){n — x — y).

19. (2 a; + yy-ix"" + 2 2/)-(m + ny(x + 2/ + zy.

20. Show that (a - 6 + c)^ = a^ + 6^ 4. c2 _ 2 a6 + 2 ac - 2 &c, when a = l, 5 = 2, and c = 3 ; when a = 4, 6 = 2, and c = — 1.

21. By substituting numbers for a, b, and c, show that (a -\-b)(b-\- c){c H- a) 4- a6c = (a + 6 + c)(ab + bc + ac).

2.	x+{y-\-m).
3.	(m — n)2 + y.
4.	n{y — m)+z.
5.	(m-\-y -{-xy — n.
6.	m{x + 2/) + 2;^
7.	2 + m2-(2/3-l).
8.	a;2 — 2/2 _ ^2 _|_ y^2

MULTIPLICATION 63

SPECIAL CASES IN MULTIPLICATION

104. The square of the sum of two numbers. Show by actual multiplication that

Also that 142 = (10 + 4)2= 102 -h 2 X 10 X 4 4- 42 = 196.

105. Principle. — The square of the sum of two numbers is equal to the square of the first number, plus twice the product of the first and second, plus the square of the second.

Since 5 a^ x 5 a^ = 26 a^, 3 a*b^ x 3 a^ft^ = 9 ^s^io^ etc., it is evident that in squaring a number the exponents of literal factors are doubled.

EXERCISES [Additional exercises are given on page 455.]

106. Write out the square, and test each result : (f) + qy. 8. 222. 15^ (5 a; + 2)2. 22. (a^+b^y.

2. (r + s)2. 9. 312. iQ (2a-\-xy. 23. (a' + b^y.

3. {a-\-xy. 10. 232. 17. {ab-{-cdy. 24. (a''4-&'*)^

4. (a; + 4)2. 11. I32. 18. (5x-\-2yy. 25. (a;« + 2/'*)^

; 5. (a + 6)2. 12. 522. 19 (7 2! + 3c)2. 26. (3a''-\-5¥y.

6. (2/ +7)2. 13. 1012. 20. (3 6 + 10a;)2. 27. (l-\-2a'by.

7. (2+1)2. 14. 2072. 21. (3a;i/2 + 4a;2?/)2. 28. (x^-^-{-y^-'y.

I 107. The square of the difference of two numbers.

j Show by actual multiplication that

I (a - b)(a - b) = a'' -2ab + b\

I Also that 192 = (20 - 1)2 = 202 - 2 x 20 x 1 + 1^ = 361.

i

I 108. Principle. — The square of the difference of two num- I bers is equal to the square of the first number, minus twice the I product of the first and second, plus the square of the second.

I

2.	(m-	- ny.	14.	292.
3.	i^-	-6)2.	15.	392.
4.	iP-	-8)^	16.	382.
5.	(Q-	-ly.	17.	492.
6.	(a-	-cy.	18.	482.
7.	(a-	-xy.	19.	592.
8.	(X-	-ly.	20.	582.
9.	(b-	-5y.	21.	792.
10.	(x-	-4)1	22.	992.

64 MULTIPLFCATION

EXERCISES

[Additional exercises are given on page 455.]

109. Write out the square, and test each result :

1. (x-my. 13. 182. 25. (2 a -05)2. 37^ (3 a; -2)2.

26. {3m-7iy. 38. {2x-5yy.

27. (4:x — yy. 39. {5m — 3ny.

28. (m-4?i)2. 40. {3 2^ - 5 qy.

29. (2)-3g)2. 41. (a^-b^y.

30. (a - 7 6)2. 42. (x""- y'^y.

31. (4 a -3)2. 43. (a2-2 62)2.

32. (5 a; -4)2. 44. {f-^xy.

33. (a6-3)2. 45. (a6 - 2 c2)2.

34. {2z-lyy. 46. (x'+'-y^y.

11. {z-3y. 23. 9972. 35. (Sx-5yy. 47. {rnx'^-ny'^y.

12. (w-9)2. 24. 9992. 36. (9w;-2'y)2. 43. (ajm-i_ ^n-i^)2^

110. The square of any polynomial.

Show by actual multiplication that

(a -h 6 H- c)2 = a2 -h 62 + c2 -f 2 a& -h 2 ac -f- 2 6c ; also that {a-\-h -\-c-^rdy = o? + h'' + c" + d''-\-2ah ^-2 ac

-f- 2 art -h 2 6c -f 2 6d -h 2 cc?.

Similarly, by squaring any polynomial by multiplication, it will be observed that :

111. Principle. — The square of a polynomial is equal to the sum of the sqiiares of the terms and twice the product of each term by each term, taken separately^ that follows it.

When some of the terms are negative, some of the double products will be negative, but the squares will always be positive. For example, since (- 6)2= 4. 62^ (a _ 6 + c)2= a2 + (- 6)2+ c^+ 2 a(- 6)4-2 ac + 2{-b)r = a2 + 52 4. c2 _ 2 a6 + 2 rtc - 2 be.

MULTIPLICATION

65

EXERCISES

112. Expand by inspection, and test each result:

1. {x + y + zf. 3. {x — y — zf. 5. (x^y — ^zy.

2. {x-\-y — zy. 4. {x — y+,zy. 6. ix — y-{-3zy.

7. {a-2h + c)\ 13. {^x-2y + 4.zy.

8. (2a-b-cy, 14. (2a-56-f3c)2.

9. (b-2a-\-cy. 16.. (2m-4 7i-r)2.

10. \ax-by-\-czy. 16. (12 - 2 a; + 3 2/)2.

11. (qb — pc—rdy. 17. (a + m + 6 + 7i)2.

12. (ac — 6d - dey. 18. (a — m + 6 — 7i)2.

113. The product of the sum and difference of two numbers.

Let a and b represent any two numbers, a -\-b their sum, id a—b their difference.

Show by actual multiplication that

(a-{-b)(a-b)=a^-b\ Also that 32 X 28 = (30 + 2) (30 - 2) = 30^ - 2^ = 896.

114. Principle. — TJie product of the sum and difference of :o numbers is equal to the difference of their squares.

EXERCISES

[Additional exercises are given on page 465.] 115. Expand by inspection, and test each result :

1. {x + y)(x-y).

2. (aH-c)(a — c).

3. (x-\-l)(x — l).

4. (x' + l)(x'-l).

5. (a)3 + l)(a^-l).

6. {x^-l){x' + l).

7. (a:s-l)(a^-fl).

8. 19x21.

9. 31x29.

10. 43x37.

11. 56x64.

12. 89x91.

13. 45x55.

14. 34x26.

15. (p + q){p-qy

16. {p-\-5){p-5).

17. (x--^y'){x"-f).

18. (a6H-5)(a6-5).

19. {cd-\-d^){cd-d^).

20. {12-^xyX12-xy).

21. {al)-\-cd){ab — cd).

MILNE'S STAND. ALG.

66

MULTIPLICATION

33. (3 mhi - b){S w}n -f- h).

34. (2x3H-5 2/2)(2flr'-52/').

35. (3x«+2/0(3a;«-2/).

36. (2a2 + 2 62)(2a2_2 62).

37. (— 5n — 6)(— 5n + ?>).

38. (_.x-22/)(-a; + 22/). -

39. (3 a?"* + 7 2/") (3 a;'" - 7 ?/").

40. (mx'^4-2?/*)(ma-« — 2/).

41. (a"6'" -f a'"6")(a"6"* — a'"^").

42. (ic'"-i + 2/"+0(.'c"'-^-2/"+^).

43. (5a362_^2af)(5a3&2_2af).

One or both numbers may consist of more than one term.

44. Expand {a-\-m — n){a — m + w). Solution

a + wi — w = a + (m — n). a— m-\- n — a— (w — n). :. [a + w - n] [a - m + n] = [« + (w — ii)] [a — (m - w)] Prin. §114, =a^-{m-nY

§ 108, = a2 - (m"^ -2 7nn + Ji^)

= a'-^ — m- + 2 mn — n^. Expand :

22.	(x'^-^fYx'-y'^).
23.	{x'-y')(x' + y'y
24.	(2a; + 4)(2a^-4).
25.	{2x' + y){2x'-y).
26.	{ah - c^){ah + c^).
27.	{^.yjrz%^y-z^).
28.	{2x + ^y){2x-^y).
29.	{pxy-Z){^xy-{-^).
30.	{Ix — 5 m){lx -\- 5 m).
31.	(3m + 4n)(3m-4?i).
32.	(4a.V+5)(4a.Y-5).

45. (a + CC — ?/)(« — a; + 2/).

46. (cc + c — d)(ic— c 4-d).

47. {r-\-p-q){r — p + q).

48. (r4-2? + g)(r-p-g).

49. {x-\-h-\- n)(x —b — 7i).

50. {y + c + d){y +c-d).

51. (a+a;4-2/)(« + ^-2/)-

52. (.^•2+2a; + l)(.^•2 4-2aJ-l).

53. {x''-{-2x-l){x''-2x-\-l).

54. (x2_p3a.'-2)(aj2_3a;4.2).

55. (7i'-27i''-hl)(n'+2n^-\-l).

56. (ic2 + aj?/ + 2/^)(a;2-a;?/ + ?/')•

57. (2a;+3 2/-4)(2a;+32/-f-4).

58. (7-2— ?'s-f3s)(r2 + rs — 3 s).

t MULTIPLICATION 6T

116. The product of two binomials that have a common term. Let 05 + a and x-^b represent any two binomials having a common term, x. Multiplying x-{-ahy x -{-b gives

x-\-a x-\-b

x^ + ax

bx-\-ab x^-{-{a-{- b)x 4- ab

117. Principle. — The product of two binomials having a common term is equal to the sum of the square of the common term, the product of the sum of the unlike terms and the common term, and the product of the unlike terms.

EXERCISES

[Additional exercises are given on page 455.]

118. 1. Expand (a; -f- 2)(a; -f 5) and test the result.

Solution, — The square of the common term is x^;

the sum of 2 and 5 is 7 ;

the product of 2 and 5 is 10 ;

.-. (a; + 2)(ic + 5) = x2 + 7 X + 10. Test. — If a; = 1, we have 3 • 6 = 1 + 7 + 10, or 18 = 18.

2. Expand (a + l)(a — 4) and test the result.

Solution. — The square of the common term is a^ ; the sum of 1 and — 4 is — 3 ; the product of 1 and — 4 is — 4 ; .-. (a + l)(a-4) =a2_3«_4.

Test. — If a = 4, we have 5 • 0 = 16 - 12 - 4, or 0 = 0.

3. Expand {n — 2){n— 3) and test the result.

Solution. — The square of the common term is n^ ;

the sum of — 2 and — .3 is — 5 ;

the product of — 2 and - 3 is 6 ; •

.-. (/i-2)(»-3) = n-'- 5n 4- 6. Test. — If w = 3, we have 1 • 0 = 9 — 15 + 6, or 0 = 0.

68 MULTIPLICATION

Expand by inspection, and test each result :

4. (x-\-5)(x + 6). 18. (a;"-5)(a;'» + 4).

5. (x + 7)(x-{-S), 19. (af-d){x^-b).

6. (x-7)(x + S). 20. (y-2a)(y + Sb).

7. (x + 7)(x-8), 21. (z-4.a)(z + 3a).

8. (x-5)(x-4l). 22. (2a;-|-5)(2a;H-3).

9. (x-S)(x-2). 23. (2a;-7)(2a; + 5).

10. (x-5Xx-l). 24. (32/-l)(3 2/ + 2).

11. (a; + 5)(a; + 8). 25. (4a^ + l)(4a^-7).

12. (p_4)(p + l). 26. (a6-6)(a&+4).

13. (r-3)(r-l). 27. (a^2/'-a)(^/ + 2a).

14. (n-8)(7i-12). 28. (3aJ2/ + 2/')(/-«2/).

15. (w-6)(w + 15). 29. (a;4-2/-l)(a;4-2/ + 2).

16. (a;2 + 5)(a;2-3). 30. (x-y-2Xx-y-S),

17. ((c^-7)(a^ + 6). 31. (x'-{-x-l)(x^-}-x-\-S).

By an extension of the method given above, the product of any two binomials having similar terms may be written.

32. Expand (2 a; -5) (3 a; + 4).

PROCESS Explanation. — The product must have a term

2x — 5 in x^, Si term in x, and a numerical, or absolute, term.

XThe x2-term is the product of 2 ic and 3 x ; the a;-term o ,A is the sum of the partial products — 5 • 3 a; and 2 x • 4,

called the cross-products ; and the absolute term is the

6 .^2 — 7 a; — 20 product of - 5 and 4.

The process should not be used except as an aid in explanation.

Expand by inspection, and test each result :

33. (2a; + 5)(3a; + 4). 36. (Sx-y)(x-Sy).

34. (3a;-2)(2a;-3). 37. (2a + 5b){5a + 2b). 36. (3a-4)(4a + 3). 38. {7 n''-2p)(2n'-7p).

I

MULTIPLICATION • 69

Algebraic Representation

119. 1. Express in the shortest way the sum of five aj's ; the product of five ic's.

2. When the multiplicand is x and the multiplier y, express the product in three ways.

3. Indicate the product when the sum of x, y, and — d is

multiplied by xy.

4. How much will a man whose wages are a dollars per day earn in b days ? in c days ? in ic days ? in a days ?

5. If a man earns a dollars per month and his expenses are b dollars per month, how much will he save in a year ?

6. Indicate the sum of x and z multiplied by m times the sum of X and y,

7. From x subtract m times the sum of the squares of

(a 4- b) and (a — b).

8. A number x is equal to (y — c) times ((? + c). Write the equation.

9. How many seconds are x days + c hours + d minutes ?

10. Express in cents the interest on y dollars for x years, if the interest for one month is z cents on one dollar.

11. How far can a wheelman ride in a hours at the rate of b miles an hour ? How far will he have ridden after a hours, if he stops c hours of the time to rest ?

12. How many square rods are there in a square field one of whose sides is 2 6 rods long ? (x — y) rods long ?

13. What is the number of square rods in a rectangular field whose length is (a + b) rods and width (a — b) rods ?

14. A fence is built across a rectangular field so as to make the part on one side of the fence a square. If the field is a rods long and b rods wide, what is the area of each part ?

15. Represent (a — b) times the number whose tens' digit is X and units' digit y.

70 • MULTIPLICATION

Equations and Problems

120. 1. Given 5(2 a; - 3) - 7(3 oj + 5) = - 72, to find the value of X.

Solution 5(2 ic - 3) - 7(3 a; + 5) = - 72. Expanding, 10 x - 16 - 21 x - 35 = - 72.

Transposing, 10 x - 21 x = 15 + 35 - 72.

Uniting terms, — 11 x = — 22.

Multiplying by - 1, ^ 11 x = 22.

.-. x = 2.

Verification. — Substituting 2 for x in the given equation,

5(4-3) -7(6 + 5) =-72.

5 -77 =-72. Hence, 2 is a true value of x.

Find the value of x, and verify the result, in :

2. 2 = 2a;-5-(a;-3). 4. 1 = 5(2a;-4) H-5a; + 6.

3. 10aj-2(aj-3) = 22. 5. 7(5-3a;)=:3(3-4a-)-l.

6. 2(a;-5)+7=aj + 30-9(a:-3).

7. 5 + 7(x-5) = 15(a; + 2-36).

8. (a;-2)(a;-2) = (a;-3)(a;-3)-|-7.

9. (aj-4)(a; + 4) = (a;-6)(a; + 5) + 25.

10. 4.a?-^{oi?-a?-\-x-2) = 4.x^.

11. 7(2a;-3&) = 26-3(2a; + 6).

12. 3(26-4aj)-(a;-6) = -66. 14. 3(a;-a- 2&) = 36.

13. 4.x-a? = x(2-x)-\-2a. 15. 5 6 = 3(2a;-6)-4 6.

16. ar^-(2a; + 3)(2a;-3) + (2a;-3)2 = (a; + 9)(a;-2)-2.

17. 3(4-a;)2-2(a; + 3) = (2a;-3)2-(a; + 2)(aj-2) + l.

18. 20(2 - a;) + 3(a; - 7) - 2[a; + 9 - 3 ; 9 - 4(2 - 7) S ] = 23.

ft

MULTIPLICATION 71

[Additional problems are given on page 456.]

19. A rectangle is twice as long as it is wide. Its perimeter is 240 feet. Find its dimensions.

20. My lawn is 45 feet deeper than it is wide. The distance around it is 270 feet. Find its dimensions.

21. The square of a number is 15 less than the product of this number and the next higher one. Find the number.

22. The difference of the squares of two consecutive num- bers is 33. Find the numbers.

23. The students of a school numbering 210 raised $ 175 with which to buy pictures. The seniors gave $ 1.50 each,^the rest $ .50 each. Find the number of seniors.

24. Two boys sold 150 tickets, the reserved seat tickets at 75^ each and the others at 50^ each. The total receipts were $ 87.50. How many tickets of each kind did they sell ?

Suggestion. — Reduce all values to a common denomination, as cents.

25. Leo has 3 times as many plums as Carl. If each had 5 more, Leo would have only twice as many as Carl. How many has each ?

26. Two boys had 350 apples. They sold the green ones for 3^ each and the red ones for 5^ each and received in all $ 11.60. How many apples of each kind did they sell ?

27. The length of a classroom is 4 feet more than twice its width. If its width is increased 2 feet, the distance around it will be 120 feet. Find its dimensions.

28. My house is 16 feet deeper than it is wide. If it were 6 feet deeper than it is, the distance around it would be 140 feet. Find its dimensions.

29. Upon the floor of a room 4 feet longer than it is wide is laid a rug whose area is 112 square feet less than the area of the floor. There are 2 feet of bare floor on each side of the rug. What is the area of the rug ? of the floor ?

DIVISION

121. In multiplication two numbers are given and their product is to be found. The inverse process, finding one of two numbers when their product and the other number are

given,	is	called division.
		10-4-	•2 =	5,	and Z>H	'.d-.	= Q
are inverses of 5 x	:2 =	10,	and q X d:	= D.

The dividend corresponds to the product, the divisor to the multiplier, and the quotient to the multiplicand.

Hence, the quotient may be defined -as that number which multiplied by the divisor produces the dividend.

In general, the quotient of any two numbers, as a divided by

6, indicated by a -4- 6, or -, is defined by the relation

a , -xb = a.

0

122. Sign of the quotient.

The following are direct consequences of the law of signs for multiplication (§ 84) and the definition of quotient :

(H-a) (+6) =+a6; .-. H-a6-f- (+6)= +a.

(-a) (+b) = -ab; .-. -ab-i-(+b)= —a.

(+a) (-&) =-a6; .-. —ab-i-(--b)= +a.

(-a) (-b) =-{-ab; .-. +a6^ (- 6) = -a.

123. Law of Signs for Division. — The sign of the quotient is + when the dividend and divisor have like signs, and — when they have unlike signs.

72

DIVISION 73

EXERCISES [Additional exercises are given on page 447.]

124. Divide each of the following by 2; then by — 2: 1. 6. 2. -6. 3. 10. 4. -10. 5. 14. 6. -12. 7. -8. 8. 4. 9. 12. 10. -18. 11. 22. 12. -16.

Perform' the indicated divisions : 13. 7) - 14. 14. -3)15. 15. -3) -12. 16. -1)9^

17. 4^(_4). 18. 22 --(-2). 19. -l--(-l). 20. -6-i-3.

-f- /-!^- -^^- -^-

125. To divide a monomial by a monomial. Since 7 a x 3 a^ = 21 a^,

by del of quotient, 21 a^ -h 3 a'' = 7 a.

The quotient may be obtained, as in arithmetic, by removing equal factors from dividend and divisor, thus :

21 a^ ^7 ' fi ' fi ' fit ' jiL • ft ' a ^rj

or ^^ = ?^a'-^ = Ta' = 7a.

3 a' 3

126. Law of Coefficients for Division. — The coefficient of the quotient is equal to the coefficient of the dividend divided by the coefficient of the divisor.

127. Law of Exponents, or Index Law, for Division. — TJie ex- ponent of a number in the quotient is equal to its exponent in the dividend minus its exponent in the divisor.

Since a number divided by itself equals 1, a^ -4- a^ = a^-^ = a" _ i- that is, a number whose exponent is 0 is equal to 1. (Discussed in § 306.)

The law of exponents for division is of general application, but for present purposes exponents will be limited to posi- tive integers. The proof tor positive integral exponents follows :

74 DIVISION

Let m and n be positive integers, m being greater than n ; and let a be any number.

By notation, § 27, a"* = a • a • a ••• to m factors,

and a" = a • a • a ••• to n factors ;

. a^ _ a • a • a ••• to m factors a" a ' a ' a '•• to n factors Remove equal factors from dividend and divisor. Then,

a"* -^ a" = a • a • a ••• to (w — w) factors "^

by notation, = a'»-^

EXERCISES

[Additional exercises are given on page 459.] 128. 1. 5)5«^ 2. l&d')-Z5c'd\ 3. -4a^)-a^ |

52 - Sc^d la"

Divide as indicated :

4. 22)2^. 10. -a;)^. 16. 4g)-12gl 22. 28 a^js ^ _ 4 ^252^

5. 22}2^. 11. a?)-a\ 17. -2n^)16?^^ 23. - 20 by --r 5 b'^y.

6. 3^)3^. 12. 22)a^. 18. 6v^)-lHv\ 24. - 16 a^?/^ -- 4 a.y.

7. 4")4^ 13. 2)4 m. 19. 7?^)-14;^ 25. -6aY^-3ay.

8. 52)5^. 14. -2)6 r^. 20. 2 7rr)4.7rr\ 26. -4a;'"2''--32 a;"^*.

a^b ' -xy^' ' 20 a^^c^* * -a(a;-2/)2*

129. To divide a polynomial by a monomial.

Since, § 93, (a-{-b)x = ax-\- bx,

if ax + 6a; is regarded as the dividend (§ 121) and x as the divisor, (ax + bx)-r-x = a-\-b; that is,

130. The quotient of a polynomial by a monomial is equal to the algebraic sum of the partial quotients obtained by dividing each term of the polynomial by the monomial.

This is the distributive law for division.

DIVISION 76

EXERCISES

[Additional exercises are given on page 459.] 131. 1. Divide 4 a^ft - 6 aW -\- 4. aW hy 2 ah -, by - 2 ah.

PROCESS PROCESS

2 a6)4a36-6a^6^ + 4a6« - 2 ah) A. a^h - e> a'h'' -\- 4. ah^

Wk 2a2 -'Sah +262 _2a2+3a6 -2h^

Test of Signs. — When the divisor is positive, the signs of the quo- tient should be like those of the dividend. When the divisor is negative, the signs of the quotient should be unlike those of the dividend.

Test of Exponents. — Since the sum of the exponents in each term of the dividend is 4, and the sum in each term of the divisor is 2, the sum of the exponents in each term of the quotient should be 4 — 2, or 2.

Divide as indicated :

'Zx'-Qx ^ 6 n^ + 12 n^ Sa*h^-6a''b*

2x ' ' -Sn ' ' 3a262 *

g 4 ?/2 _^ 8 y^ - 5 r^ + 10 r^ -Sx'y-{-4xy

-2y^ ' ' -5r2 * ' -2a^y

8. 3 xy^ — 3 x'^y by xy. 13. x'^ — 2xy-^ x^y^ by x.

9. G a262 - 9 ab^ hy S ah. 14. -a-h-c-dhj - 1.

10. 4 xY + 2 xY by 2 xY- 15. — a + a'^h — a\ by — a.

11. ahc^ — 2 a^h'^c by — ahc. 16. x^y — xy"^ + x^if by xy.

12. 9 xYz + 3 xyz^ by 3 xyz. 17. c2rZ-3 cd2_j_4 cS^^s ^y _cd.

18. 34 a^iry - 51 aV?/^ - 68 a^x^f by 17 a^a^^i/a,

19. 8 a'y" - 28 a«64 _ ^g ^555 _^ 4 ^4^6 i^^ 4 ^453,

20. a{h - c)3- 6(6 - c)2 + c(6 - c) by (6 - c).

21. (x-y)- 3{x - 2/)^ + 4 a;(a; - yf by (« - y).

22. cc" + 2 a.-«+i — 5 cc°+2 _ a;«+3 _j_ 3 a;a+4 ^j ^o^

23. ?/"+! — 2 ?/»+2 _|_ 2/«+3 _ 3 yn+4 4- ^n+5 by y^+\

24. X" - ic"-! + x""-^ — a;"-3 -h ic"-'* — a;"-^ by a^.

76 DIVISION

132. To divide a polynomial by a polynomial.

EXERCISES [Additional exercises are given on page 459.] 1. Divide 3 a;2 + 35 + 22 a; by a; + 5.

PROCESS TEST

3a;2-f.22flj + 35 3 X times {x + 5) 3 a:^ + 15 a;

a5 + 5 60 --6

3a; + 7 =10

7 a; + 35 7 times {x -f 5) 7 a;-}- 35

Explanation. — The divisor is written at the right of the dividend, and both are arranged according to the descending powers of x.

Since the dividend is the product of the quotient and divisor, it is the algebraic sum of all the products formed by multiplying each term of the quotient by each term of the divisor. Therefore, the term of highest degree in the dividend is the product of the terms of highest degree in the quotient and divisor. Hence, if 3 ic^, the first term of the dividend as arranged, is divided by a;, the first term of the divisor, the result, 3 a;, is the term of highest degree, or the first term, of the quotient.

Subtracting 3 x times {x + 5) from the dividend, leaves a remainder of 7 a; + 35.

Since the dividend is the algebraic sum of the products of each term of the quotient multiplied by the divisor, and since the product of the first term of the quotient multiplied by the divisor has been canceled from the dividend, the remainder, or new dividend, is the product of the rest of the quotient multiplied by the divisor.

Proceeding, then, as before, we find 7 a; -=- a; = 7, the next term of the quotient. 7 times (x + 5) equals 7 x -|- 35. Subtracting, we have no remainder. Hence, all the terms of the quotient have been obtained, and the quotient is 3 a; + 7.

Test. — Let a; = 1.

Dividend = 3 aj'-* + 22 « + 35 = 3 + 22 + 35 = 60.

Divisor = x + ^ =1 + 5 =6.

Quotient should be equal to 10

Quotient =3x + 7 =3 + 7 =10.

Similarly, the result may be tested by substituting any other value for X, except such a value as gives for the result 0 -r- 0, or any number divided by 0, for reasons that will be shown in § 547.

DIVISION 77

Rule. — Arrange both dividend arid divisor according to the ascending or the descending powers of a common letter.

Divide the first term of the dividend by the first term of the divisor, and write the result for the first term of the quotient.

Multiply the whole divisor by this term of the quotient, and sub- tract the product from the dividend. The remainder will be a new dividend.

Divide the new dividend as before, and continue to divide in this way until the first term of the divisor is not contained in the first term of the new dividend.

If there is a remainder after the last division, write it over the divisor in the form of a fraction, and add the fraction to the part of the quotient previously obtained.

Divide, and test each result :

2. a^ + 4:X + 4:hy x-\-2. 8. 9 + 6a; + a^ by 3 4-a;.

3. a^ — 4 a; + 4 by ic — 2. 9. a^ + 8 a; + 15 by a; + 3.

4. a^ + a;-20 by a; + 5. 10. x^ -^ 15 x -\- 54. hy x -{- 6.

5. a^ + 7 a; + 12 by a; -}- 3. 11. m^ — 18 — 3 m by m — 6. •

6. a^ — 3 a; — 18 by a; — 6. 12. a^ + 2 a^ + 2/^ by a; + ?/.

7. ;4_6p_16by ;2_^2. 13. 10-lla; + ar^by a;-10.

14. 25a^-a^-8a;-2a^by 5a^-4a;.

15. a« + a« -f aM- «^ + 3 a - 1 by a + 1.

16. 2a*-5a% + 6a'b'-4:ab^ + b^hj a^-ab + b\

PROCE 2a' - 5 a^b + 6a'b^ -4.ab^ -\-b'	ss ^2- ab + b-"	TEST 0-1
2a'-2a%-j-2a'b'	2a2-3a6 4 6'	= 0
-3a35 + 4a2&2_4a63 -Sa'b-\-3a'b'-3ab^ a%^- ab^^b'^ a%^- ab^ + b'

Note. — It will be observed from the test that 0 -=- 1 = 0. In general, 0 -7- a =0; that is, zero divided by any number equals zero (§ 542).

78

17

18 19 20 21 22 23 24 25 26 27 28 29

DIVISION

a4 + 16 + 4 a2 by 2 a + a^ + 4. a;5_ 61 aj - 60 by a^ - 2 X - 3. a5 _ 41 a - 120 by a^ + 4 a + 5.

x'-Sa^-36x''-71x-21hy x''-3-Sx. 6 a2 + 13 a6 + 6 62 by 3 a + 2 6. 3 w^ — 4 am^ 4- a^m^ by am — 1. ax^ — aV — 6a.'2 + ft^ by ax — b.

x'y-25x''-lSf-\- 27 a;?/^ hj 6y-5x. 7^ -{- 3 t^'s + S rs" -\- s" by ?-2 + 2 rs + s^. ic^ + 4 a^y -1- 6 ic^z/^ -|- 4 a;?/^ + y* by a^ + y. a? + 5 a^a; + 5 ax"^ -j-x^ by a^ + 4 aa; + ar^. a^ — 4 a^a; + 6 a^af — 4 ota^ + x* by a^ — 2 ora; -|- x^.

. 20

30. a^ + 1

a^ — a^

a^ + l

a-1

«' + a' + a + 1 + -^ a — 1

a^ + 1 a^ — a

a-\-l a-1

31.

a^-3a^H- a;2 + 2a;-l a^- a^-2x^ -2a;'' + 3aj2-f-2a; - 2 a;^ 4- 2 a;'^ + 4 a;

a?2-2aj-l

x^- x-2

— x-\-l

x^-	- x-2
x"-	-2a. + l+^ x^-	-a; 4-1 -a;-2

DIVISION

79

Divide :

32. aa^ + Slhy x-3. 35. a' 4- 6^ by a + b,

33. ic^ + 32 by a; + 2. 36. 7n^ — n^hj m + n.

34. a^ — y^ by a^ -\- y^. 37. m^ + w^ by m + n.

38. ^ + a;*y + iK2/^ .+ 2/^ by ic 4- 1/.

39. a"* -\-d?y + ay' + ?/'* by a + ?/.

40. ^ -\-?> x^y 4- 3 x?/'^ + y' by a; + y.

41. a« + 5 a^-a^ + 2 a 4- 3 by a- 1.

42. 05^ + 2 ic^ — 2 a^ + 2 a;2 — 1 by a; 4- 1-

43. 2 a.-* — aj7 4- 2 a^ — a;2 + a;5 4- 5 by a; 4- 1.

44. 2/^ + 32/* + 5?/3 4-3/4-32/H-5by2/ + l.

45. 2 ?i5 - 4 7i* - 3 n^ 4- 7 7*2 - 3 n 4- 2 by ?i - 2.

46. 1 by 1 4- a; to five terms of the quotient.

47. 1 by 1 — a; to five terras of the quotient.

48. dot? — aV — 6a^ + 62 \^y fj^rjf, _ 5.

49. a' _ 6 ^2 _^ ;12 a - 8 - 63 by a - 2 - 6.

50. y^ 4- 32 x' by 16 x' -^-y^ -2 xif - ^ a?y + ^ xhj^

51. a.'^ + 2/^ 4- 2^ — 3 xyz hj x -\- y -\- z.

52. m^ + n^ 4- x^ + 3 m^/i -\- 3 mn^ by m + ?i 4- a;.

53. a' — 2 a'^c H- 4 ac^ — aa?^ — 4 c^^. _|_ 2 ca;'' by a — x.

54. «» _ 6^ + c^ + 3 a6c by a^ 4- 6^ 4- c^ + a6 - ac 4- he.

55. ^m4 4-|m

+ L6 _ J 7^2 by I m^ — m — f .

56. I oV _ 3 ofa.-a + I a;4 - 2 a^ by f «2 4- i a^ - J ««•

57. 7'2n ^ 11 ,,n ^ 39 ]gy y" + 6.

58. ^^-^ 4- 2/3n+3 by a;«-i + 2/''+^

59. a;" 4- ?/** by a? + ?/ to five terms of the quotient.

60. 2-3 ri* + 13 n"^ + 23 n'^ - 11 n^^ 4- 6 ?i^ by 2 4- 3 n*.

61. — x-'"+y^ — 2 a;2'-+32/2s+i _ a;2r +5^28+2 ^y _xrys-l _ ic'+^l/*.

80

DIVISION

Divide, using detached coefficients : 62. x^-5x + 4:hj x^-2x + l.

PROCESS

1+0+0+0-5+4 1-2+1

2 + 1

1+2+3+4

2-	-1+0
2-	-4 + 2
	3-2-	-5
	3-6 + 3
	4-	-8 + 4
	4-	-8 + 4

= a^ + 2i»2 + 3a;+4

63. 64. 65. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81.

a^ + 8a; + 7by a;2 + 2a; + l. 66. z^ - 64: hj z - 2.

a6 + 38a + 12by a + 2. 67. w^ + 243byw + 3.

m^ — 19m — 6 by m + 2. 68. a* — 256 by a + 4.

a«+27a2-9a-10by 5-3a + a2. i

21x'-\-4.-Sa^-\-6x-29a^hySx-2.

1 6 ar' - 1 1 aj3 _}. 2 a;^ + 9 - 1 2 a; by 2 a; - 3.

30 a;* - 36 a; + 60 a;2 _ 62 a;3 _^ g by 5 a; - 2.

a;7 _ 2a;5 - a^ - 10a; - 36 by a; - 2.

y*-\-7y-10y^-f-\-15byy'-2y-3.

Ta.-^ + 2«4- 27a;2 + 16 - 8a; by a;2 + 5a; - 4.

28a;* + 6a;3_|.6^_6^_2by2 + 2a; + 4a^^.

25 v2 - 20 y3 4- 3 y^ + 16 i; - 6 by 3 -y^ - 8 V + 2.

4- 18 a; + 30a;2 - 23a;3 ^ (5^4 by 2ar^ - 5 a; + 2.

32a;3^24a;4-25a;-4-16a;2by 6a,-2-a;-4.

t^-2t^ + ^f-^it^-^^t + ihyt-l.

DIVISION 81

SPECIAL CASES IN DIVISION

133. 1. By actual division,

(x" - y^)^(x - y)= x-\-y.

(i»3 _ ^3) ^(^x — y)=x'^-{-xy-{- y\

(0)4 _ y*)^lx - y)= x^ + xh/ + xy^ + f,

(x^ — y^) -j- {x — y)= X* -j- a^y -\- xhf + xy^ + 2/^.

Observe that the difference of the same powers of two num- bers is divisible by the difference of the numbers.

' Divisible ' means ' exactly divisible.'

2. By actual division, {x^-y^)^{x-\- y)= x-y.

(a^ — y^)-^{x -f y)= x^ ~ xy + y\ rem., —2f.

{^ — y^)-i-{x + y)= a? - x^y + xy^ - f.

(a;5 _ y^)^{x + 2/)= a;4 _ a^y _j_ a;2y2 _ ^^ _l_ ^^^ ^em., - 2y\

Observe that the difference of the same powers of two num- bers is divisible by the sum of the numbers only when the powers are even.

3. By actual division,

{x" + y'^)^{x - y)= X ^- y, rem., 2y\

(a^ + y^)-^{x — y)= x"^ + xy -{■ y"^, rem., 2f.

{x!^ -\- y^)-^{x — y)= x^ -\- a^y -{- xy^ + y^, rem., 2y*.

(x^ H- y^)-i-{x — y)= X* -\- a:^y -\- x^y^ + xy^ + 2/^, rem., 2y^.

Observe that the sum of the same powers of two numbers is not divisible by the difference of the numbers.

4. By actual division,

(«2 + 2/^)-i-(a; + 2/)= a; — y, rem., 22/1

(a^ + 2/-^)^(a; + 2/)= a;2 — 0^2/ + 2/^.

(a^ 4- 2/^) -i- (.T + 2/) = *^ — ^'^.V + ^^ — y^i rem., 2 y*.

(a;5 _j_ ^5).^(o; + y)= x^ - x^y ^ xy - aj^ + y*.

Observe that the sum of the same powers of two numbers is divisible by the sum of the numbers oiily when the powers are odd.

milne's stand, alg. — 6

82 ' DIVISION

134. Hence, the preceding conclusions may be summarized as follows, n being a positive integer :

Principles. — 1. x^ — i/" is always divisible by x — y.

2. »*• — 2/" *s divisible by x -\- y only ivhen n is even.

3. x^ + 2/" is never divisible by x — y.

4. x^ -j- 2/" is divisible by x -\- y only ivhen n is odd.

135. The following law of signs may be inferred readily :

When X — y is the divisor, the signs in the quotient are plus.

When X -\-y is the divisor, the signs in the quotient are alter- nately plus a,nd minus.

\

136. The following law of exponents also -may be inferred :

The quotient is homogeneous, the exponent of x decreasing and that ofy increasing by 1 in each successive term.

EXERCISES [Additional exercises are given on page 459.] 137. Find quotients by inspection :

7. ^'-^

8.

1.	a2- 62
a-b
2.	^_g2
r-hs
3.	a'-b'
a-b
4.	a' + b^
a + b
5.	m^ — n^
m — n
a	m' 4- n^

10.

11.

x-2

«2-9

x-\-3

a'-S

a-2'

9^4-8

r-f2

n^-16

n-4:

0^ + 27

13.	n' - 1
n-1*
14.	71^ -f-1 n-\-l
15.	^2-25
1; + 5
16.	s3 + 64
s + 4
17.	a^-8l
x + 9'
1Q	W - 125

12. „ . — ^ ^

m-\-n c + 3 6 — 5

DIVISION 83

Write out the quotients :

19. (r^ + s^)-h(r + s). 24. (a^ -y^)^{x-y).

20. (l + a^)-j-(l4-«).' 25. (a^f + w')^(xy-\-a).

21. {x^-l)-i-(x-\-l). 26. (x^'^-\-y^z^)^{x'^ + yz).

22. (m^ + n7)--(7^i + n). 27. (?/3 _ ^qoo) -j- (y _ 10). ^' 23. (x5-32)-j-(a;--2). 28. (a^ + 128) -- (a + 2).

29. Find five exact binomial divisors of a^ — x^.

Solution

a^ — x^ is divisible by a •— a; (Prin. 1).

a^ — 7^ is divisible by a + a; (Prin. 2).

Since a^ — x^ = (a-^)^— (.x^)^, a^ — x^ may be regarded as the difference of two cubes, and is, tlierefore, divisible by a^ — x^ (Prin. 1).

Since a^ — x^ = (a^y — (x^)^, a'^ — x"' may be regarded as the difference of two squares, and is, therefore, divisible by a^ — x"^ (Prin. 1).

Since a^ — x^ ={a^Y — {x^Y^ a^ — x^ may be regarded as the difference of two squares, and is, therefore, divisible by a^ + x^ (Prin. 2).

Therefore, the exact binomial divisors of a^ — x^ are a — x, a + x, a^ — x2, a^ — x^, and a^ -\- x^.

30. Find an exact binomial divisor of a^ + a^.

Solution

Since a^ -\- x^ ={aP'y +(x2)3, a^ 4- x^ may be regarded as the sum of the cubes of a^ and x^, and is, therefore, divisible by ofi + X'^ (Prin. 4).

Find exact binomial divisors :

31. a? — m\ 37. x'' -f (i\ 43. a^ — b\ four.

32. a^ - m\ 38. a^" + 5^°. 44. a« - 1, five.

33. b^ 4- x". 39. «!« + bK 45. a^ - 6^ six.

34. x^ - a\ 40. a^2 _j_ 512 46. a^° - 6^", five.

35. c^ + n^ 41. a? -21. 47. a^^ - 6^^ eight.

36. a« -f 6^ 42. a^ - 27. 48. o}"^ - b^% nine.

84 DIVISION

Algebraic Representation

138. 1. Express 5 dollars in terms, of cents ; 500 cents in terms of dollars ; m dollars in terms of cents ; m cents in terms of dollars. ,

2. Find the value of x that will make 6 x equal to 48. i

3. By what number must 25 be multiplied to produce 300 ? 10 to produce a; ? r to produce s ?

4. Represent (the third power of a minus the fifth power of X) divided by (m plus w^).

5. Express the multiplicand when Imn is the product and Im the multiplier.

6. Find an expression for 5 per cent of ic ; 2/ per cent of z.

7. It takes a men c days to do a piece of work. How long will it take one man to do it ? 2 men ? x men ?

8. At a factory where N persong were employed, the weekly pay roll was F dollars. Find the average earnings o>f each person per week.

9. A train ran JW miles in H hours and m miles in the suc- ceeding Ti hours. Find its average rate per hour during each period and during the whole time.

10. A farmer has hay enough to last m cows for n days. How long will it last (a — 6) cows ?

11. Indicate the quotient of m -h n divided by the number whose first digit is a;, second digit 2/, and third digit z.

12. A dealer bought n 50-gallon barrels of paint at c cents per gallon. He sold the paint and gained g dollars. Find the selling price per gallon.

13. If it takes h men c days to dig part of a well, and d men e days to finish it, how long will it take one man to dig the well alone ?

DIVISION 85

Equations and Problems

139. 1. Find the value of x in the equation hx — W — cx — c?.

Solution hx-y^-cx- c2. Transposing, hx — cx = h^ — c^.

Collecting coefficients of a;, (Jb — c)x = b^ — c^.

7)2 _ /.2

Dividing by 6 - c, • x = = 6 + c.

0 —c

2. Find the value of x in the equation x — a^ = 2 —ax. Solution

X — a^ = 2 — ax. Transposing, ax + x = a^ + 2.

Collecting coefficients of x, (a + l)x = a^ + 2.

Dividing by a + 1, x = ^^i±-? = a^ - a + 1 +-^-

a + 1 a + 1

Find the value of a; in :

cx — (^ — d^-{-dx = 0, 6. 7 a — 10 = a^ — aajH- 5 a;.

a2_^a; — 2a& + 6a;+62 = 0. 7. x — 1 — c = cx — <^ — c*. 2n^ + 5n-\-x = n^ — nx — 2. 8. 2 m^ — ?na;-|-fia; — 2/1^=0. 9. Sab-a^-2bx = 2h^-ax.

10. a2a._a3_|_2a2 4.5^_5^^10 = 0.

11. aa;-2fca;-f3cic = a6-2624-3&c.

12. cx-c*-2c^-2c^ = 2c-x + l.

13. 9a24-4mx= — (3aa; — 16m2).

14. X + Cy n"^ — A n^ = 1 — S nx + 2 n — v?.

15. n^a;— 3mV + na; + 3m2 + a; = 0.

16. a;-362_19262c3-4ca; + 16c2a; = 0.

86 DIVISION

Solve the following problems and verify the solutions : [Additional problems are given on page 460.]

17. George and Henry together had 46 cents. If George had 4 cents more than half as many as Henry, how many cents had each ?

Solution

Let X = the number of cents George had.

Then, x— i = the number of cents George had less 4,

and 2(x — 4) = the number of cents Henry had ;

.•.x + 2(a;-4) =46.

Solving, X = 18, the number of cents George had,

and 2(x— 4) .-= 28, the number of cents Henry had.

Verification

The answers obtained should be tested by the conditions of the prob- lem. If they satisfy the conditions of the problem, the solution is pre- sumably correct.

1st condition : They had together 46 cents.

18 + 28 = 46. 2d condition : George had 4 cents more than half as many as Henry, j 18 = ^ of 28 + 4.

18. In a certain election at which 8000 votes were polled, B received 500 votes more than ^ as many as A. How many votes did each receive ?

19. A had $40 more than B; B had $10 more than J as much as A. How much money had each ?

20. Twelve years ago a boy was J as old as he is now. , What is his present age ? 1

21. In 2 years A will be twice as old as he was 2 years ago. How old is he ? |

22. A father is 4 times as old as his son. Six years ago he was 7 times as old as his son. Find the age of each.

DIVISION 87

23. A man has $ 1.80. He has twice as many quarters as dimes. How many coins has he of each denomination?

24. I bought 15 books for $ 6.60, spending 30^ each for one kind and 60y each for the other. How many books of each kind did I buy ?

25. John has $6.75. He has 3 times as many dimes as nickels, and as many quarters as the sum of the nickels and dimes. How many coins has he of each denomination ?

26. A man has $ 27.50 in quarters and half dollars, having 5 times as many half dollars as quarters. How many coins of each kind has he ?

27. Mary bought 17 apples for 61 cents. For a certain number of them she paid 5 cents each, and for the rest she paid 3 cents each. How many of each kind did she buy ?

28. A is 25 years older than B. In 20 years A will be twice as old as B. Find the age of each.

29. John is 15 years older than Frank. In 5 years Frank's age will be i John's age. What is the age of each ?

30. George is ^ as old as his father ; a years ago he was J- as old as his father. What is the age of each ?

31. Harold is n times as old as his brother ; r years ago he was m times as old. Find the age of each.

32. A had 3 times as many marbles as B. A gave B 50 marbles ; then B had twice as many as A. How many marbles had each ?

33. Separate 24 into two parts, such that one part shall be 3 less than twice the other.

34. Separate 52 into two parts, such that 2 times one part shall be 4 greater than 3 times the other.

35. The shed that sheltered an airship was 544 feet in perimeter. If twice its length was 52 feet more than 4 times its width, what was its width ? its length ?

88 REVIEW

REVIEW

140. 1. Define a term ; similar terms ; the degree of a term ; the degree of an expression.

Illustrate symmetrical expression ; homogeneous expression.

Simplify :

4. How may a parenthesis preceded by a minus sign be removed from an algebraic expression without changing the value of the expression ?

Simplify :

5. a^-{a^ — hx'^y-\-l()o^f-10a?f^nxy*-f).

6. |a-|aj-(|a--ia;)-(3&--Ua;-fa) + i-a.

7. a?-{2xy-y^-(a?^xy-y^-o?-2xy-f + ^f.

8. m + 2J2m-[n + 3p-(4p-3n)-57i + 2m]-7i)|. J

9. What are the various ways of indicating multiplication in algebra ?

i

10. {m — x){m ■\- x), 14. (a'* + 6'*)(a" — 6'*). ^

Expand

10. (m-

11. (a^4-4)(ic2-3). 15. (a-irh-\-c){a-irh-c).

12. (x' + x^ix + l).^ 16. {x + y-^z)(x-y + z),

13. (a;-l)(l + fl5). 17. {m-{-n-p){m-n-\- p).

18. Why should the terms of the dividend and divisor usually be arranged, before division, according to the ascend- ing or the descending powers of some letter ?

19. What is the advantage of using detached coefficients ?

REVIEW 89

Arrange terms and divide, using detached coefficients:

20. a:^-x + 2x'-S-2x^ + 12x'hj x + 1.

21. x^-4:X + 5a^-4:a:^ + lhyl-Sx-\-a^,

22. a^-12a2_a + 12 by a3-3 + 4a-2a2.

Simplify :

23. l-51_[a^_3_(2aj_4)2 + 3a^ + l]-(x-4)2|-l.

24. x-l5x-[6x-(7x-8x-9x)-10x2 + llxl-\-9x.

25. l-l-l-(l-x)-l-]-ll-\x-(5-3x)-7-\-x\.

Collect, in order, the coefficients of a;, y, and z :

26. ax-^ay-\-az — hx — by — bz.

27. ax — 2 y-^cz + by — 12 X + 4: z,

28. 3 mx —nx-\-by — y-\-Scz — 4:Z.

29. py — y — 4:Z-{-bz — x + mx — nx — z.

30. ca; — &?/ — 3 a2J + ic — ?/ — 4 2; + 2; — 2/.

31. State the law of signs for multiplication ; for division.

32. What is the sign of the product of an even number of negative factors ? of an odd number of negative factors ?

Expand:

33. (a-bXa + bXa' + by

34. (l_a;)(l+a^)(l4-a;2)(l+A

35. (l-x)(1-\-x)(l-x)(l-{-x),

36. (a'-{-3a'y-{-3ay')(a'-2ay-hy').

37. (a^" + 2 afy"" + 2/^") («"" - 2 a;"^/" + /").

38. (la^ + Lxy^^fy^^-.^xy^^fy

39. (.2a2_.8a-|-1.6)(.la2 + .4a + .8).

90 REVIEW

40. Give a rule for multiplying a monomial by a monomial ; for dividing a polynomial by a polynomial.

41. State and illustrate two ways of testing the correctness of a result in algebraic multiplication ; in algebraic division.

Expand, using detached coefficients ; test results :

42. (a* + a^ + «^ + « + l)(a-l).

43. (a^ — x* + x^ — x^ + x — l%x-{-l),

44. (a« + 2a^4-4a3 + 8a2 + 16a4-32)(a-2). Divide, and test results :

45. 4:-10b^-5b + b^hySb-2b^ + lr'-l.

46. m^«-6m3-f 5m-2by 2m3-24-m*-3m.

47. 12.7a3-20a4-a'-100a2+16-160a^bya«-6a2-f5a-4.

48. b''-^29b*-22-61b'-{-210b-170b^ by 6^-56+2 b'-ll.

Simplify:

49. a-(2b + 5a)(6b-3d)-2b-6[Sa^-4.ab-2b^}.

50. x-\3y + l4.x-2(y-hSx)-Syf-(5y-^2xy-Sy]. 61. (a2 4- a& - by - {a" -ab - by ~ 4 ab{a^ - by

52. o^-l-WJ^b{pb-3a)-\-3ab+a^-b{a-2a+2b)}'].. -Square :

53. 2x — 3y. 56. 10 — 3a;. 59. a + 6 — c + d

54. x^ — aa?. 57. n' — m^. 60. 2 a — 3 6 — 4 c.

55. 5a2-l. 58. Ix-Wy. 61. aj^-^-y-a^.

62. What laws are illustrated by a(bc) = b (ac) ?

63. Show that a'^ a*^ = a" ; that a" 4- a^ = a*.

64. In what respect do (a — b) and (6 — a) differ ? Expand and compare (a — 6)^ and (6 —ay.

REVIEW 91

Collect, in order, the coefficients of a;, y, and z :

65. 16 ny — I'G mx -{-ax-{-by-{-cx~2y,

66. mx -\- ny -{- az -\- 2 ax — 2 my -f 2 nz.

67. a; — 2/ — az + 8 ma? + aft^^ — ar^ -f 2/^ -f- 2J.

68. a^x -\- b^y — 2 ax — 2 cz -{- c^z -\- X -{- y -\- z.

69. m^ic — 71^2/ + ^^^ — ^^^ — 2 mna; — 2 mwy — nh -\- z.

70. 4:(ax — by-{- cz) — 2(bx — ay— dz) — 2(x — y + z).

71. Show why a broader definition is necessary for multi plication in algebra than in arithmetic.

Expand :

72. (5a-42/)(5a-32/). 75. {2a''x-^Wy){4.a'x-U^y).

73. (6fl; — 4y)(3a; + 5y). 76. (&amn + bp){Qamn—3p).

74. (3a;-|-a2/)(3a; + 62/). 77. (3a"+i-2 6«-i)(2a''+i-36"-i).

78. (a; + 2/)(a;-2/)(ar^-h2/')(a;^ + 2/'*)(^ + 2/')-

79. (m8 + l)(m* + l)(m2 + l)(m + l)(m - 1).

80. (16a;^4-l)(4a^4-l)(2a; + l)(2a;-l).

81. For what values of n is a;" + 2/" divisible by a; + 2/? by x — y? When is a;** — y"^ divisible hj x + y? hj x — y?

82. State the law of signs for the quotient when a;** + 2/** or a-" — 2/" is divided by a; + 2/ or a; — 2^ ; the law of exponents.

Divide :

83. xr^+^ — a^+^^« — 2 2/^* + 3 y^'z"'^ — is^*-^ by af+^+y'* — 0"-^

84. 6a' + ffaV-ifa/ + |i2/'bya« + ia22/-iay^ + i2/'.

85. a^c - ab' + acd - ad^ - abc + b^ - bed + bd"" - oc^ + cft^ _ c^d + cd^hy ac-b^-\-cd-dK

FACTORING

141. An expression is rational, if in its simplest form it con- tains no root sign of any kind; and integral, if in its simplest form it contains no literal number in any denominator or divisor.

a^ — 6a;2-)-iiaj_6 and ^ cc — 3 are rational integral expressions ; - + b

a

is rational but not integral ; a + y/x is integral but not rational.

Until noted farther on, the term factor (§ 24) will be under- stood to mean rational integral factor.

142. A number that has no factors except itself and 1 is called a prime number.

143. The process of separating a number into its factors is called factoring. Usually the prime factors are sought.

144. To factor a monomial.

EXERCISES i

If xy is one factor, find the other factor in :

1. 6x^y. 3. 15a^2/*- 5. 2 a^y\ 7. a'^a^bY- 9- —mxy.

2. —xy. 4. 12 xy, 6. —x'^y. 8. —c^xf. 10. —mt^y. If abc is one factor, find the other factor in :

11. a^bc. 13. ab'^c. 15. abc^. 17. —aWc\ 19. —a'^bc.

12. 2 ab&. 14. a^bc. 16. 5 abc. 18. — \ abc. 20. — 8 a6V. Separate into two equal positive factors ; negative factors :

21. x^. 23. 4:a^x\ 25. 16 al 27. 25 y\ 29. 9b\

22. 4 ni 24. 36 c\ 26. 9 xy. 28. 49 a^. 30. 4 a^b\

145. A factor of two or more numbers is called a common factor of them.

02

FACTORING 93

146. To factor a polynomial whose terms have a common factor.

EXERCISES

[Additional exercises are given on page 462.]

I. What are the factors of Sa'xy — 6 ax^y + 9 axy^?

PROCESS Explanation. — By examining the terms

o2 f\ 2|Q 2^^ ^^® polynomial, it is seen that 3 axy is a

!f y "T y factor of every term. Dividing by this

= 3 axy {a — 2x-\-^y) common factor gives the other factor.

Hence, the factors are 3 axy^ the monomial factor, and (a — 2 a: + 3 y), the polynomial factor.

Test. — The product of the factors should equal the given expression ;

thus, Zaxy{a-2x + Zy) =Z a'^xy - 6 ax'hi + 9 axy"^.

Factor, and test each result :

2. 2c-2d, 12. 5a;*-10a?3-5ar».

3. 3a2-2al 13. ^ a^ - 2 a?h -^ a^b\

4. Sa^-Sar*. 14. x^"^ -\- x^^ -\- x^"^ - x^.

5. 8a;2H-2ar*. 15. 4(a + 6) - ar(a + 6).

6. 3 a^ — 6 ah. 16. ac — bc — cy — abc.

7. 4 a^ — 6 x^y. 17. a{x — 2/) + ^(x — y).

8. 5m^ — Smn. 18. c^y -]- zy — d{y -\- zf.

9. Sa^b-Sa^b\ 19. Sx^f -3 xY + 12xy.

10. 4:^y^-6xY. 20. 6a''by -ISabY + ^^^.^bV'

II. 5m^-10mV. 21. 12 xY^ -WxYz'' -20x^fz^.

147. To factor a polynomial whose terms may be grouped to show a common polynomial factor.

EXERCISES [Additional exercises are given on page 462.]

.. Factor ax + ay -\- hx + by.

>LUTiON. ax -^ ay + hx + hy = a{x + y) + hix + y)

= {a ■^b)(x-\-y).

94 FACTORING

2. Factor ax — ay — bx-\-by.

Solution ax — ay-bx-{-by = a{x - y)— b(x - y) = ia-b)(x-y).

Observe that, when the first two terms are factored, (a; — y) is found to be the binomial factor. Since (x — y) is to be a factor of the other two terms, the monomial factor is —b, not +&, for (^—bx+by)-^(x—y) = — b.

3. Factor cx + y — dy-^cy — dx-\-x.

Solution cx-^ y - dy -{-cy — dx -{-x Arranging terms, = ex — dx -{- x + cy — dy ■}- y

= (c - d -\- l)x +(c - d + l)y = (c-d+ l)(x + y).

Factor, and test each result, especially for signs :

4. am — an -\- mx — nx, 18. x^ -\- oi^ -\- x^y -\- y.

5. be — hd -\- ex — dx. 19. 2 — 2n — 7i^ -\-n\

6. pq — px — rq -\- rx. 20. o^ — x — a-{-ax.

7. ay-by-ab + b\ 21. 3x^-lox-{-10y-2 x^y.

8. x^-xy-5x + 5y. 22. 12a^ -Sab -3 a* + 2a^b.

9. b^ — be-^ab — ac. 23. 3 m^n — 9 m?i^ -i- am — San.

10. ar^ + «2/ — Ota; — ^2^. 24. ISafe^ — 9 6^0 — 85a6H-21 6c.

11. c^ — Ae + ae — ia. 25. 16 aa; + 12 ay — 8 6a; — 6 6?/.

12. 2 a; — 2/ + 4 ar^ — 2 aji/. 26. oa;^ — aa; — aa;?/ -{-ay -{-x — 1.

13. 1 — m + n — mn. 27. aji/H-a; — 3?/^ — 3?/ — 4 y— 4.

14. 2p-i-q-{-6p^-{-S pq, 28. aa; — a — 6a; + 6 — ca; -h c.

15. ar — rs — ab -\- bs. 29. mx — nx — x — my-\-ny-\-y.

16. a;^ + a;2 + a; + l. 30. 6a^ - 6 — a;2/ — 2/ + 2/^;^ — ^a^-

17. ^4-^2 _8y — 3. 31. m^ + wi»* + »ww + ?i^ + w + n.

FACTORING 95

148. To factor a trinomial that is a perfect square.

Since by multiplication, §§ 105, 108, (a + h){a -f- 6) = a' + 2a6 + Wdjndiia - h)(a -h)^a^- 2ab + 6=, a^ + 2ab + A^ ^ (a -h 6)(a -\-b)8inda^-2ab + b^=(a- b)(a-b).

These two trinomials are perfect squares, for each may be separated into two equal factors. They are types, showing the form of all trinomial squares.

149. A trinomial is a perfect square, therefore, if these two conditions are fulfilled :

1. Two terms, as -|-a^ and + b"^, must be perfect squares.

2. The other term must be numerically equal to twice the product of the square roots of the terms that are squares.

25 x2 — 20 xy + 4 y2 is a perfect square, for 25 x^ = (5 x)2, 4 y^ = (2 2/)2, and -20xy =- 2(5 x) (2 y).

150. Every number has two square roots, one positive and the other negative. In factoring, usually only the positive square root is taken.

Thus, V25 = 5or-5, for5- 5 = 25 and (_5)(-5) =26. d^ -\- 2 ab + b'^ = (a -\- b)(a + 6) or (— a — 6)( — a — 6), but we usually factor trinomial squares in the first way only.

EuLE. — Connect the square roots of the terms that are squares with the sign of the other term, and indicate that the result is to be taken twice as a factor.

From any expression that is to be factored, the monomial factors should usually first be removed.

Thus, 2a3-4a2 + 2a = 2 a(a2 -2a + l)=2a(a- 1)2.

EXERCISES

[Additional exercises are given on page 462.]

151. Make a trinomial square by writing the missing term :

1. a;2+*4-2/2. 4. c^-2cd + *. 7. *-\-4.ab + b\

2. a^ — *^b\ ■ 5. x'^-^4:Xy-\-*. 8. * — 2pq + q\

3. 2/2 + ^ + 22. ^. r^^Srs-^*. 9. * + 6xy-{-y\

96 FACTORING

Factor, and test each result :

10. i»2 4- 2 iC2/ + y^- 23. 1 - 6 a^ + 9 a\

11. jp2-2^g-hg^ 24. 5rc2 + 30ajH-45.

12. c2 + 2cd + d2^ 25. 16p2_24i) + 9.

13. ar'-2a; + l. 26. 9 a;^ - 42 a; + 49.

14. a^ + 4ic + 4. 27. 9 + 4263 + 4956^

15. a;2 + 6a;-f-9. 28. 36^8-12 71^ + 1.

16. 4 — 4« + a2. 29. x^' + %x'z + l^z^.

17. 4 a- 4 «2 4.^3^ 30. ^^x^y"^ -20xy -^25.

18. m2-8m + 16. 31. 4. x^ + 12 xyz + ^ y'^\

19. a2-16« + 64. . 32. 2 a; + 20 a^a; + 50 a^a;.

20. 1 + 8 6 + 1661 33. 8 cfc^^ + 40 a62 + 50 61

21. 3 aj2 + 6 ar^ + 3 /. 34. 4 ar^* + 8 x'^f + 4 y"^.

22. 2m'^ — 8mH + 8 7i2. 35. a;^'* — 2 a^X^" + 2/^ "^2^

When either or both of the squares are squares of poly nomials, the expression may be factored in a similar manner.

36. Factor a;^ + 6 x{x — y) + 9(a; — y^.

Solution

x'^ + Qx{x-y)^-Q{x-yY = [x + 3(x - y)-\lx + 3(a: - y)] = (a; + 3a; — 3?/)(x + 3a; — 3 2/) = (4x-3?/)(4x-32/).

37. Factor (a - 6)2+ 2 (a - 6)(6 - c) + (6 - cf.

Solution

(a - 6)2 + 2{a - b)(b - c) + (6 - c)2 = [(a_6) + (&_c)]C(a-6) + (6-c)] = (« - 6 + 6 - c) (a - 6 + 6 - c) = (a — c)(a — c).

Test. — When a = 3, 6 = 2, and c = 1,

(a - 6)2 + 2(a - 6)(6 - c) + (6 - c)2 = 12 4. 2 . 1 . 1 + p = 4, and (a — c) (a — c) = 2 . 2 = 4.

FACTORING 97

38. x^-{-2x{x — y) + {x — yf. 41. (r + s)2 — 4(r4- s) + 4.

39. ^2-4^(^-1)4-4(^-1)2. 42. 16-24(^-0 + 9(^-02.

40. c2-6c(a-c) + 9(a-c)2. 43. 14 (a; - ?/) + (a; - 2/)^ + 49.

44. (a + &)'-2(a + 6)(5 + c)-|-(6 + c)2.

45. (a-a;)2 + 4(a-a;)(aj-6)4-4(x-6)2.

46. 16(a-a;)2 + 32(a-a;)(a; + 6)+16(a; + 6)2.

47. (a + 3 6)2 - 4 (a + 3 6)(3 6 - 2 c) + 4 (3 6 - 2 c)2.

48. (a + 6 -f c)2 + 2 (a + & + c){a -^b - c)-\-{a-\-h- c)\

152. To factor the difference of two squares.

By multiplication, (a + 6)(a — h)=a^ — ft^. Therefore, a^ - 6^ = (a + 6) (a - 6).

Rule. — i^iVic? the square roots of the two terms, and make their sum one factor and their difference the other.

Sometimes the factors of a number may themselves be factored.

[Additional exercises are given on page 462.] 153. 1. Factor ¥-y\ Solution. l)^ — y'^ ^-ih -\- y){h — y).

2. Factor x^ — 1.

Solution. x^— 1 = (x + l)(ic— 1).

3. Factor a^ — 1.

Solution. x* - 1 =(a;2 + 1) {x^ - 1) .

= (x2 + l)(a;4-l)(a:-l).

Resolve into simplest factors :

4. x^-L 9. a2-16. 14. 25a^2-l. 19. 144m4-l.

5. a2-9. 10. x'^-m\ 15. ?>V-16. 20. 4.x'^-25y\

6. x'^-y\ 11. 25 -c2. 16. a2a;2-4c2. 21. 9 a^- 49 61

7. 22 _ 4 12. a^-49. 17. 25 - 9 ajl 22. m*-16n\

8. c2-d2. 13. a^-Sl. 18. 9 62_c2d2. 23. 16 a^ - 81 6*.

milne's stand, alg. — 7

FACTORING
33. 18C2-50.	42.	a^-i
34. Sx'-Sy^	43.	0^2 -.01.
35. 36 a^- 225.	44.	^-tV
36. ajy-256.	45.	x^^-y^.
37. 3m^— 3m.	46.	4a2-.25.
38. 4.a'-4.b\	47.	a:4a _ y\
39. 5a;8 — 5.	48.	8a2»_i8^2s^
40. a}'-b^	49.	^2n-2 _ 2^4m
41. 2a^-2y\	50.	a;2n+l _ a,y^.

24. ic' — xy\

25. 8s2-2i2^

26. 169-a262.

27. 2ar4-22/^

28. 12162_a2c2,

29. 100a2-81/.

30. 640^^-6252/2.

31. 400^2 -36 2/2.

32. 144m2-16M2.

When either or both of the squares are squares of poly- nomials, the expression may be factored in a similar manner.

51. Factor 25 ci^ _ (3 a -f 2 by.

Solution

One factor is 5 a + (3 a + 2 6) and the other is 5 a — (3 a + 2 6).

5 a + {S a + 2 b)= 5 a + S a -{- 2 b = S a -\-2 b = 2(4 a + ft). da-{Sa + 2b)=^a-3a-2b = 2a-2b = 2(a-b). .'. 26 a2 - (3 a + 2 6)2 = 2(4 a + b)2{a - b) = 4(4a+6)(a-6). Factor :

52. a'-ib + cf. 56. 9b^-(a-xy.

53. a^-{a + bf. 57. 9a^-{2a-5y.

54. &2_(2a + 6)2. 58. x'-{Sx^-2yy.

55. 4c2-(6-hc)2. 59. 49 a2-(5a- 4 6)2. 60. Factor (3 a - 2 6)2 - (2 a - 5 6)2.

Solution

(3a- 2 6)2 -(2a -5 6)2 ^

= [(3 a - 2 6) + (2 a - 5 6)][(3 a - 2 6) - (2 a - 5 6)] = (3a-26 + 2a-56)(3a-2 6 -2a + 56) = (5a-7 6)(a-|-3 6).

FACTORING

99

Factor :

61. (2a-{-Sby-{a + by.

62. (5a-3by-{a-by.

63. {2x + 5y-{5-3xy.

64. (a -2 6)2 -(a -5)2.

65. (2x-3yy-(Sy-^zy.

66. (56-4c)2-(3a-2c)2.

67. (4:X-Syy-(2x-3ay.

68. (9x + 6yy-(4:X-3yy,

69. (a^ + a;2)2-(2a: + 2)2.

70. (a + 6-|-c)2-(a — 6-c)2.

71. Factor a2 + 4 — c2 — 4 a.

Solution. a^ -]- 4 — c^ — 4:a

Arranging terms, =(a2 — 4a + 4) — c^

= (a - 2)2 - c2 = (rt -2 + c)(a -2 -c).

72. Factor a2 + 62_c2_4-2a5 + 4c. Solution. a^ + b'^ - c'^ - 4 - 2 ab + ic Arranging terms, = a^ — 2 ab + b'^ — c^ + 4 c — 4

= (a2 - 2 a& + ft2) _ (c2 _ 4 c + 4)

= (a-6)2- (c-2)2

= (a - 6 + c - 2)(a - 6 - c -f 2).

Factor, and test each result :

73. a2 — 2 aa; + a;2 — w2.

74. b'^-^2by + y^-n\

75. l_4g + 452_^2^

76. r'^-2rx-{-x^-16t\

77. c2-a2

2ab.

78. b'^-x^-y^-{-2xy.

79. 4 c2 - a;2 — 2/2 — 2 a;?/.

80. 9c2-ic2-2/2 + 2a;?/.

81. 6c2-9a26-63-6a62.

82. a62 _ 4 a^ - 12 a2c- 9 acl

83. a^ -2 ab -\-b^ - c^ -\-2 cd- d\

84. ar'-2aj?/-|-?/2-m2 + 10m-25.

85. 0? - a" ^ y"" -b'' + 2 xy -2 ab.

86. 4a^-l-9-12a; + 10w7i-m2-257i2.

100 FACTORING

154. To factor a trinomial of the form x^ +px 4-9 . By multiplication,

{x + a){x + ^) = a;2 + (a + h)x + ah.

This trinomial consists of x^, an avterm, and an absolute term ; and therefore has the type form jr^ +/7jr + q.

Therefore, by reversing the process of multiplication, a tri- nomial of this form may be factored hj finding tivo factors ofq (the absolute term) such that their sum is p (the coefficient of x) and adding each factor of q to x.

Thus, ic2 + 8 x + 15 = (x + 3) (ic + 5),

a;2 - 8x + 15 = (« - 3) (x - 5), a;2 4- 2a; - 15 = (a;- 3) (x + 5), ic2 _ 2x - 15 = (x + 3) (X - 6).

EXERCISES

[Additional exercises are given on page 463.]

155. 1. Eesolve aj^ — 13 cc — 48 into two binomial faxitors.

Solution. — The first term of each factor is evidently x.

Since the product of the second terms of the two binomial factors is — 48, the second terms must have opposite signs ; and since their alge- braic sum, — 13, is negative, the negative term must be numerically- larger than the positive term.

The two factors of —48 whose sum is negative may be 1 and —48, 2 and - 24, 3 and - 16, 4 and - 12, or 6 and - 8. Since the algebraic sum of 3 and - 16 is - 13, 3 and - 16 are the factors of - 48 sought.

.•.x2_i3a;-48 = (x + 3)(x-16). Factor :

2. iB2 + 7a; + 12. 7. b^-\-b-12,

3. y^-Ty-^12. 8. c^-c-72. i

4. p^ — Sp + 12. 9. c2 — 5 c — 14. '

5. r2-f-8r4-12. 10. a.'2-a;-110.

6. a2 + 4a-21. . 11. a"" -\- 2 a - 120.

FACTORING 101

12. Factor 72 — m^ — m.

Solution. — Arranging the trinomial according to the descending powers of m, we have

72 - m2 - m = - m2 - w + 72 Making m^ positive, we have = — (m^ + w — 72)

= -(w-8)(m+9) = (-?ji + 8)(m + 9) = (8-m)(9 + w).

Separate into simplest factors :

13. 30 + ^-7-2. 20. a^-\-12a3? + 20a\

14. 15-a2_f_2a. 21. 4 aa; - 2 aa;^ + 48 a.

15. aj2 _|_ 5 o^a; 4. (5 0^2^ 22. ic^" - 11 ?> V + 24 6^

16. -a2-9a-h52. 23. 150 ?i - 5?ia^ - 65 wa;.

17. y^-4:by-12b\ 24. 11 a2a; _ 55 aa; + 66 a;.

18. z'^-anz-2ahi^. 25. Sa^^aj^ _ 3^25^ _ g ^25^

19. -a;2 + 25a;-100. 26. 20 6x + 10 6^ _ 630 a^.

27. Factor x'^-(c-{- d)x + cd. Suggestion. — Write the trinomial in the standard form, a;2 + (_ c - cZ)a; + (- c){-d). 28. Factor a;2— (a— cZ)a;— ad 29. Factor a^^— 2 (a— n) a;— 4 aw.

156. To factor a trinomial of the form ax'^ -\- bx ■\-c.

EXERCISES [Additional exercises are given on page 463.] 1. Factor3a~^ + lla;-4.

Solution. — If this trinomial is the product of two binomial factors, they may be found by reversing the process of multiplication illustrated in exercise 32, page 68,

Since 3 x'^ is the product of the first terms of the binomial factors, the first terms, each containing ic, are 3 x and x.

Since — 4 is the product of the last terms, § 84, they must have unlike signs, and the only possible last terms are 4 and — 1,-4 and 1, or 2 and - 2.

102 FACTORING

Hence, associating these pairs of factors of — 4 with 3x and x in all possible ways, gives as the possible binomial factors of 3x2 -|- 11 x — 4 :

3a; + 4| 3x-ll 3x-41 3x4-11 3x + 21 3x-2 x-lj' X + 4J' x + lj' x-4j' X-2J' x + 2

Of these we select by trial the pair that will give + 11 x (the middle term of the given trinomial) for the algebraic sum of the " cross-products," that is, the second pair.

.-.3x2+ llx-4=(3x-l)(x + 4).

Remark. — Since changing the signs of two factors of a number does not change the value of the number, 3 x^ + 11 x — 4 has also the factors ( — 3 X + 1) and (— x — 4) ; thus,

3x2 + llx-4=(-3x+l)(-x-4).

Such negative factors, however, are not usually required.

By a reversal of the law of signs for multiplication and from the above solution it may be observed that :

1. When the sign of the last term of the trinomial is 4-, the last terms of the factors must be both + or both —, and like the sign of the middle term of the trinomial.

2. When the sign of the last term of the trinomial is — , the sign of the last term of one factor must be +, and of the other — .

Factor :

2. 5aj2 + 9aj-2. 12. 2ay^-\-x-W.

3. 3x^-7x-6. 13. 6ar^-10a; + 4.

4. 6x'-13x-\-6. 14. 21a2-a-10.

5. 2x''-5x-i2, 15. 10a;4-llr^-6.

6. 5x^ + lSx-\-6. 16. 15a;2-f-22a;-h8.

7. Sx^-lTx + lO. 17. 15x' + 17x-4:.

8. 6x^-llx-35. 18. 18ar^-54aj + 36.

9. 276^-36^-14. 19. 12ar^ + 14 a;- 40.

10. 10x^-\-i2a^-\-U. 20. 2x^-{-5xy + 2y^.

11. 16 a.-2- 68 07 4-66. 21. 2x^ -\-3xy -2y\

FACTORING 103

When the coefficient of x^ is a square, and when the square root of the coefficient of x^ is exactly contained in the coefficient of X, the trinomial may be factored as follows :

22. Factor 9 aj2 ^ 30 aj + 16.

Solution. 9 x2 + 30 x + 16 = (3 a;)2 + 10 (3 x) + 16

= (3x + 2)(3x + 8).

23. ractor4aj2_5aj_6.

Solution

4 4

^(4x)2-5(4a:) -24^(4x-8)(4.r + 3)

4 4

^4(x-2)(4a: + 3)^^^_^^^^^^3^^ 4

Explanation. — Although the first term is a square, its square root is not contained exactly in the second term. Bat if such a trinomial is mul- tiplied by the coefficient of x^, the resulting trinomial will be one whose second term exactly contains the square root of its first terra.

Multiplying the given trinomial by 4, factoring as in exercise 22, and dividing the result by 4, we find that the factors of the given trinomial are (x -2) and (4 x -f 3),

24. Factor 2^x'^ + l'ix — 5.

Suggestion. — When the first term is not a square^ it may always be made a square whose square root will be contained exactly in the second term by multiplying the trinomial by the coefficient of x% or by a smaller multiplier. In this case multiply by 6, and divide by the same number to avoid changing the value of the expression.

Separate into simplest factors, testing results :

25. 9a32_42a?-f 40. 31. 9 a-^ + 43 a; - 10.

26. 25a;2 + 15aj + 2. 32. 18 a;^ - 9 a; - 35.

27. 36 a;2 - 48 a; - 20. 33. ^ :d' -10 x^ -IQ.

28. 25aj2_^25aj-24. 34. 16 aj2 4- 50 a; - 21.

29. 49a;2-42.^•-55. 35. 32 ?i2 + 28 n- 15.

30. ^x^ — 10xy-\-^y\ 36. b x^" -^ ^ x^'y — 2 y\

104 FACTORING

157. To factor the sum or the difference of two cubes. By applying the principles of §§ 134-136,

^-±^ = a2 - a6 + ft' and ^l:z^ = a^^ab-\- h\ a -\-b a — b

Then, § 121, a' + b'={a-h b){a' - ab -^ b'), and a3 - 6^ = (a - b){a^ + ab+ b"").

By use of these forms any expression that can be written as the sum or the difference of two cubes may be factored.

EXERCISES

[Additional exercises are given on page 463.]

158. 1. Factor x^ -\-f.-

Solution, y^ + y'^ = {x^y + (y2)3 _ (-^2 ^ y2^ (a;4 _ yfiyi _|. ^4).

2. Factor a^- 125 63.

Solution a9 - 125 63 =(a3)3_(5 6)3 =(a3 - 5 6)(a6 + ba% + 25 62). Factor, and test each result :

Z. o? + f. 9. a; - x^. 15. r'^ — 729 sK

4. o?-y^. 10. v' + 27 V. 16. 512 a;^ + 64 f\ j

5. m^-l. 11. a^b^ — (?d?. 17. l+(a + 6)^

6. l+m'. 12. 7^+6483. 18. {x-yf-S.

7. 3?-y\ 13. 0^2/^0^-216. 19. ^{m+nf-{-V25n\

8. r» + s». 14. 343 n^ + 1000. 20. {x - yf -(x + y)\

159. To factor the sum or the difference of the same odd powers of two numbers.

By applying §§ 134-136, as in § 157, any expression that can be written as the sum or the difference of the same odd powers of two numbers may be resolved into two factors. Thus,

fl^ + 6^ =((1 + b){a' - a^b + a'b'' - ab' + b'), and a'- b' = ia- b){a' + a^b + a^b^ + ab' + b').

FACTORING 105

EXERCISES

160. 1. Factor m^H- 32 a:^. Solution. — m^ + 32 x^ = m^ + (2 x)^

■ =(m+2x)(m4-2m3a;+4w2a;2-8mx3+16xO'

2. Factor 128 a"- 1.

Solution

128^14 _i =(2 a2)7_l

= (2 a2 - l)(64ai2 + 32 a^o + 16 a^ + 8 a^ + 4 a* + 2 a^ + 1). Factor :

3. m^4-^*. 8. l + a^ 13. m^ — m^n^

4. m^ — n^, 9. x^ — jf. 14. a^b — ab\

5. a;^-l. 10. a^ + y^*^. 15. ^-y^.

6. a;9 + ^». 11. a* + 32. 16. l-a«6^V*.

7. «^-a;. 12. 64-2a». 17. a^'' + 243a^

161. To factor the difference of the same even powers of two numbers.

EXERCISES

1. Factor a«-6«.

First Solution

§§ 134-136, a6 - 66 = ( a _ 5) (a5 + ^4^ + a%^ + a%^ + ah^ + 6^) = (a - 6) (a5 + ^253 + ^4?, + «54 4. a3&2 + 65) = (a- 6)[a2(a3 + 63) + ah{a^ + 63) + 52(0,3 4. 6-3)] = (a - 6) (a2 + a6 + 62) (a3 + 63)

§ 157, = (a - 6) (a2 + tt6 + 62) (a + 6) {a^ - a6 + 6^).

Second Solution

§ 152, a6 - 66 = (a3)2 - (63)2 ^ (^3 + 53) (^3 _ ^3)

§ 157, = (a + 6) (a2 - a6 + 62) (a - 6) (^2 + a6 + 62).

Considering the even powers as squares, as in the second solution, the process may be regarded as factoring the difference of tioo squares.

106 FACTORING

Separate into simplest factors :

2. a;«-2/^. 5. a;^-16. 8. l-h\

3. a^-1. 6. a;^-81. 9. 64-2/^

4. a^-h\ 7. a^-625. 10. 1-a^.

I

162. All the preceding methods of finding binomial factors are really special methods. The following is a general method of finding binomial factors, when they exist.

163. To factor by the factor theorem. j Zero multiplied by any number is equal to 0. J Conversely, if a product is equal to zero, at least one of the

factors must be 0 or a number equal to 0. J

If 5ic = 0, since 5 is not equal to 0, x must equal 0. 1

If 5 (a;— 3) = 0, since 5 is not equal to 0, x must have such a value as to make a; — 3 equal to 0 ; that is, x = 3.

If 5 (a; — 3), or 5 a; — 15, or any other polynomial in x re- duces to 0 when a; = 3, a; — 3 is a factor of the polynomial.

Sometimes a polynomial in x reduces to 0 for more than one value of X. For example, a;^ — 5 a; -f 6 equals 0 when a; = 3 and also when a; = 2; or when a; — 3 = 0 and a; — 2 = 0. In this case both a; — 3 and a; — 2 are factors of the polynomial. .•.a^-5a;4-6 = (a;-3)(a;-2).

164. Factor Theorem. — If a polynomial in x, having positive integral exponents, reduces to zero when r is substituted for x, the polynomial is exactly divisible by x — r, j

The letter r represents any number that we may substitute for x. 1

Proof. — Let D represent any rational integral expression containing a;, and let D reduce to zero when r is substituted for x.

It is to be proved that D is exactly divisible by x — r.

Suppose that the dividend D is divided by a; — r until the remainder does not contain x. Denote the remainder by B and the quotient by Q.

Then, D= Q{x-r)-\- B. (1)

But, since D reduces to zero when a: = r, that is, when ic — r = 0, (1) becomes 0 = 0 + jB ; whence, ^ = 0.

That is, the remainder is zero, and the division is exact.

FACTORING 107

EXERCISES

165. 1. Factor aj3 — a:^ _ 4 ^ _l_ 4^

Solution When a; = 1, x^ -x'^ -4:X +i = 1 - 1 - 4 + 4 =0.

Therefore, x — 1 is a factor of the given polynomial.

Dividing x^ — x^ — 4: x -{- i hy x — I, the quotient is found to be x^ — 4.

By § 152, x:'-4=(x + 2)(x - 2).

... x^ - x^ - ix + 4 =(x - l)(x + 2)(x - 2).

Suggestions. — 1. Only factors of the absolute term of the polynomial need be substituted for x in seeking factors of the polynomial of the form a; — r, for if x — r is one factor, the absolute term of the polynomial is the product of r and the absolute term of the other factor.

2. In substituting the factors of the absolute term, try them in order beginning with the numerically smallest.

3, When 1 is substituted for x, the value of the polynomial is equal to the sum of its coefficients ; then x — 1 is a factor when the sum of the coefficients is equal to 0.

2. Factor 17 3^-14x2 -37 a; -6.

Solution

Since the sum of the coefficients is not equal to 0, a; — 1 is not a factor.

When X = - 1, 17 a;3 - 14 rK2 - 37 X - 6 = - 17 - 14 + 37 - 6 = 0.

Therefore, x — (— 1), or x + 1, is a factor of the given polynomial.

Dividing 17 x^ — 14 x'^ — 37 a; — 6 by a; + 1 , the quotient is found to be 17 x2 — 31 X — C, which in turn may be tested for factors by the factor theorem.

Substituting factors of — 6 for x, it is found that :

When X = 2, 17 a;^ - 31 x - 6 = 68 - 62 - 6 = 0.

Therefore, x — 2 is a factor of 17 x^ — 31 x — 6.

Dividing by x — 2, the other factor is found to be 17 x + 3.

.-. 17x'5-14x2-37x- 6=(x + l)(x-2)(17x + 3).

3. Factor 2a^+a;V-5aj/+22/^ Suggestion. — When x = y,

2x^ + x'^y - bxy"^ + 2 y^ =2y^ -\- y^ - 5y^ + 2y^ = 0. Therefore, x — ?/ is a factor of 2 x^ + x^y — 5 xy^ + 2y'^.

108 FACTORING

Factor by the factor theorem :

4. 0^ - 1. 24. a^ - 7 a; + 6.

5. a^-^1. 25. sc^ -19 X +30.

6. a^-f. 26. a^-39«-70.

7. a^ + f. 27. 0^-67 x-126.

8. a;2-31a; + 30. 28. a^ - 21 xy' + 20 f.

9. a;2_27a;-28. 29. a^ - 31 a;?/^ - 30 2/^^ .

10. 4a;2-ha;-3. 30. a^-lSxy'' + 12f.

11. 3a;2-2a;-8. 31. a^ + 4a2- 11 a -30.

12. 5a;2-2a;-24. 32. a; + 9 a^ + 26 a + 24.

13. aj3-4a;2-7a; + 10. 33. 2 71^- 7^^- 7 n +30.

14. a:3_ea;2-9a; + 14. 34. ¥ -5b^-29b + 105.

15. a^-7a;2 + 7a; + 15. 35. a3 + lOa^ - 17 a- 66.

16. a^-12a;2 + 41a;-30. 36. m^ -f- 7 m2 + 2 m-40.

17. 2a^-3a;2-17a;-12. 37. b^ + 16b'' + 73b + 90.

18. a^-13x^ + ^6x-AS. 38. a^ - 15 aj^ + 10 a; + 24.

19. aj3-16a;2-h71a;-56. 39. ai^-Q a.'34-21 a^^+a;-30.

20. 2a^-9a;2-2a;+24. 40. a^+8a^+14a;2-8 a;-15. '

21. n^ +12n^ + Aln + A2. 41. aj^ - 9 a^y - 4 x?/^ + 12 2/*.

22. x^ + 2x^y-xy^-2f. 42. a^ - 9 a;y + 12 a^i/^ _ 4 ^4,

23. aT^ + Ax^y + 5xy^ + 2f. 43. a:5-4a^+19a;2-28 a:4-12.

44. a;5-18a^ + 30a;2-19a; + 30.

45. a;5-10a.'4 + 40a^-80ar^ + 80a;-32.

FACTORING 109

SPECIAL APPLICATIONS AND DEVICES

166. Factor:

1. a^^b^-\-c^r^d^-\-2ab-2ac-\-2ad-2bc-{-2bd-2cd.

Solution. — Since the polynomial consists of the squares of four num- bers together with twice the product of each of them by each succeeding number, the polynomial is the square of the sum of four numbers, § 111, and may be separated into two equal factors containing a, 6, c, and d with proper signs.

Since the ab, ad, and bd terms are positive, a and 6, a and d, and b and d must have like signs ; since the ac, he, and cd terms are negative, a and c, h and c, and c and d must have unlike signs.

Therefore, the factors are either

(a + 6-c+d)(a + 6-c + d) or (— a — 6 + c — d)(— a — 6 + c — (!).

2. 9x^-\-4:y^ + 25z^-12xy + S0xz-20yz,

5. 25m^ + S6n^-^p^-60mn-10mp + 12np, 4. a' + 16x*-{-36y^-Sax'-\-12ay-ASx'y.

6. x^-{-4.a^-j-b^ + y^ + 4:ax — 2bx-{-2xy — 4:ab-\-4:ay — 2by. 6. m^+4: n^ + a^ + 9 — 4 mn — 2 am -h6m + 4 a/i — 12 w — 6a.

167. The principle by which the difference of two squares factored has many special applications.

1. Factor a^ + a262 + 6^

Solution. — Since a* + a^b^ + ¥ lacks + a^b^ of being a perfect square, and since the value of the polynomial will not be changed by adding a^b^ and also subtracting a^b^, the polynomial may be written

V + 2 a2&2 + 64 _ 05252^ which is the difference of two squares.

.-. a* + a2&2 4. 64 = 05* + 2 05252 + 54 _ 05252

= (a2 + 62)2 _ ^262

= (a2 + ab + 62) (^52 _ ^6 + 62).

2. Factor4a*-13a2 + 9.

Suggestion. 4 a* - 13 a2 + 9 = 4 a* - 12 a2 + 9 - a2 =(2 a2 - 3)2 - a2.

110 FACTORING

3. Factor a^ + 4.

Suggestion. a* + 4 = a* +4 a^ + 4 - 4 a2 = (a^ + 2)2 - 4 a^. Factor the following :

4. a;^4-a;y + ^^ 10. x*-\-a^-^l,

5. a^-{-a'^h* + b\ 11. ?i« + n^ + l.

6. 9x^ + 20ic22/2 + 16/. 12. 16 x* + 4. x^y' + y\

7. 4 a^ + 11 tt252 + 9 6^ 13. a'b' - 21 a'b^ + 36,

8. 16a*-17aV + a;*. 14. 25 a^ - 14 a^fe* + ^s.

9. 25a^-29a^/ + 42/^ 15. 9 a^ + 26 a^fe^ .f 25 6*.

16. 5* + 64. 19. a* + 324. 22. a;* + 64 2/^

17. a^ + 4 6S 20. a«-16. 23. 4 a* + 81.

18. m8 + 4. 21. m^ + 4mn^ 24. ar'/ + 4a;y2.

168. Many polynomials may be written in the form x^ + px + q, x^ and x being replaced by polynomials.

1. Factor 9 a^ + 4 / + 12 2^ _f- 21 a;^ + 14 2/2; + 12 xy.

Solution. 9 x^ + 4: if + \2 z^ + 21 xz -h Uyz + 12 xy

= (9x2-\-12xy+4 y2) + (21 xz + 14 yz) + 12 ^j* = (Sx-h2yy-i-7 z(Sx + 2y) +4z'Sz

§154, = (3a; + 2y + 45;)(3x + 22/+30).

Factor the following :

2. a2 + 2a& + 62 + 8ac + 86c+15c2.

3. a^-6xy-{-dy- + 6xz-lSyz + 5:s\

4. m^ + ?i^ —.2 m/i + 7 mp — 7 rip — 30 p\

5. 16 w^ + 55 — 64 n — 16 m + m^ + 8 mn.

6. 9m* + A;2-30 + 39m2 + 13A; + 6m2A;.

7. 25a2 + 2/2 + 10a^ + 10a2/-35a»-7a;2/.

8. a2 + 62-j_c2 + 2a6 + 2ac + 2 6c + 5a + 56 + 5c + 6.

FACTORING 111

REVIEW OF FACTORING

169. Summary of Cases. — In the previous pages the student has learned to factor expressions of the following types :

I. Monomials ; as a^b^c. (§ 144)

III. Polynomials whose terms have a common factor ; as nx + ny + nz. (§146)

III. Polynomials whose terms may be grouped to show a com- ♦' men polynomial factor ; as

ax + ay ^- bx + by, (§ 147)

IV. Trinomials that are perfect squares ; as

a^+Zab + b' and a'-Zab^ b\ (§§ 148-151) V. Polynomials that are perfect squares ; as

a2 + 62 + c2 + 2a6 + 2ac + 26c. (§166)

VI. The difference of two squares ; as

0^-6^ (§152)

and a' + a'^b' + bK (§167)

VII. Trinomials of the form

x' + px + q, ' (§154)

VIII. Trinomials of the form

ax^^-bx + c (§156)

IX. The sum or the difference of two cubes ; as

a' + b^ova'-b^ (§ 157)

X. The sum or the difference of the same odd powers of two

' ^^ a" + 6" or a" — 6" (when n is odd). (§ 159)

XI. The difference of the same even powers of two numbers ; as

(jn _ fjn (when n is even). (§ 161)

XII. Polynomials having binomial factors. (§§ 162-165)

112 FACTORING

170. General Directions for Factoring Polynomials. — 1. Be- move monomial factors if there are any.

2. Then endeavor to bring the polynomial under soyne one of the cases II-XI.

3. When other methods fail, try the factor theorem.

4. Resolve into prime factors. J

Each factor should be divided out of the given expression as soon as found in order to simplify the discovery of the remaining factors.

EXERCISES

[Additional exercises are given on page 464.]

171. Factor orally :

1. 2a-2b. 18. a^-y*. 35. x'^-\-5x-^6.

2. x^-3x. 19. a^-1. 36. (x-yf-z^

3. x^-y\ 20. a^-S. 37. 4:a^-9I^.

4. a^ — 1. 21. m^ -h 1. 38. a;2« — y^.

5. a^ + x. 22. a2-4 52. 39. c^-fSd^.

6. 4a2 + 4a. 23. d^-9d. 40. a^"" - b\

7. Sa?-2x^. 24. a^-a^y. 41. q?^+'^-1.

8. a'^ — W- 25. 4a;^ — 4 a;. 42. ^x^ + ^^^f-

9. c2 — 4. 26. rc2 — 2 a; 4- 1. 43. x^-\-2xy + y\

10. 3^ + 2/^. 27. a^a; — 2aV. 44. x^-\-Sxy-{-2y\

11. f-y?. 28. a;2-}-3a;4-2. 45. 3 + 4a; + a^.

12. a;2 — 9. 29. a2-(6 + c)2. 46. ^ -\- ax + x + a.

13. «3 — a;. 30. ar^ — a; — 6. 47. a;^ — a; + a; — 1.

14. 2^^^— 1. 31. a^+a; — 6. 48. ab — bx -{■ ac — ex,

15. a'^x-x. 32. a;2 + a;-2. 49. oi? -{- x"^ + x -{- 1

16. aj^+i + a;. 33. a;^— 2a;-3. 50. a;^ + a^'/ + 2/^.

17. y^+^-y. 34. 5 a^^a _p 5 ^454 51, «* + a^ + 1.

5.

1

FACTORING 113

Factor :

52. 1-x^. 60. 62 _ 196. 68. a^-{-27a\

53. x'^-1. 61. a' -256. 69. 4:50- 2 a\

54. a — a\ 62. 64 — 2 2/^ 70. 125 - 8 a^.

55. a;*°-l. 63. 6 6^ + 24. 71. 4 m' -f .004.

56. Z;^ + &. 64. a^^-ab^\ 72. • 2 a;^ + x- 1.

57. i)* + 4. 65. Tn'-^Tn. 73. ic2_^9a;_90.

58. l+x^\ 66. ly^-115, 74. 3ar^-2a;-8.

59. a^-ftl 67. 8-27aV. 75. 15 + 6a;-9a^.

76. 17 - 16 a - a^. 93. W + 6y + 1/*.

77. o?h'^+ah-56. 94. aV + aV + a^.

78. Q>x+ 5x^-0^. 95. 128 a^- 250 a^

79. aV-4aa;4-3. 96. 8 a^ - 6 a.-2 - 35.

80. x^ — ax—12a\ 97. aV + a^ftV + 6^.

81. n2_o7i-90a2. 98. yi^ -h n^a^fe^ + ^458.

82. x^-xy-ld2y\ , 99. 77 - 30 a;^ - 37 a;.

83. 3a;2 + 30a;+27. 100. 5 a;^" -f 10 a;^ - 15.

84. 6a;2- 19a; + 15. 101. 16 x^ + 4 ar^/ + y.

85. 4 a - 3 aa; — aa;2. 102. 5x^ -26 xy -\- 5y\

86. a;2«H-2a;«?/^ + iy2p. 103. 8a2-21 «6 -9 d^.

87. ?/^c - 13 62c -f- 42 c. 104. a2_,_52_c2_2a6.

88. 2«3 + 28a;2-i-66aj. 105. 60 a^ + 8 aa? - 3 ar*.

89. 2/2 H- 16 a^ - 36 a2. 106. 10 a2c + 33 ac - 7 c.

90. aa;2 + 10aa;-39a. 107. 25 ^2 + 60 a;i/ + 36 /.

91. 7 a;2 - 77 a'2/ - 84 ?/2. 108. 6 aaj2 + 5 aajy — 6 o?/2.

92. 2/^ - 25 ?/a; + 136 «2. 109. a;2- ca; + 2 da; - 2 cd

milne's stand, alg. — 8

114 FACTORING

110. x^ + ix. 125. (x + yf -i-ix-yy.

111. a^"-a3*. 126. {a-2y+{a-lf.

112. 3x^-^96x. 127. 4:3^ -\- x'^ - Sx - 2.

113. (a + &)^ — 1. 128. a;2 + 5 a.' + aa^' + 5 a.

114. (a + xy — X!*. 129. X* — 119 a^y + y^.

115. l+(ir + l)3. 130. (a + &)^-(6-cy. •

116. a^ — 2 a2"+ 1. 131. m^ + m^ — mn — mw^.

117. 63 _ 4^2.^ 8. 132. (a2 4- &' - c2)2 - 4 a262_

118. a3-(aH-6)3. 133. {a?-y-^y -{x" - xyy.

119. a;7 - 2 a^ + a;. 134. 3 ah{a + 6) + a' + ^>^ ^

120. a^62 _j_ (^2^ _ 12. 135. aj2«-2 _f_ 522^2 ^ 2x''-^by.

121. 1000 :x? — 21f. 136. a6 — 6a;'* + a;"?/"* — ay"^.

122. a;^ — a;2 - a;4 + a;3_ 137 a^ — i,^ — (^a + h){a — h).

123. 17a;2 + 25a.'-18. 138. a.-^ + 15 x^ + 75 a; + 125.

124. a? — xy — x'y + y\ 139. a;^— 3 aa; + 4 6a; — 12 a6.

140. a:^-9 ax^ + 26 aa; - 24 a.

141. 12 aa; — 8 6a; — 9 a?/ + 6 by.

142. 25a;2-92/2-24?/2- I622.

143. a6a.'3 + 3 ah^ — a6a; — 3 ah.

144. 3 6ma; -J- 2 6m — 3 anx — 2 an.

145. 20 aa.-3 - 28 aa;2_j_ 5 ^2a;_ 7^2^

146. x^ — z^ + y^ — a^ - 2 xy + 2 az.

147. 2 62m - 3 a62 + 2 6ma; - 3 a6a;.

148. Factor 32 — x^ by the factor theorem.

149. Factor 16 -|- 5 a; — 11 a;^ by the factor theorem.

150. If n is odd, factor a;" — a" by the factor theorem.

151. If n is odd, factor a;'* -f- r** by the factor theorem.

152. Factor a;^ _ 6 6a;2 ^ 12 52^; _ g 6^ by the factor theorem.

153. Discover by the factor theorem for what values of /?, between 1 and 20, a;" + a** has no binomial factors.

FACTORING 116

EQUATIONS SOLVED BY FACTORING

172. 1. Find the values of a; in a;^ + 1 = 10.

Explanation. — On transposing

FIRST PROCESS ^^^ known term 1 to the second

i)^ _|_ 1 = 10 member, the first member contains

^2 _. g the second power, only, of the

^./v._Q.Q ./>.__Q unknown number. On separating

r. n ' o each member into two equal factors,

OVX'X=:— 3'— 3.'. X=— 3 ^ ^ QQ«^^^ Q Q

x-x = O' 6 or X ' X =: — 6 ' — o. .'. x= ±S Since, if X = 3, X . a: = 3 • 3, and if

X = — 3, X • ic = — 3 • — 3, the value of x that makes x^ = 9, or that makes x2 + 1 = 10, is either + 3 or -3 ; that is, x = ± 3.

Find the two values of x in each of the following:

2. a^ + 3 = 28. 6. x" -h 3= 84.

3. x' + l=50. 7. aj2-24 = 120.

4. a^-5 = 59. 8. 0^24.11^180.

5. «2_7==29. 9. 0^-11 = 110.

10. Find the values of a; in a^ + 1 = 10.

SECOND PROCESS EXPLANATION. — The first process

o -j _ -I <-w is given in exercise 1.

^^ ^ ~ In the second process, all terms

^m X J = U g^j.g brought to the first member,

^B (x — 3)(a;H-3)= 0 which is factored as the difference

^E*. £C — 3 = 0, whence X = S of the squares of two numbers.

^or ic + 3 = 0 whence x = 3 Since the product of the two fac-

. , o tors is 0, one of them is equal to 0.

Therefore, x— 3 = 0 or x + 3 = 0; .whence, x = 3orx=— 3; that is, x = ± 3.

Solve for a;, and verify results :

11. 0^ + 35 = 39. 14. a^- 312 = 0.

12. a:2_50 = 50. 15. x'-^b' = 0. .13. a;2_|_ 90^91^ ^g ^_9^2^0

116 FACTORING

17. a^-21=4. 22. 32-3^ = 28.

18. a^-56 = 8. 23. 65-a^ = 16.

19. a^-3a2==6a2. 24. 4 a;2-8 &2 = 8 61

20. x' + 5¥ = 6bK 25. «2 + 25 = 25 + m2.

21. 0^-40 = 24. 26. x" -S0==2{2b^-15).

27. Solve a;2 4_ 2 am = a^ + m^.

Solution a;2 + 2 am = a2 + wi^.

a;2 33 a^ - 2 am + m2. x-x = {a—m){a — m) or a; • ic = — (a — m) • — (a — w).

,•. x= ± (a — m). Solve for x, and verify :

28. x^-c' = d^-2cd. 34. a;2_c2 = 36_i2c.

29. a^-62 = 42,c + 4c2. 35. ar^-4 52 = 36- 24 6.

30. a)2-n2 = 67i + 9. 36. ^-o? = ^-Qa.

31. a;2 + 10a = a2 + 25. 37. x^ -h^ = 4.-4.h\

32. .'c2-a2 = 2a4-l. 38. a:2_ ^262 = 2 a6 + 1.

33. ic2-m2 = 8m + 16. 39. x^ - r^ = h^ - 2 r''b\ 40. Find the values of a: in ar* + 4 a; = 45.

FIRST PROCESS SECOND PROCESS

a^ + 4a; = 45 x' + ^.x^^^

x'-^4:X-4o= 0 x^-{-4:X-^4. = 49

(x-5){x-{-9)= 0 (a; + 2)(a; + 2) = 7.7or-7.-

.a; — 5 = 0ora; + 9=0 .*. a; + 2=7or— 7

.-. a; = 5 or — 9 .*. a; = 5 or —9

FACTORING

117

Explanation. — For the first process the explanation is similar to that given for exercise 10.

In the second process, it is seen that, by adding 4 to each member of the equation, the first member will become the square of the binomial (x+2). On solving for (x+2) as for x in previous exercises, x+2 = ±T; wlience, a; = ±7— 2=+7 — 2or — 7-2 = 5 or— 9.

Suggestion. — In the following exercises, when the coeflBcient of the first power of the unknown number is even, either of the above processes may be used ; but when it is odd, the first process is simpler.

Solve, and verify results :

41. a^-6x = 4:0.

42. a^-Sx = 4:S.

43. x^ — 5x= -4:.

44. a;2 + 4a; + 3 = 0.

45. r2 + 6r4-8 = 0.

46. a^-9a; + 20 = 0.

47. x^-3x = A0.

48. a^-9x = 36.

49. x^ + llx==26.

50. x^ - 12 x = 4:5.

51. 2/2-15 2/ = 54

52. 2/^ -21 2/ = 46.

53. x'-10x = 96.

54. 2/^ -20 2/ = 96.

55. 2/' + 12 2/ = 85.

56. 2/^ + 42 = 13?/.

57. ^24.63 = 16^.

58. t'2-60 = llv.

59. or-7x = lS,

60. a^ + 10 aj = 56.

61. a;2 + 12a; = 28.

62. w2 + lln4-30 = 0.

63. a;2_^a._i32 = 0.

64. 32 = 4w + w;2.

65. 3s = 88-s2.

66. 160 = a^-6a;.

67. 4 2/ = 2/^ -192.

68. 600 = 2/2-102/.

69. c2 + 16c-36 = 0.

70. Z2^_i5;_34 = 0.

Solve for x, y, or z, and verify results :

71. a^ + 26x + &' = 0. 73. x" - (a + b)x-{-ab = 0.

72. 22_^4a2; + 4a2 = 0. 74. x^ -{-(c ^d)x + cd = 0.

118 . FACTORING

75. x'-\-(a-{-2)x + 2a = 0. 77. x"- (a-(jr)x-ad = 0.

76. y^-(c-n)y-nc=0. ^ 78. x^- {b -\-7)x-{-7 b = 0.

79. (2ic + 3)(2a;-5)-(3a;-l)(a;-2) = l.

80. {2x-6){Sx-2)-(5x-9)(x-2) = 4.,

81. Solve 6x'-{-5x-21 = 0.

Solution 6a;2 + 5a;-21 = 0. Factoring, § 156, (2 a; - 3) (3 a: + 7 ) = 0.

.-. 2a;-3 = 0 or 3 x + 7 = 0.

.•.a; = |or-|. Solve, and verify results :

82. 8x^ + 2x-l=0.' 87. 7a^ + 6a;-l = 0.

83. 5x^-\-4.x-l = 0. 88. 2v^-9v-35 = 0,

84. 3/ + 2/-10 = 0. 89. 62/'-222/ + 20 = 0.

85. 3/-42/-4 = 0. 90. 3a;2 + 13a;-30 = 0.

86. 42/' + 92/-9 = 0. 91. 4ic2 + 13a;- 12=0.

92. Solve the equation a^ — 2ic^ — 5aj-f6 = 0.

Solution

a;3 _ 2 a;2 _ 5 X + 6 = 0. Factoring, § 163, (a; - 1 ) (x - 3) (x + 2) = 0.

/. a; - 1 = 0 or a; - 3 = 0 or a; 4- 2 = 0 ; whence, a; = 1 or 3 or — 2.

93. oi^-15a^ + 71x-105 = 0. 95. a:^ -12a; + 16 = 0.

94. a^ + 10a^ + lla;-70 = 0. 96. a3-19a;-30 = 0. '

97. x*-}-x^-21x^-x-^20 = 0.

98. x*-7x'-\-x^ + 63x-90 = 0.

99. ar*-Ua;4 + 45ar'«-85«2 + 74a;-24 = 0.

HIGHEST COMMON FACTOR

173. The sum of the exponents of the literal factors of a rational integral term determines the degree of the term.

Thus, a and 5 a are of the first degree ; 3 x^ and 3 xy are of the second degree ; 4 ahH and x^{y — 1)^ are of the fifth degree.

174. The term of highest degree in any rational integral expression determines the degree of the expression.

Thus, the expression x^ — 6 x^ + 11 x — 6 is of the third degree.

175. An expression that is a factor of each of two or more expressions is called a common factor of them.

176. The common factor of two or more expressions that has the largest numerical coeSicient and is of the highest degree is called their highest common factor (H. C. F.).

The common factors of 4 a^&2 and 6 a^h are 2, a, 6, a^, 2 a, 2 &, 2 a^, a&, 2 ah, a^b, and 2 a^b, with sign + or — . Of these, 2 a^b (or — 2 a^b) has the largest numerical coefficient and is of the highest degree, and is there- fore the highest common factor.

The highest common factor may be either positive or negative, but usually only the positive sign is taken.

The highest common factor, or divisor, of 4 a'^b'^ and 6 a^b is 2 a^b^ regardless of the values that a and b may represent. What the arith- metical greatest common divisor is depends upon the values of a and b. If a = 2 and & = 6,

H.C.F. =2^^6 = 48; but since ^ a^b'^ = 1152 and 6 a^S = 144, the arithmetical greatest common divisor = 144.

177. Principle. — The highest common factor of two or more expressions is equal to the product of all their common prime factors.

m 178. Expressions that have no common prime factor, except B, are said to be prime to each other. i 119

120 HIGHEST COMMON FACTOR

EXERCISES

179. 1. Find the H. C. F. of 12 a%^c and 32 a^ftV. Solution

The arithmetical greatest common divisor or highest common factor of 12 and 32 is 4. The highest common factor of a^h'^o, and d?-hH^ is d^-h'^c. Hence, II. C. F. = 4 (f-VH.

Rule. — To the greatest common divisor of the numerical coefficients annex each common literal factor with the least expo- nent it has in any of the expressions.

Find the highest common factor of : '

2. 10 a^/, 10 :x?f, and 15 xy^z.

3. 70 a%\ 21 a^b\ and 35 a^6«.

4. 8 mV, 28 m^n\ and m m^n\

5. 4 h\a, 6 6V, and 24 ah(?,

6. 3(a + &)2 and 6(a + 6y. ,

7. 6(a + &)2and4(a + ??X«-^)-

8. 12(a - x^, ^{a - x)\ and (a - «)*.

9. 30(a; + 2/)^ 18(0^ + 2/), and (a; + 2/)^

10. 10(a; — ?/)V and 15(2; — 2/)(a; — 2/)^

11. 3(a2-62)2anda(a-6)(a2_6^). ;

12. What is the H. C. F. of 3 a;3_3 ^^ and 6 a^-12 x^y+Qxf\

PROCESS

3 a^- 3 ici/^ =3 a;(a; + ?/)(a;- 2/)

6 cc^- 12 ar^2/ 4- 6 a;/ = 2 • 3 « (a;- ?/)(a; - y) .-. H.C.F. = 3a;(a;-?/)

Explanation. — For convenience in selecting the common factors, the expressions are resolved into their simplest factors.

Since the only common prime factors are 3, a;, and (x — y), the highest common factor sought (§177) is their product, 3 x (x — y).

HIGHEST COMMON FACTOR

Find the highest common factor of : 13. ic2 -2a; - 15 and «2 _ a; - 20.

121

14.

if, x^ — 2/^, and x -\- y.

15. a^ + 7a -f 12 and a} + 5 a -f 6.

16. V? + 2/^ and x^ -\- 2xy -\- y"^.

17. «^ — 0/'^ and o} — 2 ax -\- x^.

18. ce - W and a^ 4. 2 a6 + h\

19. ic^ + x^t/^ + y^ and x^ 4- xy + 2/^.

20. a:^ + ?/^, a;^ + 2/^, and x^y -|- a-i/^.

21. «4 + a^ft* 4- &^ and Sa^ - 3a62 + 36*.

22. a^ — a;^, o? -\-2 ax + aj^, and a^ + a;'.

23. aa; — 2/ H- a72/ ~ ^ and ax^ -\- x^y — a — y.

24. a^h — h — a^G -\- c and ab — ac — b -{■ c.

25. 1 - 4 a^*, 1 + 2 a;, and 4 a - IGaa.-^.

26. (a - 6)(6 - c) and (c - a){a^ - b^),

27. 24 a^2/' + 8 a^^^s ^nd 8 xhf - 8 a^2/'.

28. 6 a^ + a; - 2 and 2 x"- 11 a; + 5.

29. 16a.-2-25 and20a;2-9,a;-20.

30. x'^ -\- xif and x^y -\- xy"^.

31. 17 ab&d'' - 51 a'^hc'd' and ab&d} - 3 a^ftc^d.

32. x'y -\- xy^ and 2 a;52/ — 2 a:^2/^H- 2 a'2/^.

33. 6?-7 + 10r«s-4rVand2r"^ + 2r«s-4rV.

34. (Sx'-- 54, 9(a; + 3), and ^Q{^ -x- 12).

35. 8 a - 8 a\ 12 a(a2 - 1)^, and 18 a^ - 36 a + 18.

36. 9 a\x?- 8 a; + 16) ahd 3 a^x + 6 aa; - 12 a^ _ 24 a. i37. a.-^ - a:^ - 2 a;2, a^ - 2 a;3 _ 3 r^i^ and a^ - 3 a.-^ - 4 a;^^

{8. 7 Ff + 35 W + 42 If and 7 ^^^ + 21 IH^ - 28 Pe - 84 /^.

J9. a? 4- a'-62 +2 aa;, x^-a" + ?>2^2 6a;, and x^-a''-b''-2 ab.

122 HIGHEST COMMON FACTOR

Apply the factor theorem when necessary.

40. x'-6x-^5siiidx^-5x'-\-7x-3,

41. x'-4:?Lnda^-10x'-h31x-S0.

42. 3^-4:X-\-3sinda^-haf-37 x + 35.

43. 3x^-12x^8Liid6x*-\-30s(^-96a^-\-24:X.

44. a'b - a'W and a% + 2 a^V^ + 2 a^h^ + ah\

45. 9 — 71^ and w^ — w — 6. Suggestion. — Change 9 — w^ to — (n^ - 9) = — (w + 3) (n — 3).

46. l-a^andar^-6a^-9aj4-14.

47. 4-a2anda* + a3-10a2-4a + 24.

48. (9-a;2)2anda;*4-5a^-3a;2-45a;-54. Suggestion. (9 - x'^y = (x^ - 9)2.

49. (4-c2)2andc3 + 9c2 + 26cH-24.

50. {x - a;2)3, (a;^ _ ^^s^ ^^j^^ ^ _ ^^^3^

51. (1 - fY and (2/ + 1)^(1 - y)\f -ly + 6).

52. aji/ — /, — (2/^ — x-y), and a;^ — a^^/^-

53. 16-s^2s-s2, ands2-4s + 4.

54. y^ — x'^^x^ + f, and y'^-\-2yx + o^.

55. iB2-(2/ + 2)2, (?/-a;)2-22^and2/'-(a;-2y.

56. {y — x)\n — m)^ and {x^y — jf) (m^n — 2 mn^ + ?i^.

When some of the given expressions are difficult to factor, their factors may often be discovered by dividing by those of the more easily factored expressions.

57. (m + 2)(m2-9)andm*-3m3 + 3m3yi4-37nV-9m2n + mn^ — 9 mn^ — 3 n^.

68. 6x^-3x-4:5,9x^-33x-\-lS,and6a^-3a^-39x-lS.

59. 2ai^-a^-oi^,2a^-^x-3,a.nda^-x^-x-\-l.

60. a^-4ar^+2a^+a; + 6, 2 x^-9 x^ + 7 x-\-6, and ar^-5iB+6.

61. s«-8, .93 + s2 + 2s-4, ands4 + 2s3-s2-10s-20.

62. xf^ + fa.ndxf^-2x^y + 2a:^y^~2x^y^ + 2xy*-f.

LOWEST COMMON MULTIPLE

180. An expression that exactly contains each of two or more given expressions is called a common multiple of them.

6 dbx is a common multiple of a, 3 6, 2 x, and 6 ahx. These numbers may have other common multiples, as V2ahx^ Ga^b^x, IS a^bx^, etc.

181 . The expression having the smallest numerical coefficient and of lowest degree that will exactly contain each of two or more given expressions is called their lowest common multiple.

6 abx is the lowest common multiple (L. C. M.) of a, 3 6, 2 x, and Qabx. The lowest common multiple of a and b is ab, regardless of the values that a and 6 may represent. What the arithmetical least common multiple is depends upon the values of a and b. If a = 6 and 6 = 2, the least common multiple is not 12, the value of ab, but 6.

The lowest common multiple may have either sign.

In §§ 180, 181, only rational integral expressions are included.

182. Principle. — The lowest common multiple of two or more expressions is the product of all their different prime factors, each factor being used the greatest uumher of times it occurs in any of the expressions.

EXERCISES

183. 1. What is the L. C. M. of 12Q^yz\ Qa'xy^ and ^axyz" ?

Solution. — The lowest common multiple of the numerical coefficients is found as in arithmetic. It is 24.

The literal factors of the lowest common multiple are each letter with the highest exponent it has in any of the given expressions (Prin.). They are, therefore, a'^, x^, ^/2, and 0*.

The product of the numerical and literal factors, 2\i a'^x'^-'ip-z!^ , is the lowest common multiple of the given expressions.

123

124

LOWEST COMMON MULTIPLE

2. What is the L. C. M. of 0^-2x2/ + /, 2/^- a^, and a^ + 2/^?

PROCESS

y^ — a^=-(x^ — y^) = — (x + y){x--y) a^+f =(a;-|-2/)(«2-a^ + 2/2)

L. C. M.

z=(x- yf(x + y)(QiP-xy-\- y"^

Rule. — Factor the expressions into their prime factors.

Find the product of all their different prime factors, using each factor the greatest number of times it occurs in any of the given expressions.

The factors of the L. C. M. may often be selected without separating the expressions into their prime factors.

Find the lowest common multiple of:

3. a^x^y, a^xy^, and aa^y.

4. 10 a^b'c", 5 ab% and 25 W(?d^.

5. 16 d'Wc, 24 (^de, and 36 a'bH^^.

6. 18 a^bi^, 12fq% and 54 abVq.

7. a?"*/, of"- V> a;'"~y, and a;'"+Y

8. x^ — y^ and x^-{-2xy -\- y\

9. x^ — y'^ and x^ — 2xy + y^.

10. Qi? — 'if,x^-{-2xy-{- 2/^ aad oi? — 2xy -\-y^.

11. a^-n2and3a3-|-6a2n + 3an2.

12. a;^-landaV + a2-6V-62.

13. a^ + l, a6 — 6, a'^ + a, and 1 — a^.

14. 2x-\-y,2xy — y^, and 4 a;- — y'^.

15. 1 +a;, x — 7?, 1 + ar^, and iB^(l — x).

t

LOWEST COMMON MULTIPLE 125

16. 2x-\-2,5x — 5,3x-S,^ndx^ — l.

17. 16&2_1, 12 62^36, 206-5, and 26.

18. l-2x^ + x*,{l-xy,Sindl-\-2x-\-a^,

19. xy — y% Q? + xy, xy + y'^, and x^ + y^.

20. 2/^ — ic^, a;^ + aj?/ + 2/^ and y? — xy,

21. 6^-56 + 6, 6^-76 + 10, and 62-106 + 16.

22. ar^4-7aj-8, ^ -1, x^y?, and 3 aaj^ _ 6 ao; + 3 a.

23. ar^-a^, a-2a;, a^ _|. 2 aa;, and a» - 3 a^a; + 2 aa^.

24. m^ — ar', m^ + wia;, m^ + mx + a^, and (m + a;)a^.

25. 2-3a; + a;^ a^ + 4 a; + 4, a;2 + 3 a; + 2, andl-o^^

26. x"^ — 2/^, a;'' + x^y^ + y*, ^4- 2/^ and x^ + xy-\-y^,

27. a^ + x'^y -{-xy--{-y^ and x^ — x^y + xy- — y^,

28. a2 + 4a4-4, a2-4, 4-a2, anda^-16.

29. a2-(6 + c)^ 62-(c + a)2, andc2-(a + 6)2. 0. m — n, (m? — n^y, and (m + ny.

31. a«-63and-a8 + a*62 + 6*.

32. x^ + 2/^ and aV - by + aY - 6V.

33. a*-a^-\-l,a^-{-l,a*-\-a^ + l,SiTida^-l,

34. 2(aa;2 _ ^y2^ 3 ^^((12^ _ ^^3^ and 6(a^a^ - a').

35. (2/2^ — a;2/2j)2, 2/^(^^^ — ^)} ^-nd a;V + 2 a;?^ + 2;*. Suggestion. — In solving the following, use the factor theorem.

36. a^-6aj2 + lla;-6andaf-9a.'2 4-26aj-24.

37. a^-5«2-4aj + 20anda;3 + 2x2-25a;-50.

38. aj3_4aj2_^5a;-2anda^-8a^4-21a;-18.

39. ar^ + 5a;2 + 7a;4-3anda^-7aj2-5a; + 75..

40. a^4-2ic2-4aj~8, ar^-a?2-8 x4-12, a.-^ + 4 a:^ _ 3 ^ _ ;^8.

FRACTIONS

I

184. A fraction is expressed by two numbers, one called the numerator, written above a line, and the other the denominator, written below the line.

If a and b represent positive integers, as 3 and 4, the fraction

o

- is equal to - ; that is, it represents 3 of the 4 equal parts of 0 4

anything. This is the arithmetical notion of a fraction.

But, since a and b may be any numbers, positive or negative,

integral or fractional, - may represent an expression like — .

Since a thing cannot be divided into 5| equal parts, algebraic fractions are not described accurately by the definition com- monly given in arithmetic. But, since an expression like ^^-^ regarded as 20 fourths, is equivalent to 5, or 20 -f- 4, it is evi- dent that the numerator of a fraction may be regarded as a dividend, and the denominator as its divisor ; and this inter- pretation of a fraction is broad enough to include the fraction

- when a and b represent any numbers whatever. Hence,

The expression of an unexecuted division^ in which the dividend is the numerator and the divisor the denominator j is an algebraic fraction.

The fraction ^ is read, ' a divided by 6.'

0

185. The numerator and denominator of a fraction are called its terms.

186. An expression, some of whose terms are integral and some fractional, is called a mixed number, or a mixed expression.

a — ^~ ^, ^ — 24-—, and a — 6 + — are mixed expressions.

126

FRACTIONS 127

Signs in Fractions

187. The sign written before the dividing line of a fraction I is called the sign of the fraction.

I It belongs to the fraction as a whole, and not to the numera- I tor or to the denominator alone.

! In — — the sign of the fraction is — , while the signs of x and 3 z are + .

188. An expression like ^^— indicates a process in division,,

— b

in which the quotient is to be found by dividing a by 6 and , prefixing the sign according to the law of signs in division ; that is, -a_^a -ha__^a

'^b-'^V +b~^V

— a_ a 4-ot_ «

-\-b~ b[ -b~ b

By observing the above fractions and their values the fol- lowing principles may be deduced :

189. Principles. — 1. The signs of both the numerator and the denominator of a fraction may be changed without changing

I the sign of the fraction.

2. The sign of either the numerator or the denominator of a fraction may be dianged, provided the sign of the fraction is changed.

When either the numerator or the denominator is a polynomial, its sign is changed by changing the sign of each of its terms. Thus, the sign of a — 6 is changed by writing it — a + 6, or 6 — a.

EXERCISES

190. Reduce to fractions having positive numbers in both terms : ,

-2-m

2-{-n

8 -4(a + &) 5{-x-y)

i 1.	-3 -4	3.	— a — x	5.	-a-b
2x	c + d
; 2.	2 -5	4.	-4.C -b-y	6.	-2 -a-y

128 FRACTIONS

191. By the law of signs for multiplication, the product of two negative factors is positive ; of three negative factors, nega- tive; of /owrnegative isiCtovB^ positive ; and so on. Hence, J

Principles. — 3. TJie sign of either term of a fraction is changed by changing the signs of an odd number of its factors.

4. Tlie sign of either term of a fraction is not changed bu changing the signs of an even number of its factors.

EXERCISES

192. 1. Show that (a-b)(d-c)^(a-b)(c-d)^

(c — a)(6— c) {a — c)(b — c)

Solution or Proof

Changing (d — c) to (c — d) changes the sign of one factor of the numerator and therefore changes the sign of the numerator (Prin. 3).

Similarly, changing (c — a) to (a - c) changes the sign of the denomi- nator (Prin. 3).

We have changed the signs of both tenns of the fraction. Therefore, the sign of the fraction is not affected (Prin. 1).

2. Show that {h-a){d-c)^_{a- b){c - cZ)^

(c '-'b){a — c) * (^ — c)(a — c)

Solution or Proof <fl

Changing the signs of two factors of the numerator does not change the sign of the numerator (Prin. 4).

Changing the sign of one factor of the denominator changes the sign of the denominator (Prin. 3).

Since we have changed the sign of only one term of the fraction, we must change the sign of the fraction (Prin. 2).

3. Show that -^ — may be properly changed to

b — a a —

4. From — ^^ — derive by proper steps.

6_a-f c a-b-c ^ ^ ^ ^

5. Prove that ^-^ = --^; that - ^ ^

1' 4-a^ 0^-4:

6. Prove that

7. Prove that

8. Prove that

9. Prove that

FRACTIONS 129

2 2

aih — a) a{a — h)

5x 5x

(« +y){y — «) (« + y){^ — y)

2a ^ 2a

9_a2~(a + 3)(a-3)*

1 1

(6_a)(c-6) (a-6)(6-c)

tlO. Prove that (m-n)(m + n) ^ -^; + ^\^ . (a — c) (6 — a) (a — c) (a — 6)

Bll. Prove that (^-&)(&-a + c) ^ (c^-&)(«- ^>-c) (y-x)(z-y)(z-x) (x-y)(y -z){x-z)

REDUCTION OF FRACTIONS

193. The student will find no difficulty with algebraic frac- tions, if he will bear in mind that they are essentially the same as the fractions he has met in arithmetic. He will have occa- sion to change fractions to higher or lower terms ; to write in- tegral and mixed expressions in fractional form; to change fractions to integers or mixed numbers; to add, subtract, multiply, and divide with algebraic fractions just as he has learned to do with arithmetical fractions.

194. The process of changing the form of an expression without changing its value is called reduction.

195. Principle. — Multiplying or dividing both terms of a

fraction by the same number does not change the value of the

fraction: that is,

a _ am am _ a

b bm bin b

196. A fraction is in its lowest terms when its terms are prime to each other.

Milne's stand, alg. — 9

130 FRACTIONS

197. To reduce fractions to higher or lower terms.

EXERCISES

1. Keduce to a fraction whose denominator is a^ — HK

PROCESS

Then,

(a2_62)_^(a + 6) = a-6. __ a(a — b) _a?—ah

a + h (a + 6)(a-6) a'-W

Explanation. — Since the required denominator is (a — h) times the given denominator, in order that the value of the fraction shall not be changed (§ 195) both terms of the fraction must be multiplied by (a — &)»

5a

2. Reduce — to a fraction whose denominator is 42.

6

3a;

3. Reduce — — to a fraction whose denominator is 55 h,

11 h

4. Reduce --— to a fraction whose denominator is 84 xu,

14a; ^

5. Reduce — — to a fraction whose denominator is 20 ifi,

5y ^

x — S

6. Reduce to a fraction whose denominator is (x — ly.

x—1 ^

7. Reduce ^ "~^ to a fraction whose denominator is (2 « + 5)^.

2a;+5 ^ ^

8. Reduce to a fraction whose numerator is 3 a -f- a^. I

o — a \

9. Reduce ^ to a fraction whose numerator is a:^ — v*.

2a; + y ^

10. Reduce ~ ^ to a fraction whose denominator is 4 — aj*.

x — 2

11. Reduce a; — 5 to a fraction whose denominator is a; + 5.

FRACTIONS

131

12. Reduce — ^ to its lowest terms.

30 a^xz

PROCESS

30 a^xz 10 az

Explanation. — Since a fraction is in its lowest terms when its terms are prime to each other, the given fraction may be reduced to its lowest terras by removing in succession all common factors of its numerator and denominator (§ 195), as 3, a, a, and x\ or by dividing the terms by their highest common factor, 3 aH.

Reduce to lowest terms 16 m^nx^z^

13.

14.

15.

16.

n.

40 am^y^

210 hcH 750 ab'c'

35 a'bcd^ 42 ab^cd'''

77 qV6«v 121 a^b'c''

- 25 a^fz'

18.

19.

20.

21.

28.

29.

30.

31.

32.

33.

100 a;y

a'-b' a^^2ab + b''

a'~2ab-^b^ a'-b'

4a^-9a^ 80^ + 270^'

Sa^ + Sab

a^-j-ab^

3x^y — 6 xy x'^y — 8xy

Sa'b-S b^ 2a^b-2b*'

22.

a^xy

amhi*

a'b^x' Wxy^

34.

35.

36.

37.

38.

39.

23.

24.

25.

26.

27.

x^'^^y

x

m—n+l

ax Sa^b' 2 a'y^

2a^f-Sy\ 4 0^2/ -32 2/**

10 nic + 10 ny

25 71x^-25 71/

ft"+^ — aV x'y-Qi?f-\-f ^

132 FRACTIONS

.40. y'-^\.. 54.

a^-10a;2-h9

41. «'^-llct + 24 gg^ x^ + a^y + y^

x^ + f

20 - 21 a; + a^

42. ^ -^ — —' 56.

43. :^- -^^^ — ^. . 57.

/-81

2/2 + 72/-18

a2._ 11^.^24

a^'-a-e

a^-6x-7

a;2_ila;_(_28

2a!2_2a;-12

6a;2-10a;-44

a^_6ar^ + 5x

a^ + 2a^-35a;

7a._2a;2_3

2x2 + 11 aj-6

62 _^ ft _ 12

362 + 96-54

x^-2x'-^a^

a^-7a: + 6

44. -^^ :^ . ^ -^ , gg^

45. -u:^ — ^ir^ ^. 59.

46. „ _^„ ^^^^^. 60.

a^-26a~^ + 25

cd^ — c c2# + c2r72+c2*

3 g^ + 4 eta; — 4 a;2 9a2_l2aa; + 4»2*

x^ + 3x^-\-Sx + l

4 + 4 a; — a;2 — a^

5r2-26r8 + 582 r^ — 5 r^s + r — 5 s '

48 o^M^Hh^^. 62 ^ln^-H^^i^. g6(g2-4&2)2* • 3^4 _i

49 a^ + 5ar^-6a; ^3 ,3^25 ^3^52 _ 53

2a52-2 ' ' 3g&2-3g26 *

_^ g' + 2g26 + a62 ^^ 2g.T - g.?/- 46a; + 26y

g^ - 2 aW -\-ab* 4:ax-2ay-2bx + by

51. (a + 6)2-1 gg 9a:3- 13g2a:-4g' a2c + abc + dc

52. m-m^-n + myi^ ^^

(,„ am — a7i — m-\- n ^^

OQ, • d7.

m — mn + n2 — ri

am — an — m-\- n

(m — CLn-f-m — n ^" 24: a^d — ^Oabd-h 16 b^d

3bx-\-3xy-	-4g6	-4 ay
a^ + 5a^-	-9a;-	45
ar' + 3a;2-	25 a;-	-75
9 g^ - 13	a^¥ + ib*

FRACTIONS 133

198. To reduce a fraction to an integral or a mixed expression.

EXERCISES

1. Reduce — ^^ to a mixed number.

X

PROCESS Explanation. — Since a fraction may be regarded

as an expression of unexecuted division, by perform- ^^~r 0 _ ^ _j_ ^ ing the division indicated the fraction is changed into ^ ^ the form of a mixed number.

2. Reduce «^ - ^ ^'^ - ^ + ^ to a mixed number.

Solution, a^- 3a^ - « + 1 ^ ^ _ 3 _^ - « + 1 ^ ^ _ 3 _ g^ll.

The division should be continued until the remainder is lower in degree than the denominator or no longer contains the denominator.

Reduce to an integral or a mixed expression :

3.

2

ah -{-d

^ xy + m

y

AiX^ -\-x

^ abc + 2 ah ^^^ a^ - a; - 15 ah ' ■ a;-4

8.

:x? + 2x + l

. 12.

X

x^—Qx^^ x-1

6.

15.

16.

2x

ftV — ax^ — x — l ax

x-2

a^j^^x' + ^x-^

x-\-2

g-^- 7. -1^-4x4- 40 x^-Z

52 a + h

x-y

19.

ic^ — 6 ic?/ -f 4 1/'^

2 a;?/

20. a;'-a?y-3y'^-g. a;H-2/

21. 22.

a;-3 a3^3^25_q^2_^^5 a^ + ft

134 FRACTIONS

/SO. • /&5. •

a^ + 62 x + y

.. 4.x^-^22x + 21 „^ m^-2m'n-3mri'-2mn

Z'k. — • AXi, — — — •

Zx-\-4: w? — mn

199. To reduce dissimilar fractions to similar fractions.

200. Fractions that have the same denominator are called similar fractions.

201. Fractions that have different denominators are called dissimilar fractions.

202. Principle. — The lowest common denominator of ttvo or more fractions is the lowest common multiple of their denominators.

The abbreviation L. C. D. is used instead of lowest common denominator.

EXERCISES [Additional exercises are given on page 465.]

to fractions having their low-

Qab

Explanation. — Since the L. C. D. of the given fractions is the lowest common multiple of their denominators (Prin.), the lowest common multiple of their denominators must be found. It is 6 ahc.

To reduce the fractions to equivalent fractions having the common denomi- nator 6a6c, the terms of each fraction (§ 195) must be multiplied by the quo- tient of 6 dbc divided by the denominator of that fraction.

E-ULE. — Find the lowest common multiple of the denominators of the fractions for the lowest common denominator.

Divide this denominator by the denominator of the first fraction^ and multiply the terms of the fraction by the quotient.

Proceed in a similar manner with each of the other fractions.

All fractions should first be reduced to lowest terms.

203.	1. Reduce	'' ai 3 6c
est common denominator.
	PROCESS
^ a 3 be	a x2a 3bcx2a	2 a'' 6abo
c Qab	ex c Gab xc	c^ (jabc

FRACTIONS 136

2. Eeduce 2 m and — ^t_^ to fractions having their L. C. D.

m — 7i

Suggestion. — First write 2 m as a fraction with the denominator 1. Reduce to similar fractions having their L. C. D. :

3. r and -f' 7. , 2, - — - —

2 5 a m-\-n

4. — and ox, 8.

56 " .a^-1' x+1' x-1

a?h -, a6^ ^

3-6 3

10.

a^ a	2a
a^_16' 0? + ^'	4.-0?
4.a 3b	1

x'y^' Q^f' xY ' a-b' b+a d' -b^

111

11.

12.

ar^ + 7a' + 10' x' + x-^' x' + 4.x-5 a-\-5 a — 2 a-fl

a2_4a + 3' a2-8a + 15' a?-Qa-{-6

ADDITION AND SUBTRACTION OF FRACTIONS

204. It has been learned in arithmetic that only similar frac- tions may be united into one fraction by addition or subtraction.

The method of addmg and subtracting similar fractions is much the same in algebra as in arithmetic. In algebra, how- ever, subtraction of fractions practically reduces to addition of fractions, for every fraction to be subtracted is really added with its sign changed (§ 56, Prin.).

The usual method of changing the sign of a fraction, in such cases, is to change the sign of its numerator (§ 189, Prin. 2).

Thus, g + ^_^+...=" + ^-''+-. That is,

XXX X

205. Principle. — If fractions have a common denominator , their sum is the sum of their numerators divided by the common

I denominator.

136 FRACTIONS

EXERCISES

[Additional exercises are given on page 465.]

206. 1. Add ?^, ^, and ||. 4 ' 10 12

PROCESS

4 "*" 10 12 ~ 60 60 60

_ 87 a; + 25 ?/ ' 60 *

Explanation. — Since the fractions are dissimilar, they must be made similar before they can be united into one term. The L. C. D. = 60.

Then, 3x^45^ 7^^42^ ^^^ 6y^25y

4 60 ' 10 60 ' 12 60

The sum = ^ + ^ + ^ = 45 a; + 42 x + 25 y ^ 87 a; + 25 y 60 60 60 60 60

2. Subtract ^^ from ^-^^ + ^. 7 8 4

Solution

6a; — Ix X— 2_ 35 a;— 7 .14 a; 8a;- 16 8 4 7 ~ 56 56 56

^ 35 X - 7 + 14 X - (8 X - 16) 56

_35x-7 + 14x-8x + 16_41x + 9 56 ~ 56 *

Suggestion. — When a fraction is preceded by the sign — , it is well for the beginner to inclose the numerator in a parenthesis, if it is a poly- nomial, as shown above.

EuLE. — Reduce the fractions to similar fractions having their lowest common denominator.

Change the signs of all the terms of the numerators of fractions preceded by the sign — , then find the sum of all the numerators, and write it over the common denominator.

Reduce the resulting fraction to its lowest terms, if necessary.

	FRACTIONS
Add:	Subtract :
'■¥-¥■	'•T^--x-
'1f-o¥-	8. ^from-|.
^ 2a -,3a 5. — and - — • 36 26	9. -2from§. 3 3^
6. -^ and ~^. 7x 3x	10. «t*from«-* 3 J

137

Simplify :

11 2a;4-l . a;-2 a;-3 5 — x

12 a?-2 a;-4 2-3a; 2g;4-l

6 9 4 12 '

13 ^ — ^ _ ^ — 2 _ 4 a; — 3 1 — a;

" 3 18 27 6 *

''• -5- + -2 6--^-'

-^ a;H-3 a; — 2 , a; — 4 a; + 3

4 5 ^ 10 6

16 l-^ct 2a-l 2a-a2-f-l

17.

5 4 8

34-a;-a;^ l-a^ + a^g l-2a;-2a;g 4 6 3 '

18. Eeduce ^ "^ „ — 2 to a fraction, a^ — 6^

Solution

ba^+b"^ 2 ^ 5 g^ 4. &2 _ 2 (q2 _ 52) a2 - 62 1 flj2 _ 52

_5q2-|-52_2a2 4.2 62

ff2 - 6-2 ^ 3 (a2 + 62)

138 FRACTIONS

Reduce the following mixed expressions to fractions :

19. a + l- 23. g-^'-^^

20. x-y. 24. a-"*-^"^

2 --- ^ 2

« — c^ . (? ^- . ar

21. hSc. 25. a + a;

c a — a;

22. l^-4a;. 26. a'-ah + h^ ^'

3 a + 6

Perform the additions and subtractions indicated :

37. 4 + i + J_^_2. X 1 -i-x

38. 2•a-36-i^!±ll^ 2a + 36

o« o o 8 a^ — 4 a;2

39. 3 a — 2ic —

3a + 2a;

1 12

27.	a — b b — G ab ' be
28.	a-i-b a — b a — b a-\-b
29.	b—c a—c be ac
30.	a-\-b a — b a—b a+b
31.	x + y ^ + 2/^ x-y
32.	X x — 2
x—2 x+2

40.

X— 1 x-\-l x^

41. -1 ^+ 2«

a + 6 a — 6 a^ — W 42 ^ + a; , g — a; 4 aa;

a~x a-^x a^ — x^ 33. 'l^ + ^-S + l. 43. ^ + ^ 4- ^-^

5 2 6 a2_^a + l a^-a + l

34. a;4-l + ^!lI-5. 44. 3a; + A_/'2a;+ ^

("23;

a; — 1 ax \ ax

35. m-"^' + "' + ». 45. «-& ^o^ + V «

wi — n 2 (a H- &) a^ — 6^ a — ^

3g ^_axj-bx±oh a4-33 6 10

x" * " a2-9 a-3 a + 3

FRACTIONS

139

47.

a-2

a-2 Suggestion. — By Prin. 1, § 189

+

a+2 4-a2

3 -3

48.

49.

a-f 1

+

4 - a2 a2 - 4

4a

a-1 a+1 1

5a; + 2 2 4 "^aj-2

a;2._4 aj-2 2-a; 50. ^(^+^)-^^^-^' + 4a.

51. 52.

a — x 1

» — a

Q I J

a^ + 8 8

5(a;-3) 2(a; + 2) a;-l

a^-x-2 x'-\-4:X-\-S 6-x-a^'

53. Simplify ^^±^±1-1 +-T^^

Solution By reducing the first fraction to a mixed number,

x^ + x-\-l

X2 — X + 1

-1 +

2x

x^ -{-x+l

= 1 +

2x

a;2 — ic + 1

2a;

-1 +

2a;

2a;

a;2 + a; + 1

_ 4 a;(a;2 + 1)

a;2-x + l a;2 + a; + l a;* + a;2 + l

Suggestion. — Frequently, by reducing one or more of the given frac- tions to mixed numbers, the integers cancel each other and the numerators are simplified.

54.

M

55.

56.

57.

a^-\-2ab + b'	1 -t '	2ab
a'-hb^	a	'-b'
a'-\-3ab-\-2b^	0?-	13 b'
a'^Sab -46^	d'-	16 62
x+1 x-1	x + 2	x-2
x-1 OJ + l	x-2	x-\-2
x+3 x-3	x-\-4c	X-4:

3 x-^3 x-4. a; + 4

140 FRACTIONS

X a 2 ah 4tab^

58.

a — b a-\-b d^ -{- b^ a^-{-b^

Suggestion. — Combine the first two fractions, then the result and the third fraction, then this result and the fourth fraction.

a-i-b a — b 4 a& 8 a6^

a-b a + b a^ + b^ a' + b*

60. ^ L.__ll_+ 263

T . T <) . T O '

a-b a-i-b a^-{-b^ a'^-^b*

-, a + ic,a^ + ic^ a — x a^ — a^ 4: a^x -{• 4: aoc^ a — x a^ — x^ a + x a^-\-ai? a'^ — x'^

x-\-y y 4-g , ^-\-x

(jj-z)(z-x) (x-z){x-y) (y-x)(z-y) Solution

62.

Sum = ^+i^ . + y±^ + ?-±^

Of - «)(^ - x) iz -x)(x- y) (X - y){y - z)

63.

64.

65.

66.

67.

		= =0.
	1	1^1^
(6-	-c)(a-	-c) ' (c-a)(a-b) {b-a)(b-c)
	a + 1	, 6+1 c+1
(«■	-b)(a-	-c) (6-c)(6-a) (a-c)(6-c)
	c'ab	6^ca a^bc
(c-	-a)(6-	■c) (6_a)(6-c) (a-6)(a-c)
	b-e	c — a a-\-b
(b-	-a){a-	-c) (6_c)(a-6) (a-c)(6-c)
	c + a	6+c , a+b

(a— 6)(6 — c) (c — a)(6 — a) (c — b)(a — c)

FRACTIONS 141

MULTIPLICATION OF FRACTIONS

207. Fractions are multiplied in algebra just as they are in arithmetic.

Thus, M=r4-

4 2 4x2

In general, - X - =— ,. That is,

0 a od

Principle. — The product of two or more fractions is equal to the product of their numerators divided by the product of their denominators.

EXERCISES

[Additional exercises are given on page 466.]

208. 1. Multiply ^^ by a;2_25.

x-B

Solution. ^-=1^ .^^'"^^= (x - 6^ = x^ - 10 x + 26.

-aj-^FiJ" 1

2. Multiply ^±1 by 1 4- "^

x + 2 x + 1

Solution

\x + 2)\ x + l) x + 2\a; + l x + l)

x + S -arT^ *x+ 1

x+l

General Suggestions. — 1. Any integer may be written with the de- nominator 1.

2. After finding the product of the numerators and the product of the denominators, the resulting fraction may be reduced to lowest terms, in many cases, by canceling common factors from numerator and denomi- nator. It is, however, more convenient to remove the common factors before performing the multiplications.

3. Generally, mixed numbers should be reduced to fractions.

142 FRACTIONS

Simplify :

3.^x2. l.o^x!^- 11. ISik^pL z xy^ 4:xy 6a^

4. aftx^- 8. Axa^ft. 12. if^X^.

a 2a^ 10 c^ a^

5. 2zx^- 9. cd3x-^. 13. '|^x^«^

z"^ c'd'^ 2ac 10?/2

15. 5^ X^'. 23. ^A^X^X^. ^

16. ^£?X^- 24. ^X^^X-^-

ad z^ a¥ y^z &x

a'"^^ 6"*"^^ „- a; — 4 4 — a^

' 6^^"^* ' x + 2 ' 16-a^"

18. -^x-^. 26. ^/ ,25- 10 a:,

a + & « — ^ 20 — 8 a? x-^/

19 a^-6^^ a-6 g? l-6a; + 5a^ 2 - a;

a + 6 a2 + 62* " a:2_3^^_2 'l_a;*

20 cr^x-^^ 28 '^' + ^ ^'+2a + 4

* 4x a-+i62n 'a^.g ^2_2a4.4

4 mn 15 6a; ^^ a'^ + 3a;-f2 _ ar^— 6a; + 5

'6xy IGm^' * a;--3a;-10 V + 8a; + 7

22 .?J^x-i^. 30 (liz:^ . _&_ . («dL?0!.

* 12 6?/ X? ' a+6 a2-a6 a--62

31.

32. 33. 34.

p-^2 do? -21 ^ 4 a;-3 " 2y _8 ' px-^Zp p* — q* p — q p^

{p — qf p^ -{-pq p- + g2

a^ + o^^a?^ H- ^ , a;

a^ — aa? a^ — ax-^-a?

a^ + 4 a^ + a H- 1

a^ + a' + 1 «' + 2 a 4- 2

FRACTIONS 148

35 a''-hab-^2a-\-2b ^ x^-2xy ax — 2 ay-^2x—4:y (a + by '

36 '^^-^ . 2a ^ 4 g;'^ - y^ 2x + y' 4.a}-ab' 4

37 <^^ — ^ . « + a; ^ g'^ — «a; 4- a;^ a^^a?' a^-x"' (a + xf

3 a; 2/^ *

38. (l—i\A^i^.1±A

\^x J 4:-9x^ 4y

1^ x^ + 6x+5j\ x'-h7x-^12J

DIVISION OF FRACTIONS

209. The reciprocal of a number is 1 divided by the number.

The reciprocal of 5 is - ; of &, - ; of (a + &) , — i— . 5 b a + b

210. Since - is contained d times in 1, (c times - ) is con- d \ dj

tained - of d times, or - times, in 1 ; that is, c c

■\_^<±_d ' d c'

Principle. — Tlie reciprocal of a fraction is the fraction inverted.

211. Since l-f-- = -, and a=l • a,

d c

it follows that a-i-- = - - a ov a --.

d c c

Principle. — Dividing by a fraction is equivalent to multi- plying by its reciprocal.

144 FRACTIONS

EXERCISES [Additional exercises are given on page 466. ] 212. Divide:

1. 1 by J.	3.1by|.	5.	Iby	^^ 7. Iby"-^ X -^ a+6
2. Iby?.	^-.ibyj.	6.	Iby	-.V. 8. lby«-^ m "^ 0+3
Write the	reciprocal of:
«•-:•	11. 3».	13.	3a; 5y	15. '• + *.
10. ™. n	12. 1. 3m	14.	4 a6*	16. «-». 0—1/
17. Divide ^'-f by ^ + 2. x^-1 '' x — 1
Solution.	a;2 _ 4 X + 2 _ CiM^)(a; - 2) x2_i • a;_i (a;+i)(^B-=rr)	jt-<r rK-2 j^-rz' « + 1
Simplify :
io 5mn '	LO m^n		a^ + 1	/ . x''-\-xy-hy^

18. !iiii!f^±lii^\ 24.

6 6a; 3 ax^ x^ — y^ ^ — y

19. 120^^1^. 25.

25 ac 15c2 a'-2ah + ¥ a'-ab \

20. ^^^^ : abx. 26. ^^^ + ^^ . »^^^^ -mx\ ■

7 ' m^a; — moi? m^a^ — a;^

21 -'"y-.v' ■ y' 27 fx^V\^fv'^l.\

{m + yf m'-y'' ' \ ' y)'\ ' ^)

22. (^^^.Z^. , 28. (t^1,y(<l^^ab\

23. (4a + 2)^2a + l. ^^ („ + c)^(^^

oa \ 1 +

FRACTIONS 145

3^ a« + 27 . a + 3 ^

a2 - 62 _ c2 + 2 6c ' a - & + c' Suggestion. — Reduce the dividend to a fraction,

»■ ('-S)K'-7-f)-

Complex Fractions

213. A fraction one or both of whose terms contains a frac- tion is called a complex fraction. It is simply an expression of unexecuted division.

EXERCISES

a I 214. 1. Simplify the expression — •

-

! 9

a Solution,

		a
►N,		b X y		,_x_ ' y	a b	X	X	bx
milne's	STAND	. ALG. -	-10

146 FRACTIONS

Simplify :

x_±y 2 + —

(lb . 46

x^

	^-f
	aV
3	c
	6 + ^ a
4	X

8	"-i
	1-
9.	-f
	1-

a!_l g + y ic + y

7. ?. 10. 2^

._£ 1+1 1-i

m a; 3( a;

11. Simplify the expression ^ — ^

^ + - + 1

y y

Solution. — On multiplying the numerator and denominator of the fraction by y^, which is the L. C. D. of the fractional parts of the nu- merator and denominator, the expression becomes

Simplify :

x^-l a^ + /

x^ — xy + y^

12. — — ^. 14.

xy

x-{-l x^ — xy-\- y^

xy

1+ 1

13. ^JLtf. 16. " + ^

X y + z a-\-l

a;2 + xy + y2
16	^+y'-x 2y
	y X
	1
17.	1-a a a-1

FRACTIONS 147

Simplify :

a;-2 + -i-- 6a-l--

18 ^

x-2

14 4

- + - + — X a^ a^

X or

20.	a
	2a-l
	3a
21,	.-5 ,^24 2 X
	9-3a;

22. -iLtl_+^±l+lz:f,

23.

l + x 1 — a; l + a?

3a;yg x y z

yz + zx + xy 1_^1_^1

1^2 9 a^-\-(a + b)x-\-ab

24 a; + y x-y 3x-y x^ - (a + b)x -{- ab

-Sy ' ' a^-b'

y^—9ix^ 01?— a?

26. "^ ^ + ^

1 \_ ' ^ ^ b'-^(?-a'

a b-{-c 2 be

^ (x^ + f-zy 27. ^/

2/2 4

215. A complex fraction of the form is called a

continued fraction. b -{■ ^

d +

148 FRACTIONS

28. Simplify 1

1+ ^

1 + ^-

X

Solution

X-\-l JC+ 1

1+ ^ 1 1 ^ x + l + x 2a; + l 1+1 x+1

X

Suggestion. — In the above exercise, the part first simplified is the last

complex part • it is simplified by multiplying both terms by the last

^ + ~ denominator, x.

Every continued fraction may be simplified by successively simplifying its last complex part by multiplying both terms by the last denominator.

Simplify :

29. ^ 33 ^-^

x+ 1__- x-2 ^

3-x x-2

30. L^ — . 34 ^

a+-i- a+l+ ^

a-h- a + 1--

a a

31. =— 35. 1 +

1 + C+ 2c

2 — x c

32. ^ 36. a +

ic + l a — 1

a^ + -^ 2 + i

«— 1 a

REVIEW 149

REVIEW 216. 1. When is a fraction in its lowest terms ?

2. Define factoring ; prime factors ; reciprocal of a number.

3. When may the factor theorem be used to advantage ?

4. Define highest common factor ; lowest common multiple.

5. Give a rule for finding the lowest common multiple of two or more expressions.

6. Show that the product of o? — h^ and a^ — 6^ is equal to the product of their highest common factor and lowest common multiple.

7. Distinguish between an integral and a fractional alge- braic expression.

8. By what must a fraction be multiplied in order to obtain the lowest possible integral expression ?

9. Why must a broader definition of fraction be given in algebra than in arithmetic ? Give the algebraic definition.

10. Under what conditions may the sign of the numerator or of the denominator of a fraction be changed ?

5 -5

11. Show that

9-a^ a?

12. Show that the sum of a and b divided by the sum of their reciprocals equals ah.

13. Distinguish between a complex fraction and a continued fraction.

14. Show that dividing - by a; is the same as multiplying

X. 1 y

- by -. y X

Simplify :

,v.r»+4 ^5-2n 03x+l

15. ^. 16. ^. 17. ^

a;— 4 a^ -21

150

REVIEW

Eeduce to lowest terms :

18 5 0)^-11 a; -12

10a;2 + 23a; + 12'

a^ + 3a;2 + a;4-3' Simplify : 22.

20.

21.

a^-\-b^

a^ + aW + b*

cx—cd ^

cx-\-3x — 3d — cd

0-

y

y-x

2y-l 22/ + 1 1-42/2

23. — ^? ^ ^ +-J: lr^\

4(l-a)2 V8(l-a) 8(a + l) 4(a + l)y

24.

25.

26.

27.

28.

29.

30.

1 +

i)(

m^ + m — 2

m— lyv m'^ + m a + ft a-& 4.ab \a'^-b^

a-b a-\-b a^-^-by Sb^

1 +

?/2 2/ icy

y.

a. + l + - + i

a + l

a; a;V

31. 1

a-{-b_^ a^_b^\fa±b _ a^ + b^ a — b a^ — b^

a?-3xy-2f^-^^\J3x-Qy--^\ x-\-3yJ \ x + 3yj

nj \n m J

3-3a;\ (^ + 1)V

m — 3 yi\ /-. 4?! m-\-n J\. m +

2a; + 5ar2

2.-?

2(a; + l)2

32. fl \ ^ Y "^ 2x^ + 2ax-a^\

\ a — x)\x-\-a a;'^ + 3 aa? + 2 a?)

REVIEW 161

Expand by inspection:

Simplify :

87. p-j-.

« ar

88. V ^ J \n mj

41.	6a 4
	1
	a2 + 3a4-2
42.	a2 + 2a + l
a^4_7a+12

a' + 5 a 4- 4

+

m** —

a , 1 ^ , 2mn

-\-n^\ m-\-n)

+ 7-^7Tl M

(a + 1)' {a + iy ^2_„^^^^2 40. ?^^ 44. L^

1-

1--^ 4- ^

1 — a; 6— a;

' \x — y x + yj ' \2{x — y) 2x + 2yj

Var a; a a^J\xr x ay 46. ^ 3. ^^. ^-

/ a;y a^y^ y ^^.

a:«y^^a:22^2>^^^ _4_^ajy + l a;?/ ar^i/^

SIMPLE EQUATIONS

ONE UNKNOWN NUMBER

217. The student has already learned what an equation is (§ 4), and he has solved many equations and problems. In this chapter and the next, however, he will find a more com- plete and comprehensive treatment of the subject, extended to some kinds of equations that are new to him.

218. An equation all of whose known numbers are expressed by figures is called a numerical equation. |

219. An equation one or more of whose known numbers is expressed by letters is called a literal equation.

220. An equation that does not involve an unknown number in any denominator is called an integral equation. J

a; + 5 = 8 and j- 6 = 8 are integral equations. Though the second ■'.

o

equation contains a fraction, the unknown number x does not appear in the denominator.

221. An equation that involves an unknown number in any denominator is called a fractional equation.

jK + 6 = - and — :^ = 7 are fractional equations. X X — 1

222. An equation whose members are identical, or such that they may be reduced to the same form, is called an identical equation, or an identity.

05 + 6 = a + & and a'^ — h^ = (a -{- b) (a — b) are identical equations. An equation whose members are numerical is evidently an identical equation.

10 = 6 + 4 and 8 x 2 = 6 +12 — 2 are identical equations.

162

SIMPLE EQUATIONS 153

A literal equation that is true for all values of the letters involved is an identical equation, or an identity.

(x + yy = x^ + 2 xy+ y'^ is an identity, because it is true for all values of X and y.

I. An equation that is true for only certain values of its letters is called an equation of condition.

An equation of condition is usually termed simply an equa- tion.

X + 4 = 10 is an equation of condition, because it is true only when the value of X is 6. x^ = 9 is an equation of condition, because it is true only when the value of x is + 3 or — 3.

224. When an equation is reduced to an identity by the substitution of certain known numbers for the unknown num- bers, the equation is said to be satisfied.

When a; = 2, the equation 3 a; + 4 = 10 becomes 6 -f 4 = 10, an identity; consequently, the equation is satisfied.

225. Any number that satisfies an equation is called a root of the equation.

2 is a root of the equation 3 x + 4 = 10.

226. Finding the roots of an equation is called solving the equation.

227. An integral equation that involves only the first power of one unknown number in any term when the similar terms have been united is called a simple equation, or an equation of the first degree.

3 X + 4 = 10 and x + 2y — z=:S are simple equations.

For reasons that will be apparent later on, simple equations are sometimes called linear equations.

228. Two equations that have the same roots, each equation having all the roots of the other, are called equivalent equations.

x + S = 7 and 2 x = 8 are equivalent equations, each being satisfied for X = 4 and for no other value of x.

164 SIMPLE EQUATIONS

229. By the axioms in § 68, if the members of an equation are equally increased or diminished or are multiplied or divided by the same or equal numbers, the two resulting num- bers are equal and form an equation. But it does not neces- sarily follow that the equation so formed is equivalent to the given equation.

For example, if both members of the equation oj -I- 2 = 5, whose only root is j; = 3, are multiplied by a; — 1, the resulting numbers, (a; + 2)(a! — 1) and 6(x — 1), are equal and form an equation,

(x + 2)(x-l) = 5(ic-l), which is not equivalent to the given equation, since it is satisfied by « = 1 as well as by x = 3 ; that is, the root x = \ has been introduced.

In applying axioms to the solution of equations vre endeavor to change to equivalent equations, each simpler than the pre- ceding, until an equation is obtained having the unknown number in one member and the known numbers in the other.

The following principles serve to guard the student against introducing or removing roots without accounting for them :

230. Principles. — 1. If the same expression is added to or subtracted from both members of an equation^ the resulting equa- tion is equivalent to tlie given equation.

2. If both members of an equation are multiplied or divided by the same known number, except zero, the resulting equation is equivalent to the given equation. ^

3. If both members of an integral equation are multiplied by the same unknown integral expression, the resulting equation has all the roots of the given equation and also the roots of the equa- tion formed by placing the multiplier equal to zero.

It follows from Prin, 3 that it is not allowable to remove from both members of an equation a factor that involves the unknown number^ unless the factor is placed equal to zero and the root of this equation preserved.

Thus, if a; — 2 is removed from both members of the equation (a; - 2)(a; + 4) = ?(« - 2), the resulting equation x + 4 = 7 has only the root x = 3 ; consequently, the root of x — 2 = 0, removed by divid- ing by the factor x— 2, should be preserved.

SIMPLE EQUATIONS 166

Clearing Equations of Fractions

231. The process of changing an equation containing frac- tions to an equation without fractions is called clearing the equation of fractions.

EXERCISES

232. 1. Solve the equation ^^ = 6 - f •

Explanation. — Since the first fraction PKOCESS ^jj become an integer if the members of the

jg _ g a; equation are multiplied by 2 or a multiple of

2 ^^ S ^' ^^^ since the second fraction will become

an integer if the members of the equation 3 a? — 24 = 36 — 2 a? ^j-e multiplied by 3 or a multiple of 3, the

5 if = 60 equation may be cleared of fractions in a

a; = 12 single operation by multiplying both mem-

bers by some common multiple of 2 and 3, as 6, or 12, or 18, etc. It is usually best to use the L.C.M. of the denomi- nators, which in this case is 6.

Then, multiplying both members by 6, transposing terms, and dividing by the coeflBcient of a;, we obtain a; = 12.

Verification. — When 12 is substituted for x, the given equation be- comes 2 = 2, an identity ; consequently, § 224, the equation is satisfied by X = 12.

2. Solve the equation ^^ - ^^ = - - ^^f^ . ^ 2 3 3 4

Solution x-\ a;-2 2 a;-3

2 3 3 4

Multiplying both members of the equation by the L.C.M. of the de- nominators, which in this case is 12, we obtain

6 (x - 1) - 4 (x - 2) = 8 - 3 (x - 3).

Expanding, 6x-6-4x + 8 = 8-3x + 9.

Transposing, etc., 5x=16.

Hence, x = 3.

Verification. — When x = 3, the given equation becomes f = f , an identity ; consequently, the equation is satisfied by x = 3.

156 SIMPLE EQUATIONS

To clear equations of fractions :

E/ULE. — Multiply both members of the equation by the lowest common multiple of the denominators.

1. To simplify the work and to avoid introducing roots, reduce all frac- tions to their lowest terms and unite fractions that have like denomi- nators before clearing.

2. If a fraction is negative, the signs of all the terms of the numerator must be changed when the denominator is removed.

3. Roots are sometimes introduced in clearing of fractions. Such roots may be discovered by verification. Those which do not satisfy the given equation are not roots of it, and should be rejected

Solve, and verify each result :

3. 2x4-- = — • 7. --f- = — .

^3 3 2^6 3

• 4. - + 10 = 13. 8. 71-1^ = -.

5.

4 ■ "14

X

6

6. 3a;

x = 26.				9. | + f=24.
!=''■				10. 2-_5^£ 3 6 4
11.^+	X	-!+	3x 10	-12='-

25 a; 5 a; 2a; 5a;_9

2 a; 7a;5a; a;_4 ' 3 T"^18 24~9'

3a; 7a; x 9a;_l ' 4 16 2 16 ~ 8'

,^ 15 a; , 5 a; 11a; , 19 a; «

"•— +"6— 3"+ir=^-

16 5^ 4.^ _i^' 4-^ = 1

* 16 25 9 6""9*

SIMPLE EQUATIONS 167

Sx 7a;^lla; 8a; 3 * 4 12 36 9 2*

18 y-1 ■ y-2 y-3_5y-l

19. -^ __ + ___io.

20. I^±2_12-^^M:2^g^

21 ^ — 3 ■ ^4-5 ^ + ^^4 * 7 "^ 3 6

„-3^-5 7^-13 Q ^4-3

<&>*. ' ' : — O •

4 6 2

-(._3^1) = 3^_(»_±l + 5)

-- 5a; + 2 / 3a;-l\ 3aj + 19 /a; + l 23 -^ -

24. 1.07a; + .32 = .15ic + 10.12 + .675a;. Suggestion. — Clear of decimal iEractions by multiplying by 1000.

25. .604 a; -3.16 -.7854 a; + 7.695 = 0.

26. 3.1416 X - 15.5625 + .0216 x = .2535.

27 -^^ -1^ .la; .4a; _ .3 7 4 T" T"~i4*

__ n + 4 , 2-2n n + 1 10

^^•^3~"^-:6~=r2-"T

gg 9a; + 5 8a;-7^36a; + 15 lOjr 14 6a;+2 56 14*

Suggestion. — The equation may be written

9x _6^ 8a;-7 _ 36 a; :

14 14 6x + 2 56 56 ' 66

130 ^ + A _L 8a^-7 _ 36jK , 15 , 41^

168 SIMPLE EQUATIONS

^^ 3»-2 , 3a;-21 6a;-22 30. =

2x-5 5 10

^- 4a; + 3_8a; + 19 7a;-29 31» "^ —

32.

5

8 a; + 19

18

2p-4:

33.

34.

5ic-12

6p + l 2j9-4 ^2p-l 15 7p-13 5 *

lOg + 17 5g-2^12^-l 18 9 llg-8

6rH-3 3r-l ^2r-9 15 5r-25 5

35. Solve the equation + = H •

x—2 x—7 x—6 x—3

Solution. — It will be observed that if the fractions in each member were connected by the sign — , and if the terms of each member were united, the numerators of the resulting fractions would be simple. The fractions can be made to meet this condition by transposing one fraction in each member.

Consequently, it is sometimes expedient to defer clearing of fractions.

x—1 X— 2 X — S x — 6

Transposing, Uniting terms,

x-2 x-3 x-6 x-7 -1 -1

Since the fractions are equal and their numerators are equal, their denominators must be equal.

Then, a;2 - 5 a; + 6 = a;2 - 13 a; + 42.

.-. X = ih

36. x~l_^^_^^x-5^x-S

37.

3S.

a; — 2 x — S x—6 a; — 4

a; — 3 a;— 7_a; — 6 .a; — 4

a; — 4 a; -8 a? — 7 x — 5

v-\-2 v-\-S_v-\-ov + 6

v + 1 v-^2~v + 4: v-\-5

SIMPLE EQUATIONS

169

39. g + 1 , 8 + 6^8 + 2 ^ 8 + 5

s+2 s+7 s+3 s+6

40.

41.

a^ + 1 a^-1

a: + l a^-1

-f2a;.

a^ + 2 of

1^ o

a; + 1 a; - 1 42. r(2_.)_I(3-2r) = ?:±12,

43.

44.

45.

46.

47.

3n — 4 /4n,7i + 2

(a; -3)^ (a; + 4y

^0•

3 _7j.-c c/'e 2 ~ 3 "^2

24 60 ~5"'

48. l.-2(V^-3)=4-|g+l).

49 (2g + l)^ (4a;-iy^l5 3(4x + l) .05 .2 .08 .4

17+:

1 + 18 21_,

50.

51.

+

X X

+

100 5 X 3

9 ' 15 2x

i(a;-4) 4a;-16_3 I 6 ~5

+ 5

L60 SIMPLE EQUATIONS

Literal Equations

233. 1. Solve the equation ^^^' = ^^=-^ for as.

a b

Solution

a b

Clearing of fractions, bx — b^ = ax — aK

Transposing, etc., - ax — bx = a^ — 6*.

(a - b)x = a^ - 68. Dividing by (a — 6), x=a^ -{- ab ■\- h^.

Verification. — When a = 2and& = l,a; = a2+a&+&2_4^2 + l=7^

7—1 7—4

and the given equation becomes = , or 3 = 3, an identity

Ji 1

consequently, the equation is satisfied for x = a^ + ab + b^. i

Solve for x, and verify each result : ^

^ (?—x ^ n^^l^ ^ x — a 2a;^g 6&^

na; cflj c 6 a a

3 i_^^^_i9. 8 «' . 6^^a + & 3(a+&)^''

a; a6 a6a; ' 6ic aa; ad x

' afe^"^^"" ~~^' • 2bx ~ 2bx^~x

^ X x-\-2b a Q ^^ 2x — a x — a ^

5. — = - — c>. 10. = 1.

0 a b x — a x-\-a

6 «^ — 2^^1__^ — 3c_ a; — 2a a? _a^ 4- 6^^

ca? a; a6a; * a 6 a6

12. 6a; + 18/'l-|^ = a(a;-a).

13. 5(2a;-9c-146) = c(c-a;).

14. a(a; - a — 2 6) + b(x —b)-\- c(x + c) = 0.

SIMPLE EQUATIONS 161

15. (a — x){x —b)-\-{a-\- x){x —b) — {a- h)\

16. {a — b){x — c) — (h — c)(x — a) = (c — a){x — 6). a— b-i-c b — a-\- c

17

x-{- a x — a

18. --I_4.-L— _1_ = 0.

19. 20.

a(b — x) b(c — x) a(c — x) 07 — 1 a - 1 Qi? — a?

a-1 x-1 {a-l)(x-l)

1 2 mn m x — n

m-\-n {m-\- nf (m -\- ny (m + ny 2 J x-\-a x + c x-^b ^a b <^ _^i b a c b c a

22. H =a^-}-b^ + c'-\-2ab.

a-{-b -{-c a-\- b — c

2„ a-\-x 2x x\x—a) _1 a a -\- X a(a'^ — a?) 3 Suggestion. — Simplify as much as possible before clearing.

_ . x^-ax-bx + ab xf^ — 2 bx -\- 2 b^ &

^4. — .

X — a x — b x — c

Algebraic Representation

14. 1. What part of 8 is 5? of m — w is p ?

2. From what number must 3 be subtracted to produce 7 ? to produce n ?

3. How much less is $f than $3? — dollars than m pilars? ''

Indicate the sum of 5 and 6 divided by 2, and that result multiplied by 4 ; the sum of I and m divided by 2, and that I result multiplied by n.

5. Indicate the product of s and (r — 1) divided by the nth. I k power of the sum of t and v.

6. A boy who had m marbles lost - of them. How many marbles had he left ? "

milne's stand, alg. 11

162 SIMPLE EQUATIONS

7. By what number must x be multiplied that the product shall be 2! ?

8. Indicate the expression for \ the product of g and the square of t.

9. Indicate the square of x, plus twice the product of x and y, plus the square of y, divided by the sum of x and y.

10. By what number must the sum of x and — 2/ be multi- plied to produce the square of x minus the square of ?/ ?

11. Indicate the result when the sum of a, b, and — c is to be divided by the square of the sum of a and b.

12. It is t miles from Albany to Utica. The Empire State Express runs s miles an hour. How long does it take this train to go from Albany to Utica ?

13. A cabinetmaker worked x days on two pieces of work. For one he received ^• dollars, and for the other iv dollars. What were his average earnings per day for that time ?

14. A train runs x miles an hour and an automobile x — y miles an hour. How much longer will it take the automobile to run s miles than the train ?

15. Indicate the result when b is added to the numerator and

subtracted from the denominator of the fraction -.

c

16. A farmer had - of his crop in one field, — in a second,

and - in a third. What part of his crop had he in these three fields?

17. A won m more games of tennis than B, and B won w

more games than C. How many more games did A win

than C?

1 1

18. A student spends — of his income for room rent, - for

m n

board, - for books, and - for clothing. If his income is x s r

dollars, how much has he left ?

SIMPLE EQUATIONS 163

Problems

[Additional problems are given on page 467.]

235. Review the general directions for solving problems given in § 77.

1. A grocer paid $ 8.50 for a molasses pump and 5 feet of tubing. He paid 12 times as much for the pump as for each foot of tubing. How much did the pump cost ? the tubing ?

Suggestion. — If we knew the cost of a foot of tubing, we could com- pute the cost of the pump. Therefore, let x represent the number of cents one foot of tubing cost.

2. Four wagons drew 38 logs from the woods, one wagon holding 2 logs more than each of the others. How many logs did each wagon hold ?

3. A man paid $ 300 for a horse, a harness, and a carriage. The carriage cost twice as much as the harness, and the horse as much as the harness and carriage together. Find the cost of each.

4. A shipment of 12,000 tons of coal arrived at Boston on 3 barges and 2 schooners. Each schooner held 3^ times as much as each barge. Find the capacity of a barge ; of a schooner.

5. The powder and the shell used in a twelve-inch gun weigh 1265 pounds. The powder weighs 15 pounds more than \ as much as the shell. Find the weight of each.

6. A merchant bought 62 barrels of flour, part at $ 4J per barrel, and the rest at % h\ per barrel. If he paid $ 320 for the flour, how many barrels of each grade did he buy ?

7. Three pails and 6 baskets contain 576 eggs. All the pails contain \ as many eggs as all the baskets. How many eggs are there in each pail ? in each basket ?

8. The sum of ^ of a number and 4 of it is 90. Find the number.

9. If the sum of ^ of a number and \ of it is subtracted from the number, the result is 25. Find the number.

10. The sumof ^of anumberand^ of it is a. Find the number.

11. If i of a number is multiplied by 3 and from this \ of the number is taken, the result is 24. Find the number.

164 SIMPLE EQUATIONS

12. The cost per mile of running a train was 14 ^ less with electrical equipment than with steam, or f as much. What was the cost per mile with electricity ?

13. A rectangle is 9 feet longer than it is wide. A square whose side is 3 feet longer than the width of the rectangle is equal to the rectangle in area. What are the dimensions of the rectangle ?

14. A field is twice as long as it is wide. By increasing its length 20 rods and its width 30 rods, the area will be in- creased 2200 square rods. What are its dimensions ?

15. An acre of wheat yielded 2000 pounds more of straw than of grain. The grain weighed .3 of the total weight. How many 60-pomid bushels of wheat were produced ?

16. A reporter saved -J of his weekly salary, or $ 1 more than was saved by an artist, whose salary was $ 5 greater, but who saved only -^ of it. How much did each earn ?

17. In a purse containing $1.45 there are I as many quarters as 5-cent pieces and ^ as many dimes as 5-cent pieces. How many coins are there of each kind ?

18. The value of a fraction is f and its denominator is 15 greater than its numerator. Find the fraction.

19. The value of a fraction is f and its numerator is 3 greater than half of its denominator. Find the fraction.

20. The numerator of a certain fraction is 8 less than the denominator. If each term of the fraction is decreased by 5, the resulting fraction equals ^. What is the fraction ?

21. The sum of ^ and ^ of a number multiplied by 4 equals 88. Find the number.

22. Separate 54 into two parts such that -^ of the difference between them is ^.

23. Separate m into two parts such that 1 of the difference

1 ^

between them is -. r

SIMPLE EQUATIONS 165

P

24. A can do a piece of work in 8 days. If B can do it in

10 days, in how many days can both working together do it ?

Solution Let a; = the required number of days.

Then, - = the part of the work both can do in 1 day,

\ = the part of the work A can do in 1 day, ^ = the part of the work B can do in 1 day;

.•• - = -H lOr — .

X 8 10' 40

Solving, X = Y-, or 4|, the required number of days.

25. A can do a piece of work in 10 days, B in 12 days, and C in 8 days. In how many days can all together do it ?

26. A can pave a walk in 6 days, and B in 8 days. How long will it take A to finish the job after both have worked 3 days ?

27. A can do a piece of work in 2^ days and B in 3^ days. In how many days can both do it ?

28. A can paint a barn in 12 days, A and B in 5 days, and B and C in 4 days. In how many days can all together do it ?

29. A and B can dig a ditch in 10 days, B and C can dig it in 6 days, and A and C in 7^ days. In what time can each man do the work ?

Suggestion. — Since A and B can dig ^jj of the ditch in 1 day, B and C 1 of it in 1 day, and A and C ^ of it in one day, xV + i + t% is twice the part they can all dig in 1 day.

30. A and B can load a car in 3 hours, B and C in 2i hours, and A and C in 2| hours. How long will it take each alone to load it ?

31. One pipe can fill a tank in 45 minutes and another can fill it in 55 minutes. How long will it take both to fill it ?

32. A tank can be filled by one pipe in a hours, by a second pipe in c hours, and emptied by a third in b hours. If all are open, how long will it take to fill the tank ?

166 SIMPLE EQUATIONS

33. The units' digit of a two-digit number exceeds the tens' digit by 5. If the number increased by 63 is divided by the sum of its digits, the quotient is 10. Find the number.

'	Solution
Let	X — the digit in tens' place.
Then,	a; + 5 = the digit in units' place,
and	10 a; + (x + 5) = the number ;
	. 10k + (« + 5)+63
	2x + 5 ~^"'
whence,	x = 2,
and	a; + 5 = 7.

Therefore, the number is 27.

34. The tens' digit of a two-digit number is 3 times the units' digit. If the number diminished by 33 is divided by the difference of the digits, the quotient is 10. Find the number.

35. The tens' digit of a two-digit number is \ of the units' digit. If the number increased by 27 is divided by the sum of its digits, the quotient is 6J. Find the number. ^1

36. In a number of two digits, the tens' digit is 3 more than the units' digit. If the number less 6 is divided by the sum of its digits, the quotient is 6. Find the number.

37. In a two-digit number, the tens' digit is 5 more than the units' digit. If the digits are reversed, the number thus formed is I of the original number. Find the number.

38. In a two-digit number, the units' digit is 3 more than the tens' digit. If the number with digits reversed is multi- plied by 8, the result is 14 times the original number. Find the number. j

39. The sum of the digits of a two-digit number is 11. 63 j added to the number reverses the digits. Find the number. ]

40. The &um of the digits of a two-digit number is 5. If the number is multiplied by 3 and 1 is taken from the result, the digits are reversed. Find the number.

SIMPLE EQUATIONS 167

41. Mr. Eeynolds invested $800, a part at 6%, the rest at 5%. The total annual interest was $45. Find how much money he invested at each rate.

Suggestion. — Let x = the number of dollars invested at 6%. Then, 800 — x = the number of dollars invested at 5 % ;

-'•lU^ + Th (800 -X) =46.

42. A man has | of his property invested at 4%, ^ at 3%, and the remainder at 2%. How much is his property valued at, if his annual income is $ 860 ?

43. A man put out $ 4330 in two investments. On one of them he gained 12 % and on the other he lost 5 %. If his net gain was $251, what was the amount of each investment ?

44. Mr. Bailey lent some money at 4 % interest, but received $48 less interest on it annually than Mr. Day, who had lent J as much at 6 %. How much did each man lend?

45. A man paid $80 for insuring two houses for $6000 and $ 4000, respectively. The rate for the second house was \ (fo greater than that for the first. What were the two rates ?

46. Mr. Johnson had $ 15,000 invested, part at 6 % and part at 3 %. If his annual return was 5 % of the total investment, what amount was invested at each rate ?

47. A man invests $5650, part at 4 % and the remainder at 6 %. His annual income is $298. How much has he invested at each rate ?

48. A man desires to secure an income on $12,000 which shall be at the rate of 41^ %. He buys two kinds of bonds which yield 6% and 4%, respectively. How much does he invest in each ?

49. A bank invests s dollars, part at 6 % and the remainder at 5 % . If the annual income is m dollars, how much is in- vested at each rate ?

50. My annual income is m dollars. If - of my property is invested at 5 % and the remainder at 6 %, what is my capital ?

168 SIMPLE EQUATIONS

61. At what time between 5 and 6 o'clock are the hands of a clock together ?

Solution

Starting with the hands in the position shown, at 5 o'clock, let x represent the number of minute spaces passed over by the minute hand after 5 o'clock until the hands come together. In the same time the hour hand passes over ^^ of x minute spaces.

Since they are 25 minute spaces apart at 5 o'clock,

.-^ = 25;

.'. x= 27 j\, the number of minutes after 5 o'clock.

52. At what time between 1 and 2 o'clock are the hands of a clock together ?

53. At what time between 6 and 7 o'clock are the hands of a clock together?

54. At what time between 10 and 11 o'clock do the hands of a clock point in opposite directions?

55. At what two different times between 4 and 5 o'clock are the hands of a clock 15 minute spaces apart?

56. At what time between 2 and 3 o'clock are the hands of a clock at right angles to each other ?

Suggestion. — When the hour and minute hands are at right angles they are 15 minute spaces apart.

57. Find two different times between 6 and 7 o'clock when the hands of a clock are at right angles to each other.

58. Eind at what time between 1 and 2 o'clock the minute hand of a clock forms a straight line with the hour hand.

59. I have 6^ hours at my disposal. How far may I ride at the rate of 9 miles an hour, that I may return in the given time, walking back at the rate of 3^ miles an hour?

60. A man rows downstream at the rate of 6 miles an hour and returns at the rate of 3 miles an hour. How far down- stream can he go and return within 9 hours ?

SIMPLE EQUATIONS 169

61. A steamboat that goes 12 miles an hour in still water takes as long to go 16 miles upstream as 32 miles downstream. Find the velocity of the stream.

62. A motor boat went up the river and back in 2 hours and 56 minutes. Its rate per hour was 17^ miles going up and 21 miles returning. How far up the river did it go?

63. A yacht sailed up the river and back in r hours. Its rate per hour was s miles going up and t miles returning. How far up the river did it sail ?

64. A train traveling 20 miles an hour starts 30 minutes ahead of another train traveling 50 miles an hour in the same direction. How long will it take the latter to overtake the former ?

65. An express train whose rate is 40 miles an hour starts 1 hour and 4 minutes after a freight train and overtakes it in 1 hour and 36 minutes. How many miles does the freight train run per hour ?

66. If an airship had taken 1 minute longer to go a mile, the time for a trip. of 15 miles would have been 1 hour and 45 minutes. How long did it take the airship to go a mile ?

67. If an automobile had taken m minutes longer to go a mile, the time for a trip of d miles would have been t hours. How long did it take the automobile to go a mile?

68. In 1000 pounds of beans in the pod, ^ of the weight of the pods equaled the weight of the beans. Find the weight of the beans ; of the pods.

69. In making 5000 pounds of brass, 8|- times as much copper as tin was used and twice as much tin as zinc. How many pounds of each metal were used ?

70. In p pounds of bronze, the amount of tin was m times that of the zinc and n pounds less than - that of the copper. How many pounds of zinc were there?

170 SIMPLE EQUATIONS

71. United States silver coins are y^^ pure silver, or ^ ^^ fine.' How much pure silver must be melted with 250 ounces of silver | fine to render it of the standard fineness for coinage ?

72. In an alloy of 90 ounces of silver and copper there are 6 ounces of silver. How much copper must be added that 10 ounces of the new alloy may contain | of an ounce of silver ?

73. If 80 pounds of sea water contain 4 pounds of salt, how much fresh water must be added that 49 pounds of the new solution may contain 1| pounds of salt ?

74. In an alloy of 75 pounds of tin and copper there are 12 pounds of tin. How much copper must be added that the new alloy may be 12 J % tin ?

75. In an alloy of 100 pounds of zino and copper there are 75 pounds of copper. How much copper must be added that the alloy may be 1 0 % zinc ?

76. In a solution of 60 pounds of salt and water there are 3 pounds of salt. How much water must be evaporated that the new solution may be 10 % salt ?

77. Of 24 pounds of salt water 12 % is salt. In order to have a solution that shall contain 4 % salt, how many pounds of pure water should be added ?

78. It is desired to add sufficient water to 6 gallons of alcohol 95 % pure to make a mixture 75 % pure. How many gallons of water are required ?

79. Four gallons of alcohol 90 % pure is to be made 50 % pure. What quantity of water must be added ?

80. How much pure gold must be added to 180 ounces of gold 14 carats fine (|| pure) that the alloy may be 16 carats fine?

81. How much pure gold must be added to w ounces of gold 18 carats fine that the alloy may be 22 carats fine ?

SIMPLE EQUATIONS 171

82. A body placed in a liquid loses as much weight as the weight of the liquid displaced. A piece of glass having a volume of 50 cubic centimeters weighed 94 grams in air and 51.6 grams in alcohol. How many grams did the alcohol weigh per cubic centimeter ?

83. Brass is 8f times as heavy as water, and iron 71 times as heavy as water. A mixed mass weighs 57 pounds, and when immersed displaces 7 pounds of water. How many pounds of each metal does the mass contain ?

Suggestion. — Let there be x pounds of brass and (57 — x) pounds of iron. Then, x pounds of brass will displace (x h- 8^) pounds of water.

84. If 1 pound of lead loses -^^ of a pound, and 1 pound of iron loses -f-^ of a pound when weighed in water, how many pounds of lead and of iron are there in a mass of lead and iron that weighs 159 pounds in air and 143 pounds in water ?

85. If tin and lead lose, respectively, -fj and -f^ of their weights when weighed in water, and a 60-pound mass of lead and tin loses 7 pounds when weighed in water, find the weight of the tin in this mass.

86. If 97 ounces of gold weigh 92 ounces when weighed in water, and 21 ounces of silver weigh 19 ounces when weighed in water, how many ounces of gold and of silver are there in a mass of gold and silver that weighs 320 ounces in air and 298 ounces in water ?

87. An officer, attempting to arrange his men in a solid square, found that with a certain number of men on a side he had 34 men over, but with one man more on a side he needed 35 men to complete the square. How many men had he ?

Suggestion. — With x men on a side, the square contained x'^ men ; with ic + 1 men on a side, there were places for (a; + 1 )2 men.

88. A regiment drawn up in the form of a solid square was reenforced by 240 men. When the regiment was formed again in a solid square, there were 4 more men on a side. How many men were there in the regiment at first ?

172 SIMPLE EQUATIONS

Solution of Formulae

236. A formula expresses a principle or a rule in symbols. The solution of problems in commercial life, and in mensura- tion, mechanics, heat, light, sound, electricity, etc., often de- pends upon the ability to solve formulae. ■

EXERCISES ^

237. 1. The circumference of a circle is equal to tt (= 3.1416) times the diameter, or

C=TrD.

Solve the formula for D and find, to the nearest inch, the

diameter of a locomotive wheel whose circumference is 194.78

inches.

Solution

From G = wD, wD = C.

TT 3.1416 Hence, to the nearest inch, the diameter is 62 inches.

2. Area of a triangle = ^ (base x altitude), or

Solve for &, then find the base of a triangle whose area is 600 square feet and altitude 40 feet.

3. The area of a trapezoid is equal to the product of half the sum of the bases and the altitude ; that is,

2

The bases are b and b' ; b' is read ' 6-prime.'

Solve for h, then find the altitude of a trapezoid whose area is 96 square inches and whose bases are 14 inches and 10 inches, respectively.

4. The volume of a pyramid = ^ (base x altitude), or

Solve for B, then find the area of the base of a pyramid whose volume is 252 cubic feet and altitude 9 feet.

SIMPLE EQUATIONS 17B

5. In the formula / = /> • -^ • f,

denotes the interest on a principal of p dollars at simple iterest at r % for t years.

Evaluate for t, when p=% 300, r = 5, i = $ 60. Evaluate for r, when i = $ 900, p = $ 4500, t = 5 years. Evaluate for p, when r = 3|^, ^ = $ 210, i = 1 year.

6. In § ■ 42 was given the formula for the space (s) passed over by anything that moves with uniform velocity {y) during a given time {t). It is

Solve for v, then find the velocity of sound when the condi- tions are such that it travels 8640 feet in 8 seconds.

7. The formula for converting a temperature of i'' degrees Fahrenheit into its equivalent temperature of C degrees Centi- grade is

C = |(f-32).

Solve for F ; express 25° Centigrade in degrees Fahrenheit.

8. If a steel rail at 0° C. is heated, for every degree it is heated it will expand a certain part of its original length. Let Lq (read, ^L sub-zero ') denote its original length at 0° C, L its length at t degrees C, and a the certain fractional multiplier, or 'coefficient of expansion.'

Then, L^U^L^at.

Solve for a. A steel rail 30 feet long at 0° C. expanded to a length of 30.001632 feet at 50° C. Find the value of a.

Solve :

9. .s = i at"^, for a. 14. Mt\ = mv^, for m.

10. F^ Ma, for a. 15. E^^Mv"^, for M.

11. F=ix, W, for /i. 16. PoFo = PV, for P.

12. W= F,% for s. 17. s = Vo^ -f i af^, for a.

13. P = PR, for R. 18. s = i a(2 t - 1), for t.

174 SIMPLE EQUATIONS

19. Any kind of bar resting on a fixed point or edge is called a lever ; the point or edge is called the fulcrum. -

A lever will just balance when fl

the numerical product of the power (p\ F ^ '

(p) and its distance (d) from the T d A ^ I

fulcrum (F) is equal to the numeri- cal product of the weight ( W) and its distance (D) from the fulcrum ; that is, when

pd=WD.

Solve for W and find what weight a power of 150 (pounds) will support by means of the lever shown, if d = 7 (feet) and i) = 3 (feet).

Find for what values of p, d, W, or D the following levers will balance, each lever being 8 feet long :

20. P ^ 144 22. 600 ^ W'

I 6 sn

21. 300 F 100 23.

\ d J^ 8=d \ '

700

24. A pressure of 50 pounds was exerted upon one end of a 5-foot bar in order to balance a weight of 200 pounds at the other end of the bar. How far was the weight from the fulcrum ?

25. Philip weighs 114 pounds, and William 102 pounds. They are balanced on the ends of a 9-foot plank. Neglecting the weight of the plank, how far is Philip from the fulcrum ?

26. The figure illustrates the lever of a safety valve, the

power being the steam pressure

1 3 A		1
^t		F
i	5	A

k- -2) = 50--

®

//! (p) acting on the end of the pis-

ton above. The area of the end of the piston is 16 square inches. What weight ( W) must be hung on the end of the lever so that when the steam pres- sure rises to 100 pounds per

square inch the piston will rise and allow steam to escape ?

SIMPLE EQUATIONS 175

27. The number of pounds pressure (P) on A square feet

of surface of any body submerged to a depth of h feet in a

liquid that weighs w pounds per cubic foot is given by the

formula

P=wAh.

Fresh water weighs about 62 1 pounds per cubic foot, and ordinary sea water about 64 pounds per cubic foot.

Find the pressure on 1 square foot of surface at the bottom of a standpipe in which the water is 30 feet high; at the bottom of the ocean at a depth of 3000 feet.

28. Solve P=wAh for h and find the value of h when P = 5000, m; = 621-, and ^ = 8.

29. At what depth in fresh water will the pressure on an object having a total area of 4 square feet be 2000 pounds ?

30. A cistern is 6 feet square. For what depth of water will the pressure on the bottom be 36,000 pounds ?

31. How deep in the ocean can a diver go, without danger, in a suit of armor that can sustain safely a pressure of 140 pounds per square inch (20,160 pounds per square foot) ?

32. If the pressure per square foot on the bottom of a tank holding 18 feet of petroleum is 990 pounds, what is the weight of the petroleum per cubic foot ?

33. The side of a chest lying in 25 feet of water was 5 square feet in area and sustained a pressure of 8000 pounds. Was the chest submerged in fresh or in salt water ?

Solve :

34. § = I; , for R. 38. F= V,(l + ^, for t.

35. I=-±^,fovr. 39. ; = i- + i-, f or /, ; /j.

E E'		ioT E.
1=	E	- , for r. r
a =	t	^», for V,.
E_ e ~	R + r	-, for R ;

f fxf2

36. a = ^^^^,ioTVy, 40. | TFZ = — , f or IT; /S ; - .

c c

37. ^ = ii-:^,fori2; r. 41. (7= i+ i, for ^ ; O^.

SIMULTANEOUS SIMPLE EQUATIONS

TWO UNKNOWN NUMBERS

238. In the equation x -\- y = 12, X and y may have an unlimited number of pairs of values, as X = 1 and y = 11; or X = 2 and y = 10; etc.

For since 2/ = ^^ — ^j

if any value is assigned to x, a corresponding value of y may be obtained.

An equation that is satisfied by an unlimited number of sets of values of its unknown numbers is called an indeterminate equation. J

239. Principle. — Any single equation involving two or more unknown numbers is indeterminate.

240. The equations 2a;-h2^ = 101 and Zx -\-'dy = 1^ .

express but one relation between x and ?/; namely, that their sum is 5. In fact, the equations are equivalent to

x-}-y = 5

and to each other. Such equations are often called dependent

equations, for eitlier may be derived from the other.

241. The equations x -\- y = 5

x-y = l

express two distinct relations between x and y, namely, that

176

SIMULTANEOUS SIMPLE EQUATIONS 177

their sum is 5 and their difference is 1. The equations cannot be reduced to the same equation ; that is, they are not equivalent. Equations that express different relations between the un- known numbers involved, and so cannot be reduced to the same equation, are called independent equations.

242. Each of the equations

satisfied separately by an unlimited number of sets of values

X and 2/, but they have only one set of values in common,

.namely,

a? = 3 and ?/ = 2.

Two or more equations that are satisfied by the same set or its of values of the unknown numbers form a system of lultaneous, or coiisistent, equations.

243. The equations

a;4-y = 5|

ive no set of values of x and y in common. [Such equations are called inconsistent equations.

244. The student is familiar with the methods of solving ^mple equations involving one unknown number. The gen- ' eral method of solving a system of two independent simul- taneous simple equations in tiuo unknown numbers,

as a; + 2/ = 5"

x-y = S.

\ is to combine the equations, using axioms 1-4 in such a way |> as to obtain an equation involving x alone, and another in- I volving y alone, which may be solved separately by previous methods.

The process of deriving from a system of simultaneous equa- tions another system involving fewer unknown numbers is called elimination.

milne's stand, alg. — 12

178 SIMULTANEOUS SIMPLE EQUATIONS

Elimination by Addition or Subtraction

245. In solving simultaneous equations we may apply axi- oms 1-4, subject to the restrictions mentioned in § 230 in regard to the introduction or removal of roots in multiplying or dividing both members by expressions involving unknown numbers.

5x + 2y = 24: 5x-\-2y = 24:

5x-2y = 16 5x-2y = 16

Adding, 10 a; =40 Subtracting, 4ty= 8

1

In the two given equations the coefficients of y are numeri- cally equal land 'opposite in sign. Therefore, if the equations are added (Ax. 1), the resulting equation will not involve y. This method of eliminating y illustrates elimination by addition.

If one equation is subtracted from the other (Ax. 2), the resulting equation will not involve x. The second process illustrates elimination by subtraction.

EXERCISES [Additional exercises are given on page 471.]

246. 1. Solve the equations 2 a; 4- 3 2/ = 7 and 3 a; + 4 2/ = 10.

Solution

(2x+ 3y = 7, (1)

\3x+ 4y=10. (2)

(l)x4, Sx+\2y = 2S. (3)

(2)x3, 9 a; + 12 2/ = 30. (4)

(4) -(3), x = 2. (6)

Substituting (5) in (1), 4 + Sy = 7 ; .-. y = 1. To verify, substitute 2 for x and 1 for y in the given equations.

Rule. — If necessary, multiply or divide the equations by such numbers as will make the coefficients of the quantity to be elimi- nated numerically equal.

Eliminate by addition if the resulting coefficients have unlike signs, or by subtraction if they have like signs.

SIMULTANEOUS SIMPLE EQUATIONS

179

Solve by addition or subtraction, and verify results :

2.

3.

4.

5.

7x — 5y = 52,

2 a; + 5 2/ = 47.

(Sx-{-2y = 23,

'3x-4:y = 7y .x + 10y = 25.

2a;-102/ = 15, [2x-4:y = lS.

U-f-3v=-2.

^^ Ux-y = 19, \x-{-Sy = 21.

8.

Z + 2r = 5, [2l + r = l.

2a + 36 = 17, I3a + 26 = 18.

10. .( '

U + 2v = 14.

jj ri3^-M = 20,

14^4-2^ = 20.

^^ |3d + 4y = 25, Ud + 32/ = 31.

13. pP + 6g = 32, [7p-3q = 22.

j^ r3a4-6« = 39, l9a-42 = 51.

jg^ (Sx-Sy = U, \7x-5y = 29.

16.

6a; -5?/ = 33, , 4 a; + 4 2/ = 44.

^^ r3m-flln = 67, l5m — 3/1 = 5.

I Elimination by Comparison

I 247. If x = S-y, (1)

; and also x=2~\-y, (2) by axiom 5, the two expressions for x must be equal.

I .•.8-2/ = 2 + 2/.

I By comparing the values of x in the given equations, (1) and

j (2), we have eliminated x and obtained an equation involving

i y alone.

I This method is called elimination by comparison.

180 . SIMULTANEOUS SIMPLE EQUATIONS

EXERCISES

248. 1. Solve the equations 2 a; — 3 2/ = 10 and 5x-\-2y = 6.

Solution (2x-Sy = 10, (1)

L5a; + 2y = 6. (2)

From(l), x=l^. (3)

From (2), x = ^-^^^. (4)

5

Comparing the values of x in (3) and (4),

10 + Sy^6-2y 2 5

Solving, y = — 2.

Substituting — 2 for y in either (3) or (4),

x = 2. To verify, substitute 2 for x and — 2 for y in the given equations.

Rule. — Find an expression for the value of the same unknc number in each equation, equate the two expressions, and solve tJie equation thus formed.

Solve by comparison, and verify results :

^ ^Sx-2y==10, ^ |2s + 7^ = 8,

ic + 2/ = T0. ' l3s + 9^ = 9.

(5x-{-tj = 22, g Uu + 6v = 19,

[x-{-5y = U. ' \3u-2v = i.

2x + 3y = 24:,

5.

6.

g (4:V + Sw=M,

[5x-Sy = lS. ' [llv-\-5w==S7,

(Sx + 5y = U, ^^ {4.x-lSy = ^,

\2x-Sy = 3. l3a; + ll2/ = -17.

(Sv-\-2y = S6, ^^ nSx-Sy = iy,

\5v-9y = 23. ' 1 l-4a; + 32/ = 27.

SIMULTANEOUS SIMPLE EQUATIONS

181

249. Given and

Elimination by Substitution X— w = 4.

(1)

(2)

r On solving (2) for a?, its value is found to be a; = 4 + y.

If 4 + ?/ is substituted for x in (1), 3 x will become 3(4 + y), and the resulting equation

3(4 + y) +2 2/ = 27 (3)

ill involve y only, x having been eliminated.

Solving (3), y = 3.

Substituting 3 for y in (2), x = 7.

This method is called elimination by substitution.

EuLE. — Find an expression for the value of either of the un-

m numbers in one of the equations. Substitute this value for that unknoivn number in the other 'equation, and solve the resulting equation.

EXERCISES -

[Additional exercises are given on page 471.] 250. Solve by substitution, and verify results :

4.

5.

|aj-2/ = 4, [ 4 ?/ — a; = 14.

42/-

aj + 2/ = 10, 6aj-72/ = 34.

= 26, 22.

(

{^x-^y \x-^y=

r62/-10a; = 14, \y-x = d.

{y^l = 3x, \5i« + 9 = 32/.

6. \

7.

10.

ri7 = 3a; + i2,

\l = Sz-2x.

4 2/ = 10 — a;, 2/ — a; = 5.

|'72-3aj = 18, 2^-5a; = l.

3 — 15 2/ = — a;, 3 + 152/ = 4a;.

ri-aj = 32/, l3(l-a;) = 40-2/.

182

SIMULTANEOUS SIMPLE EQUATIONS

251. Three standard methods of elimination have been given. Though each is applicable under all circumstances, in special cases each has its peculiar advantages. The student should endeavor to select the method best adapted or to invent a method of his own.

EXERCISES

252. Solve by any method, verifying all results:

2.

[x — z = 5.

fSx-\-y = 10, x-\-3y = 6.

x + S = y-3, 2(x + 3)=6-y.

5x-y = 12,

(5x-y = rz, U + 32/ = 12.

3.

4:X-^5y=-2, 5x-]-4:y = 2.

5x-y=:2S, 3x + 5y = 2S.

U{2-x)=3y, [2(2-x) = 2(y-2).

(x-\-l)-{-(y-2) = 7,

Ux

(x + l)-{y-2) = 5.

Eliminate before or after clearing of fractions, as may be more advantageous :

9.

^ + 1=11,

12.

X 2y_

= -2,

5^ + ^ = 12. 2 ^3

10.

3x Ay

21,

^4-^ = 17 3^5

13.

"2 3~" '

11.

3"^^ 2' 3 7

14.

2 3 ,

2a;-l 3y-l_^5 2 ' 3 6'

SIMULTANEOUS SIMPLE EQUATIONS

183

15.

16.

3 2'

^-^ = 1 3 3

^ + ^ = 20,

^ + ^ = 17. 2 4

19.

20.

x — 1 x-\-y 3

=0,

»-2/

+ 3 = 0.

--12 =

_y + 32

4

^ 3^-2^ _ 25. 8^ 5

17.

a;-l 4

x-1

+ y = 3, H-42/ = 9.

18. ^

1 + 4^ = 15, 6^ 3

21.

22.

5 4 -^ '

5^-7 4^--3^j3_5 2 6

.2y+.5^.49a;-.7

1.5 ~ 4.2 '

.5a?-.2^41 1.5y-ll

1.6 16 8

23.

24.

25.

a; + y , x-y _2x-y Ay-x ^

x = 2y.

a; + K3a.'-2/-l) = i + |(2/-l),

^(4x + 3y)=3-V(7 2/ + 24).

'6a; + 9 , 3a; + 5w oi , 3a; + 4 4 4a; — 6 ^

8y + 7 6a;-3y^^ 4y-9_ 10 2(2/ -4) 5

3fl;-5y 2x-Sy-9_'Sl

26.

1(M+^'*)-

12

12'

8 j 6

184

SIMULTANEOUS SIMPLE EQUATIONS

27.

(

jij_i(^ + l)=li

Jf

(E-l)-ir = ^.

28.

r2.4.

[Ju

2Ad-Mu = .62, , 4- .15 r? = 1.795. l

29.

30.

(u + .S)(v + .5) = (u-.S)(v + 2),

1(2.. + .

(2 u + .1)(3 V + .5) = 6 v(u + .3).

23 -aj 2

59

2/

Alll._30 = aj-18

_3y-73

31.

fa; , 16 — oj QA . 5.v + 2a;

2 + -T~ = ^^ + ^o3^'

4(0,-6) 83-8^^ ^^_ I 2/ + 8 ^ 8 . ^

Equations of the form - + - = c, though not simple equa- tions, may be solved as simple equations for some of their

roots by regarding - and - as the unknown numbers. X y

32. Solve the equations .	r4 3 14 X y 5' 4 10 50 x'^ y 3*
Solution. (2) — (1),	13 208 y 16 . 1_16 " y 15*
Substituting (3) in (1),	4 48 14 « 16 6* . 1 3 " X 2
From (4) and (3),	x= - and y -. 3	15 16

(1)

(2) (8)

1

SIMULTANEOUS SIMPLE EQUATIONS

Solve, and verify results:

33.

34.

5_	3_	= -2
X	y
^5	1
	+-	= 6.
X	y
2_	3^	5,
X	y

5_2

X y

= 7.

37. {

38.

^ + 5 = 64,

« 2/

6 5 X y

2x ~y- ^'

^+2=23. 2aj 2/

185

35.

r4 , 3_9

a; 2/ 8

3 4^11

la; 2/ 12'

39.

15 + 6 = 20,

5 + 15 = 57i. a; 2/

36.

^I + ? = 30, X y

^ + - = 30.

12/ a; Solve the following as if

40.

x — 1 2/+1 numbers, and then find the values of a.* and 2/ :

6 a; 11 y

, etc., were the unknown

41.

1

x-1 y-\-l 2 _^ 3

= 5,

43.

aj-1 2/4-1

12.

y '^-^

5^_6_

V 2— a;

+ 9.

I

42.

r 5			3
X —	1	y	—	1
2			1
a; —	1	y	—	1

= 14,

6.

44.

X y + 3

7^ 3 X 2/ + 3

-10.

186 SIMULTANEOUS SIMPLE EQUATIOxNS

	Literal Simultaneous Equations
253. 1.	Solve the	( ax-{- by = equations [ cx-{-dy= Solution ax + by = m cx + dy = n	m, in.	(1) (2)
(1) X d,	adx + bdy = dm	(3)
(2)X6,		hex + bdy = bn		(4)
(8) -(4),		{ad — bc)x — dm — bn		1 (5)
		. a; — ^^ ~ ^^. ad — be
(l)xc,		acx + bey = em		(6)
(2) X a,		acx + ady = an		(7)
(7) -(6),	{ad — bc)y = an — em	J
		. jj^cin — cm

In solving literal simultaneous equations, elimination is usually per- formed by addition or subtraction.

Solve for x and y, and test results by assigning suitable values to the other letters

2.

3.

4.

fax-\-by = mj fa(x-y) = 5, M

\bx — ay = c. ' \bx — cy = n. m

(ax—by = mj ( a{a — x) = b(y — 6),

\cx — dy = r. ' \ax = by.

(ax = by, (x-\-y = b-a,

] x-\y = cLb. \bx — ay -{-2 0^ = 0.

r m(x + y) = a, (x-y = a-b,

1 n(x — y)=2a. ' \ax-^by = a^ — b^.

SIMULTANEOUS SIMPLE EQUATIONS

187

10. ^	a b
	bx — ay = 0.
11.	X y a 1_1_1^ X y b
12.	a b -, b a __ ^ X y~
13.	a b
	ab oib
14.	- + ! = !' a b X y_l b a 2*
20. C	riven ]

15.

16.

17.

18.

19.

x-{-l^a + b-^l y + 1 a-64-l' x — y = 2b.

1 1

= >

x—a a—y

^x-y

a.

a b b c

« + 6 = c,

X y

??^ + 2 = e. a; y

i- + i = c,

ax by

t

bx ay F=Ma,

s = iat\

Tind the values of F and a when M=15, s = 72, and t = 6. ' l = a + (n — l)d,

21. Given

. = |(a-fO.

Find the values of a and Z when n = 50, d = 2, and s = 2500 ,^,the values of d and a when 1 = 50, n = 25, and s = 650.

rl — a

^ = 731-

ind the values of a and Z when r = 2, n = 11, and s = 2047.

22. Given

188 SIMULTANEOUS SIMPLE EQUATIONS

Problems [Additional problems are given on page 472.] 254. Find two numbers related to each other as follows :

1. Sum = 14 ; difference = 8.

2. Sum of 2 times the first and 3 times the second = 34; sum of 2 times the first and 5 times the second = 50.

3. Sum = 18; sum of the first and 2 times the second = 20.

4. The difference between two numbers is 4 and I of their sum is 9. Find the numbers.

5. A grocer sold 2 boxes of raspberries and 3 of cherries to one customer for 54^, and 3 boxes of raspberries and 2 of cherries to another for 66^. Find the price of each per box.

6. A druggist wishes to put 500 grains of quinine into 3- grain and 2-grain capsules. He fills 220 capsules. How many- capsules of each size does he fill ?

7. On the Fourth of July, 850 glasses of soda water were sold at a fountain, some at 5^ each, the others at 10^ each. The receipts were $ 55. How many were sold at each price ?

8. A fruit dealer bought 36 pineapples for $2.50. He sold some at 12^ each and the rest at 10^ each, thereby gain- ing $ 1.50. How many did he sell at each price ?

9. The receipts from 300 tickets for a musical recital were $ 125. Adults were charged 50 j^ each and children 25^ each. How many tickets of each kind were sold ?

10. A boy deposited 38 bills valued at $50 in the bank. Some were 1-dollar bills, and the rest 2-dollar bills. How many bills of each kind did he deposit ?

11. A merchant has 100 bills valued at $ 275. Some are 2-dollar bills and the rest 5-dollar bills. How many bills of each kind has he ?

SIMULTANEOUS SIMPLE EQUATIONS 189

12. A paymaster has 110 coins valued at $40. Some are quarters and the remainder half-dollars. How many coins has he of each ?

13. In a plum orchard of 133 trees, the number of Lombard trees is 7 more than | of the number of Gage trees. Find the number of each kind.

14. If 5 pounds of sugar .and 8 pounds of coffee cost $ 2.70, and at the same price 9 pounds of sugar and 12 pounds of coffee cost $ 4.14, how much does each cost per pound ?

15. A factory employed 1000 men and women. The aver-, age daily wage was $ 2.50 for a man and $ 1.50 for a woman. If labor cost $ 2340 per day, how many of each were employed ?

16. A card on which a photograph is mounted is 6 inches longer than it is wide. It is bound with 60 inches of tape. Find its dimensions.

S 17. A lieutenant of the U. S. navy, receiving $ 1620 yearly, earned $ 150 a month while on sea duty and $ 127.50 a month while on shore duty. How many months was he on land ?

^ 18. A farmer bought 80 acres of land for $ 4500. If part of it cost $ 60 per acre and the remainder ^ as much per acre, how many acres did he buy at each price ?

19. If 8 baskets and 4 crates together hold 8 bushels of tomatoes, and 6 baskets and 8 crates together hold 9f bushels, find the capacity of a basket ; of a crate.

I 20. A 5-dollar gold piece weighs ^ as much as a 10-dollar gold piece. If the combined weight of 3 of the former and 2 of the latter is 903 Troy grains, what is the weight of each ?

I 21. If Rio coffee costs 20^ per pound and Java coffee, 32^ f per pound, how many pounds of each must be bought to fill a i 120-pound canister making a blend worth 28,^ per pound ?

I 22. If a bushel of corn is worth r cents, and a bushel of \ wheat is worth s cents, how many bushels of each must be I mixed to make a bushels worth b cents per bushel ?

190 SIMULTANEOUS SIMPLE EQUATIONS

23. If 1 is added to the numerator of a certain fraction, its value becomes f ; if 2 is added to the denominator, its value be- comes I". What is the fraction ?

Suggestion. — Let _ = the fraction.

y

I

24. If the numerator of a certain fraction is decreased by 2, the value of the fraction is decreased by \ ; but if the denomi- nator is increased by 4, the value of the fraction is decreased by |. What is the fraction ?

25. The sum of the digits in a number of two figures is 9 and their difference is 3. Find the number. (Two answers.)

26. A number expressed by two digits equals 7 times the sum of its digits. If 27 is taken from the number, the digits are reversed. Find the number.

Suggestion. — The sum of x tens and y units is (10 cc + y) units ; of y tens and x units, {lOy + x) units.

27. The sum of the two digits of a certain number is 12, and the number is 2 less than 11 times its tens^ digit. What is the number ? J

28. If a certain number of two digits is divided by their ^ sum, the quotient is 8; if 3 times the units' digit is taken from the tens' digit, the result is 1. Find the number.

29. If a rectangular floor were 2 feet wider and 5 feet longer, its area would be 140 square feet greater. If it were 7 feet wider and 10 feet longer, its area would be 390 square

» feet greater. What are its dimensions ?

30. A crew can row 8 miles downstream and back, or 12 miles downstream and halfway back in li hours. What is its rate of rowing in still water, and the velocity of the stream ?

31. A man rows 12 miles downstream and back in 11 hours. The current is such that he can row 8 miles downstream in the same time as 3 miles upstream. What is his rate of row- ing in still water; W^ what is the velocity «f the stream ?

i

SIMULTANEOUS SIMPLE EQUATIONS 191

32. When weighed in water silver loses .095 of its weight and gold .051 of its weight. If an alloy of gold and silver weighing 12 onnces loses .788 ounces when weighed in water, what is the amount of each in the piece ?

33. When weighed in water tin loses .137 of its weight and copper .112 of its weight. If an alloy of tin and copper weigh- ing 18 pounds loses 2.316 pounds when weighed in water, what is the amount of each in the piece ?

' 34. When weighed in water tin loses .137 of its weight and lead loses .089 of its weight. If an alloy of tin and lead weigli- ing 14 pounds loses 1.594 pounds when weighed in water, what is the amount of each in the piece ?

35. Two pumps are discharging water into a tank. If the first works 5 minutes and the second 3 minutes they will pump 2260 gallons of water; if the first works 4 minutes and the second 7 minutes they will pump 3280 gallons. Find their capacity per minute.

36. A and B together can do a piece of work in 12 days. After A has worked alone for 5 days, B finishes the work in 26 days. In what time can each alone do the work ?

37. If 4 boys and 6 men can do a piece of work in 30 days, and 5 boys and 5 men can do the same work in 32 days, how long will it take 12 men to do the work ?

38. A and B can do a piece of work in a days, or if A works m days alone, B can finish the work by working n days. In how many days can each do the work ?

39. A and B can do a piece of work in a days ; A works alone m days, when A and B finish it in n days. In how many days can each do it alone ?

40. A can build a wall in c days, and B can build it in d days. How many days must each work so that, after A has done a part of the work, B can take his place and finish the wall in a days from the time A began ?

192 SIMULTANEOUS SIMPLE EQUATIONS

41. At simple interest a sum of money amounted to $ 2472 in 9 months and to $ 2528 in 16 months. Find the amount of money at interest and the rate.

42. A man invested $ 4000, a part at 5 % and the rest at 4 %. If the annual income from both investments was $175, what was the amount of each investment ?

43. A man invested a dollars, a part at r per cent and the rest at s per cent yearly. If the annual income from both invest- ments was b dollars, what was the amount of each investment ?

44. A sum of money at simple interest amounted to b dollars in t years, and to a dollars in s years. What was the principal, and what was the rate of interest ?

45. A certain number of people charter an excursion boat, agreeing to share the expense equally. If each pays a cents, there will be b cents lacking from the necessary amount ; and if each pays c cents, d cents too much will be collected. How many persons are there, and how much should each pay ?

46. A mine is emptied of water by two pumps which together discharge n gallons per hour. Both pumps can do the work in b hours, or the larger can do it in a hours. How many gal- lons per hour does each pump discharge ? What is the dis- charge of each per hour when a = 5, 6 = 4, and m = 1250 ?

47. Two trains are scheduled to leave A and B, m miles apart, at the same time, and to meet in b hours. If the train that leaves B is a hours late and runs at its customary rate, it will meet the first train in c hours. What is the rate of each train ? What is the rate of each, if m = 800, c = 9, a = If , and 6 = 10 ?

48. A man ordered a certain amount of cement and received

w it in c barrels and d bags ; a barrels and b bags made — of the

total weight. How many barrels or how many bags alone would have been needed ? Find the number of each, if c = 16, d = 15, a = 6, 6 = 15, m = 1, and n= 2.

\

SIMULTANEOUS SIMPLE EQUATIONS 193

THREE OR MORE UNKNOWN NUMBERS

255. The student has been solving systems of two independ- ent simultaneous equations involving two unknown numbers. In general,

Principle. — Every system of independent simultaneous simple equations involving the same number of unknown numbers as there are equations can he solved, and is satisjied by one and only one set of values of its unknoivn numbers.

EXERCISES

■x + 2y + Sz = U, (1)

256. 1. Solve 2aj + 2/ + 22 = 10, (2) .3aj + 42/-32 = 2. (3)

SoLunoi*. — Eliminating z by combining (1) and (3), l) + (3), 4a; + 6y = 16. (4)

Eliminating s by combining (2) and (3),

(2) X 3, 6x+Sy + 6z = S0

(3) X 2, 6 ic + 8 y - 6 2 = 4

Adding, 12 re + 11 y ^34 (5)

Eliminating x by combining (5) and (4),

(4) X 3, 12 x + 18 y = 48 (6) (6)-(5), 1y=U; .:y = 2.

Substituting the value of y in (4), 4 x + 12 = 16 ; .-. a; = 1.

Substituting the values of x and y in (1),

1 + 4 + 3 0 = 14 ; .-. 0 = 3.

P Verification. — Substituting a; = 1, y = 2, and z = S in the given equations, ^^^ becomes 1 + 4 + 9 = 14, or 14 = 14

(2) becomes 2 + 2 + 6 = 10, or 10 = 10 ; and (3) becomes 3 + 8-9= 2, or 2= 2

that is, the given equations are satisfied f or aj = 1 , y = 2, and z = 3. Milne's stand, alg. — 13

104

SIMULTANEOUS SIMPLE EQUATIONS

Solve, and test all results : 'x-{-3y-z = 10, 2x-^5y-\-4:Z = 57, 3x-y + 2z = W.

10. <

4 a; — 5 2/ + 3 2; = 14, x-\-7y — z = 13, 2x-{-5y-\-5z = S6.

5. <

6.

x-\-y-{-z = 53, x-{-2y-\-3z = 105, x + 3y-{-4.z = 134:.

x-y-\-z = 30, 3y — x — z = 12, [7z — y + 2x = 14:l,

'Sx-5y + 2z = 53j x-\-y — z = 9, 13x-9y-\-3z = 71,

x + 3y + 4z = S3,

x-\-y-\-z = 2%

6a; + 8?/ + 32 = 156.

'2a; + 3yH-42 = 29,

7. {3x + 2y + bz = 32, [4a; + 3y + 22 = 25.

3x-2y-\-z = 2,

8. \2x-\-by-\-2z = 27, .x + 3y + 3z = 25.

11.

12,

13.

14.

2x-\-y — 3z + i.ic — ^j 3x — 2y-\-z — vj= — 1, A:X — y-\-2z-\-ic = 55, 5a; — 32/4-42; — w; = 39.

'7x-l = 3y,

11^-1 = 7^,

4^-1 = 72/, Ll9a;-1 = 3?;.

x + \y-\-\z = 32, \x + \y + \z==l^, \x + \y + \z = 12.

15. {

\x-\y + \z = 3, \x-\y-\-\z = l, \x-\y + iz=^5.

4^ + 3. = 29,

^^ + 2. = 22, 3x-y=^3{z-l).

9. \

'2x — 3y-i-4:Z — v = 4:, 4:x-i-2y-z-^2v = 13, x-y + 2z + 3v = 17, 3a; + 22/ — «-|-4v = 20.

16.

3x + y-z + 2v=:0, 3y-2x-\-z-4:V=21, x — y-\-2z — 3v = 6, l4a; + 22/-3« + v = 12.

SIMULTANEOUS SIMPLE EQUATIONS

195

17. Solve the equations

u-{-v-{-x-y = 2f u-\-v — x-{-y = 4:, u — v-\-x-\-y = 6, .v — u-\-x-\-y = S. Solution. — Adding the equations, 2 u -\- 2 v + 2 x -{- 2 y = 20. Dividing by 2, u + v -{-x + y = 10.

Subtracting each of the given equations from this equation, 2y = 8, 2x = 6, 2u = 4, 2wz=2; svhence, y = 4, x = 3, u = 2, w = L

Solve, and test all results	:
	'x + y = %				aj + 32/ + 2; = 14,
18.	y + z = 7, z-{-x = 5, 'v + x + y = 15,			22.	^2/ 4- 2 + V — a; = 22,
19. •	x + y-{-z = lS, y-\-z + v = 17, .z-\-v + x=16. X y			23.	z-i-v-hx-y=18y v-\-x-{-y — z = 14:, .X -{- y -{- z — V = 10, fi+i-i=o, a; y
20. i	y z Z X x + y 5			24. .	-4-i-2 = 0. Z X \ xy _1
21.	yz 1			25.	t/^ _1
y + z 6'	2/ + ;^ 4'
	zx 1				;2a; 1
	z + x 7				24-3; 2
Sugg	ESTION. If ^^ = x-hy	1 ^5'	xy	4w	tience, - + - = 6. y X

196

SIMULTANEOUS SIMPLE EQUATIONS

	a . b . c - H h - = a, X V z		(1)
26. Solve for x, y, and z, ■	bzx — cxy-{-ayz =	= 6a;y0,	(2)
	a b c_		(3)
Solution. (1) + (3), — = a + c. X
		W
(2)^:,,., rl-^l = '-		(6)
(5) - (3),	21 = 6-c.		J

«/ =

2b

b-c Substituting the values of x and y in (1), and solving, z

2c

Solve for x, y, z, and v : aa^ — a; — 2/=0,

27. <62a; — 2 — a; = 0, cyz — y — z = 0.

28.

29.

30.

x-\-y-z = 0, x-y = 2b, x-\-z = Za-\-b.

v-\-x = 2 a, x-\-y = 2a — Zy y-\-z = a-\-by Lv — z = a 4- c.

'y + 2! — 3 a; = 2 a, 24-a;-32/ = 2 6, a;-f2/ — 32; = 2c,

.2 a; H- 2 y + v = 0.

31.

32.

33.

Ia6a;2/2; -f cxy — a?/2; — bzx = 0, 6ca:2/2! + ayz — bzx — cxy = 0, caxyz + 62a; — cxy — a?/2; = 0

x + y + z = a-\-b + c, x-\-2y-\-Sz = b + 2c, a; + 32/ + 42 = 6 + 3c.

''y + a; + y = a-f-264-c,

x + y + z = Sb,

2/4-2 + 'y = a + &, Lz-h'y4-a; = a + 3& — c.

aa; -|- 6?/ + 02; = 3,

34.

aj + ?/ = 2/4-2; =

ab b +c be '

i

SIMULTANEOUS SIMPLE EQUATIONS 197

Problems [Additional problems are given on page 476.]

257. 1. The sum of three numbers is 38 ; the sum of i of the first, J of the second, and J of the third is 12 ; of J of the first, ^ of the second, and ^ of the third is 9. Find the numbers.

2. Separate 800 into three parts, such that the sum of the first, i of the second, and f of the third is 400 ; and the sum of of the second, f of the first, and J of the third is 400.

3. A and B can do a piece of work in 10 days ; A and C can do it in 8 days ; and B and C can do it in 12 days. How long will it take each to do it alone ?

4. A and B can do a piece of work in r days ; A and C can do it in s days ; and B and C can do it in t days. How long will it take each to do it alone?

5. A certain number is expressed by three digits whose sum is 14. If 693 is added to the number, the digits will appear in reverse order. If the units' digit is equal to the tens' digit increased by 6, what is the number ?

6. The sum of three numbers is 162. The quotient of the second divided by the first is 2 ; of the third divided by the first is 3. Find the numbers.

7. A, B, and C have certain sums of money. If A gives B S 100, they will have the same amount ; if A gives C $ 100, G will have twice as much as A ; and if B gives C $ 100, C will have 4 times as much as B. What sum has each ?

8. A quantity of water sufficient to fill three jars of different sizes will fill the smallest jar 4 times ; the largest jar twice with 4 gallons to spare ; or the second jar 3 times with 2 gal- lons to spare. What is the capacity of each jar ?

9. A contractor used three scows, a, b, and c, to convey sand from his dredge to the dumping ground. He was credited for :

April 20, scows a, &, c, a, &, c, a, and 6, 8 loads, 3230 cu. yd. April 21, scows c, d, h, c, a, &, and c, 7 loads, 2820 cu. yd. April 22, scows a, 6, c, «, &, c, and a, 7 loads, 2870 cu. yd.

Find the capacity of each scow.

GRAPHIC SOLUTIONS

SIMPLE EQUATIONS

258. When related quantities in a series are to be compared, as for instance the population of a town in successive years, recourse is often had to a method of representing quantities by lines. This is called the graphic method.

By this method^ quantity is photographed in the process of change. The whole range of the variation of a quantity, pre- sented in this vivid pictorial way, is easily comprehended at a glance ; it stamps itself on the memory.

259. In Fig. 1 is shown the population of a town throughout

its variations during the first 13 years of the town's exist- ence.

The population at the end of 2 years, for example, is repre- sented by the length of the heavy black line drawn upward from 2, and is 4000 ; the popu- lation at the end of 6 years is 7000 J at the end of 10 yearsj 6300 approximately; and so on.

260. Every point of the curved line shown in Fig. 1 exhibits a pair of corresponding values of two related quantities, years and population. For instance, the position of E shows that the population at the end of 4 years was 6000.

Such a line is called a graph.

198

n '									f
									/
'S '1-
Ah								y
o								/
' -o		^			Y		y
5 1	-/
, 1
. ^,	r
J /j	3
li
e 1	2 3 4	I	fl	'y	8 ( ears	1	0 1	1 12	13 1	4 1	6

FiG.l.

GRAPHIC SOLUTIONS

199

t3fraphs are useful in numberless ways. The statistician uses them to present information in a telling way. The broker or merchant uses them to compare the rise and fall of prices. The physician uses them to record the progress of diseases. The engineer uses them in testing materials and in computing. The scientist uses them in his investigations of the laws of nature. In short, graphs may be used whenever two related quantities are to be compared throughout a series of values.

261. The graph in Fig. 2 represents the rate in gallons per day per person at which water was used in New York City during a certain day of 24 hours.

-i

/

'^

^

/

\

/

^1

f

•—

-^

4

/

\

\,

/

^

~

~

~

~

~

~

-

>.

e-

§

-

11

)urs fror

I) Mid

light

Miduigbt

5 0

6 A.M.

12 !Noou

Fia. 2.

15 lo

6 P.M.

a 24 Miduigbt

Thus, if each horizontal space represents 1 hour (from mid- night) and each vertical space 10 gallons, at midnight water was being used at the rate of about 84 gallons per day per per- son; at 6 A.M., about 91 gallons; at 1 p.m., the 13th hour, about 108 gallons ; etc.

1. What was the approximate consumption of water at 2 A.M. ? at noon? at 1 : 30 p.m. ? at 2 : 30 p.m. ? at 6 p.m. ?

2. What was the maximum rate during the day ? the mini- mum rate? at what time did each occur?

3. During what hours was the rate most uniform ? What was the rate at the middle of each hour ?

4. What was the average increase per hour between 6 a.m. and 8 a.m.? the average decrease between 4 p.m. and 8 p.m.?

200

GRAPHIC SOLUTIONS

EXERCISES

262. 1. Letting each horizontal space represent 10 years and each vertical space 1 miLion of population, locate points from the pairs of corresponding values (years and millions of popula- tion given below) and connect these points with a line, thus constructing a population gra-ph of the United States :

1810,7.2; 1820,9.6; 1830,12.8; 1840,17.1; 1850,23.2; 1860, 31.4; 1870, 38.6; 1880, 50.2; 1890, 62.6; 1900, 76.3.

2. From the graph of exercise 1, tell the period during which the increase in the population was greatest ; least,

3. The average price of tin in cents per pound for the months of a certain year was : Jan., 23.4 ; Feb., 24.7 ; Mar., 26.2 ; Apr., 27.3; May, 29.3; June, 29.3; July, 28.3; Aug., 28.1; Sept., 26.6; Oct., 25.8; Nov., 25.4; Dec, 25.3.

Draw a graph to show the variation in the price of tin during the year with each horizontal space representing 1 month and each vertical space 1 cent.

4. Construct the graph of exercise 3, letting each horizontal space represent 1 cent and each vertical space 1 month.

263. Let X and y be two algebraic (Quantities so related that y = 2x — 3. It is evident that we may give x a series of values, and obtain a corresponding series of values of y; and that the number of such pairs of values of x and y is unlimited. All of these values are represented in the graph of y = 2 x — 3. Just as in the preceding illustrations, so in 4he graph of y=2x — 3, Fig. 3, values of x are represented by lines laid off on or parallel to an x-axis, X'X, and values of y by lines laid off on or parallel to a y-axis, Y'Y, usually drawn perpendicular to the jp-axis. Fig. 3.

GRAPHIC SOLUTIONS

201

For example, the position of P shows that ?/ = 3 when a; = 3 ; the position of Q shows that ?/ = 5 when a; = 4 j the position of B shows that 2/ = 7 when x = 6\ etc.

Evidently every point of the graph gives a pair of corre- sponding values of x and y.

264. Conversely, to locate any point with reference to two axes for the purpose of representing a pair of corresjjonding values of x and y, the value of x may be laid oif on the avaxis as an ic-distance, or abscissa, and that of y on the y-axis as a y-distance, or ordinate. If from each of the points on the axes obtained by these measurements, a line parallel to the other

is is drawn, the intersection of these two lines locates the oint.

Thus, in Fig. 3. to represent the corresponding values a; = 3, ?/ = 3, a )int P may be located by measuring 3 units from 0 to M on the a;-axis id 3 units from 0 to iV on the ?/-axis, and then drawing a line from M

rparallel to 0 F, and one from N parallel to OX, producing these lines

^until they intersect.

265. The abscissa and ordinate of a point referred to two irpendicular axes are called the rectangular coordinates, or

limply the coordinates, of the point.

Thus, in Fig. 3, the coordinates of P are 0M(= NP) and MPC= ON').

266. By universal custom positive values of x are laid off rom 0 as a zero-point, or origin, toward the right, and 7ieg- Hve values toward the left. Also

'positive values of y are laid off up- ward and negative values doivnward.

The point A in Fig. 4 may be designated as "the point (2, 3)/' or by the equation A =(2, 3).

Similarly, B={-2, 4), C=(-3, -1), and D=(l, -2).

The abscissa is always written first. Fig. 4.

"

-

y

-

B

'a

X'

.^

X

k-

2-

0

1

2

3

l_4

c'

D

Y'

202

GRAPHIC SOLUTIONS

267. Plotting points and constructing graphs.

EXERCISES

Note. — The use of paper ruled in small squares, called coordinate paper, is advised in plotting graphs.

Draw two axes at right angles to each other and locate :

1. ^ = (3,2).

2. i5 = (3, - 2).

3. C=(4, 3).

4. D = {i,-3).

9. i = (0, 4).

10. Jf = (0, - 5),

11. ]^=(3,0).

12. P = (_6, 0).

5. ^ = (5,5).

6. F=(-5,5),

7. G^ = (-2,6).

8. H={-S, -4).

13. Where do all points having the abscissa 0 lie ? the ordinate 0?

14. What are the coordinates of the origin ?

15. Construct the graph of the equation 2y ^x = 2.

Solution

Solving for y, y = ^ (a; + 2).

Values are now given to x and computed for y by means of this equation. The numbers substituted for x need not be large. Con- venient numbers to be substituted for x in this instance are the even integers from — 6 to +6.

When X = — 6, y = — 2. These values locate the point ^ = ( — 6, — 2).

When X = — 4, y = — 1. These values serve to locate 5 = (— 4, — 1).

Other points may be located in the same way.

A record of the work should be kept as follows :

Y

^^ilE

^^S

x; 3^-- -

^A

85	y	Point
-6	-2	A
-4	-1	B
-2	0	C
0	1	D
2	2	E
4	3	F
6	4	0

Fig. 5. A line drawn through A, B^ C, 2>, etc., is the graph of 2 y

— X

GRAPHIC SOLUTIONS 203

Construct the graph of each of the following:

16. y = Sx-7. 19. 3x-y = 4:. 22. Sx = 2y.

17. 2/ = 2 a; + 1. 20. 4 a; — y = 10. 23. 2 a; + 2/ = 1.

18. y = 2x — l. 21. aj-22/ = 2. 24. 2x + 3y = 6,

268. It can be proved by the principle of the similarity of triangles that :

Principle. — The graph of a simple equation is a straight line. For this reason simple equations are sometimes called linear :equations.

269. Since a straight line is determined by two points, to plot the graph of a linear equation, plot two points and draw a straight line through them.

It is often convenient to plot the points where the graph intersect? the axes. To find where it intersects the a;-axis, let = 0 ; to find where it intersects the y-axis, let a; = 0.

Thus, iny = ^(x + 2), when y =0,x = — 2, locating C, Fig. 6 ; when = 0, y = 1, locating D.

Draw a straight line through C and D.

If the equation has no absolute terra, a; = 0 when y = 0, and this method gives only one point. In any case it is desirable, for the sake of accuracy^ to plot points some distance apart, as A and G^ in Fig. 6.

EXERCISES

270. Construct the graph of each of the following :

1. y = x-2, 8. 2x-3y = 6. 15. Sx-Sy=-6.

2. y = 2 — x. 9. 3 a; + 4 2/ = 12. 16. —2x + y=—3.

3. y = 9-4:X. 10. 5x-2y = lQ. 17. -3a; + 4?/ = 8.

4. 2/ = 4 a; — 9. 11. 7 a; — 2/ = 14. IS. 5x-\-3y = 7^.

5. y = 10 — 2x. 12. 4 -a; = 2?/. 19. a; — 1|/ = 3. G. y = 2x-10. 13. 2 a; 4-3 2/ = 0. 20. i a; -f i 2/ = 2.

,,7. y=-2x — 4. 14. x-4:y-3 = 0. 21. .7x-.3y = A.

i

204

GRAPHIC SOLUTIONS

-

N

/

c

\

nX

y

D

\

/

k

i^

N

\

H

/

"n

V

B

/

^

/

\

/

A

0

i

5M

\

/

\

/

L

-

L

L_

L

Fig. 6.

271. Graphic solution of simultaneous linear equations.

1. Let it be required to solve graphically the equations

(y = 2^x, (1)

\y = 6~x. (2)

As in § 267, construct the graph of each equation, shown in Fig. 6.

1. When x = — l, the value of y in (1) is represented by AB, and in (2) by AC.

Therefore, when x = — 1, the equations are not satis- fied by the same values of y.

2. Compare the values of y when x = Oj when a; = 1 ; 2.

3. For what value of x are the values of y in the two equa- tions equal, or coincident ?

4. What values of x and y will satisfy both equations ? The required values of x and y, then, are represented graphi- cally by the coordinates of P, tJie intersection of the graphs.

IL Let the given equations x + y = T, 2x-\-2y = U.

5. What happens if we try to eliminate either x ov y?

6. Since y = T —x in both equations, what will be the relative positions of any two points plotted for the same value of X? the relative posi- tions of the two graphs?

7. The algebraic analysis shows that the equations are indeterminate.

The graphic analysis also shows that the equations are inde- terminate, for their graphs coincide.

i

be

^

5

^

^<^-

w

-^

X

3s r-

^ _

'^

Fig. 7.

GRAPHIC SOLUTIONS

206

be

III. Let the given equations 2^ = 6-0;, (1)

y = 4-x. (2)

8. When x= —1, how much greater is the value of y in (1) than in (2), as shown both by the equations and their graphs ?

9, Compare the ^s for other values of x. 10

Y

Y'

Fig. 8. For every value of x the values of y in the two equa- tions differ by 2, and the graphs are 2 units apart, vertically. In algebraic language, the equations cannot be simultaneous ; that is, they are inconsistent.

In graphical language, their graphs cannot intersect, being parallel straight lines.

272. Principles. — 1. A single linear equation involving two unknoivn numbers is indeterminate.

2. Two linear equations involving two unknown numbers are determinate, provided the equations are independent and simultaneous.

Tliey are satisfied by one, and only one, pair of common values.

3. 77ie pair of common values is represented graphically by the coordinates of the intersection of their graphs.

EXERCISES

	Y
	y
s	*l^
\	f,p
"^s	- ^y
	^^^
	;:^S
	^^4 ^i^
t -.^'	- J ^i: X
i	^^
7^
	Y'

273. 1.

equations

Solve graphically the r42/-3ic = 6, I 2 a; 4- 3 ?/ = 12.

Solution. — On plotting the graphs of both equations, as in § 267, it is found that they intersect at a point P, whose coordinates are 1.8 and 2.8, approximately.

Hence, a; = 1.8 and y = 2.8.

Fig. 9. The coordinates of Pare estimated to the nearest tenth.

20Q GRAPHIC SOLUTIONS

Note. — In solving simultaneous equations by the graphic method the same axes must be used for the graphs of both equations.

Construct the graphs of each of the following systems of equations. Solve, if possible. If there is no solution, tell why.

^ (x-y = l, ^^ (2x-5y = 5,

\x + y = 9. ' [l0y = 2x-{-l.

8.

11.

12.

x-{-y = S, (8y = 2x — 7,

14.-1 x + 2y = 4:. [2x = 6-{-3y,

I

[y = 3 + x. ' U(y + 6) = dx.

(2x-y = 5, ^^ |10aj4-?/ = 14,

\4.x + y = 16. ' {Sx-5y=-2.

(Sx = y + 9, ^^ (2x + 3y = S,

[2y = 6x-lS. ' \3x-}-2y = S.

y = 4:X, (4:y + 3x = 5,

x-y = 3. ' \4.x-3y = 3.

[y = 2{x-2), [2x-4.y=-12.

(x + y=^3, ^^ Ux-10y==0,

\x-2y=-12. ' [2x + y = 12.

10. : ' 21. ' ^ '

{y = 2-x.

2y-6x = 3.

ra; = 2(2/ + l), f3a; + 42/ = 10,

l21 = 2(2a; + y). ' [6x-\-Sy = 20.

fx + y = S, |fa; + |2/ = 3i,

l2«-6?/=-9. ' [10x^2y = l^

INVOLUTION

274. The process of finding any required power of an ex- pression is called involution.

275. By the definition of a power, when w is a positive [integer a** means a-a-a •■-to n factors.

The following illustrate powers of positive numbers, of nega- tive numbers, of powers, of products, and of quotients, and [show that every case of involution is an example of multipli- [cation of equal factors.

^m POWERS OP A	POWERS OF A	^POWERS OF A
^m POSITIVE NUMBER	NEGATIVE NUMBER	POWER
■ 2=2'	-2 = (-2y	4 = 22
1 ■ 4 = 2'	■ -2 4 = (-2)^	4 16 = (2^ = 2*
1	-2 -8 = (-2)3	4 64 = (22)3 = 2«
	-2 16 = (-2)*	4 256= (2^ = 28
r	POWER OF A PRODUCT

(2.3)2= (2. 3) X (2. 3) =2. 2. 3. 3 = 22. 3*

POWER OF A QUOTIENT

'2y 2 2 22

(i)

3 3 32

The last two examples illustrate the distributive law for involution.

207

208 INVOLUTION

276. Principles. — 1. Law of Signs for Involution. — All powers of a positive number are positive ; even powers of a nega- tive number are positive, and odd powers are negative.

2. Law of Exponents for Involution. — The exponent of a power of a number is equal to the exponent of the number multiplied by the exponent of the power to which the number is to be raised.

3. Distributive Law for Involution. — Any power of a product is equal to the product of its factor^i each raised to that power.

Any power of the quotient of two numbers is equal to the quo- tient of the numbers each liaised to that power. I

The above laws may be established for positive integral expo- nents as follows :

Let m and n be positive integers.

1. Principle 1 follows from tbs law of signs for multiplication.

2. By notation, § 27, (a™)'* = a"* • a™ . a** ••• to n factors

§88,	_ Qm+m+m+"' to m terms-
By notation, •	= 0"*".
3. By notation.	(a&)" = ahx abx ab'-ton factors
§82,	= (aaa "')(hbb •••) each to n factors
By notation,	= a"&".
Also	f«V = ?x^x^... ton factors \bJ b b b
§207,	aaa •••to n factors bbb • • • to n factors
By notation,

277. Axiom 6. — The same powers of equal numbers are equal Thus, if a; = 3, x2 = 3'^, or 9 ; also x* = 3*, or 81 ; etc.

278. Involution of monomials.

EXERCISES

1. What is the third power of 4 a^b ? Solution. (4 a^by = 4 a^ft x 4 a^ft x 4 a^6 = 64 a^ft*

1

INVOLUTION

209

2. What is the fifth power of - 2 ah^ ? Solution [- 2 a62)5 =_ 2a&2 x - 2 a&2 x - 2 afe"^ x - 2 a62 x - 2 a&2 = _ 32 a^ftio.

To raise an integral term to any power :

Rule. — Raise the numerical coefficient to the required power \nd annex to it each letter tvith an exj)07ieyit equal lo the product b/ its exponent by the exponent of the required power.

Make the power positive or negative according to the law oj tigns.

Raise to the power indicated :

3	(a6V)2.	12.	(-4c2/)3.	21.	(-1)»
4.	(a'b'cy.	13.	(-2aV)*.	22.	(-1)-
5	(2 a'cf.	14.	(abcxy.	23.	(-1)2».
6.	(7 aVy	15.	(2 eV)«.	24.	(-bf'^K
7	(-1)^.	16.	(3 bey.	25.	(-by^^K
8.	i-aby.	17	(2 aV)«.	26.	(— a'b''c''-^dy.
9.	(Scy.	18.	{-2lVd'y,	27.	i-a'y^z'y.
10.	(-10x'2)3.	19.	(-aV^"-i)2.	28.	{-a^-^b^-hy.
11.	(-6a2a^)2.	20.	(_a^2/V*-3)3^	29.	[-2(a^&)^J.
30.	What is the	square	of ^«'^? 7b'c Solution
	/ 5a%2\2_	5 a%2 ^^ 5 a%2	25a6a4

49 6*c2

To raise a fraction to any power :

Rule. — Raise both numerator and denominator to the re- ared power and prefix the proper sign to the result.

milne's stand, alg. — 14

LO	INVOLUTION
Kaise to the power	indicated :
- (rj-	»■ (-ij-	41.	(-¥)•■
" (Ml-	"• (-A)'	42.	(-S)"
"■ m-	"■ (-If)"	43.	(f)--
- (If)"	- (-i)-	44.
- {£=-:)'■	- {-?]■	45.

279. Involution of polynomials.

The following are type forms of squares of polynomials : § 105, (a + a;)2= a^ + 2 aa; + x^^

§ 108, (a -xy= a^-2ax + x\

§111, {a-x + yf^a''-^x' + y^-2ax-\-2ay-2xy,

EXERCISES

280. Eaise to the second power:

1. 2a + &. 5. 3a; — 4y8. 9. « — & + « — t/.

2. 2 a — 6. 6. 5m^ — 11. 10. a"* + aj"— y"^^

3. (r— 3 6". 7. l-3a6c. 11. 2a4-36-4c.

4. a2-2iB2«. 8. 4 a;* + 5. 12. ^a^-l-^4.n\

Raise to the required power by multiplication :

13. (x + y)\ 15. {x + yy. 17. {x+yf.

14. (x-yf. 16. (x-yy, 18. {x-yy.

INTOLUTION 211

281. Involution of binomials by the Binomial Theorem (§ 549). By actual multiplication,

(a + xy = a^ + 3a^x + Sax^-{-a?.

(a — xy = a^ — 3 a^x + 3ax^ — a:^.

(a + o;)^ = a* + 4 a^x + 6 aV + 4 oa^ + it**.

(a — xy=a* — 4: a^x -\- 6 a^a^ — 4 aaj' + x*.

(a 4- aj)' = a* + 5 a^a; + 10 aV + 10 a^a? + 5 aa;< + a^*-

(a -a;)* == a«- 5a^a; + 10 aV- 10 a2a^+ 5 aa;^-aj«.

From the expansions just given the following observations ay be made in regard to any positive integral power of any nomialj a standing for the first term and x for the second:

1. The number of terms is one greater than the index of the equired power,

2. The first term contains a only; the last term x only ; all her terms contain both a and x.

3. The exponent of a in the first term is the same as the index >fthe required power and it decreases 1 in each succeeding term; the exponent of x in the second term is 1, and it increases 1 in each succeeding term.

4. In each term the sum of the exponents of a and x is equal to the index of the required power,

5„ The coefficient of the first term is 1 ; the coefficient of the ^^second term is the same as the index of the required power,

^f 6. The coefficient of any term may be found by multiplying the coefficient of the preceding term by the exponent of a in that term, and dividing this product by the number of the term.

7. All the terms are positive, if both terms of the binomial are positive.

8. The terms are alternately positive and negative, if the second .^term of the binomial is negative.

1

212 INVOLUTION

EXERCISES a

n

I. 1. Find the fifth power of (h — y) by the binomial theorem.

Solution Letters and exponents, 6^ 54^ ^3^2 52^3 j)y^ yb

Coefficients, 1 5 10 10 5 1

Signs, + - + - + -

Combined, fcs _ 5 ^4^ + 10 53^2 _ 10 b^y^ -\- 6 by^ - y

Expand :

2. (m + ny, 10. (x-yy, 18. (a; + 4)8.

3. {m-ny. ^ 11. (c-n)^ 19. (aj + 5/.

4. (a-c)l 12. (a; -ay. 20. {x-2y.

5. (a + 6)3. 13. {d-yy. 21. (a + ^^c^.

6. (6 + dy. 14. {h + yy. 22. (a6-cy.

7. (q-ry. 15. (m + w)*. 23. (m-pw)*.

8. (c + dy. 16. (a; + 2y. 24. (ax-byy.

9. (a; + 2/)8. 17. (a + S/. 25. (ax-byy.

26. Expand (a — ic)*; then (2 c^ — 5)^ by the same method.

Solutions (a - a;)4 = a* - 4 a% + 6 a^x"^ - 4 ax^ + x*. (2 c2 - 5)* = (2 c2)4 - 4 (2 c2)3 5 4- 6 (2 0^)2 52 - 4 (2 c2)58 + 5* = 16 c8 - 160 c6 + 600 c4 - 1000 c2 + 626.

27. Expand (1 + x^y.

Solution

(1 + a;2)8 = 13 + 3 (1)2(X2) + 3(1)(X2)2 + (X2)8

= 1 + 3 x2 + 3 x* + x6.

Test.— When x = 1, (1 + x2)8 = 8, and 1 + Sx* + 3x* + x« = 8 ; hence, (1 + x2)3 = 1 + 3x2 + 3x4 + x6, and the expansion is correct.

i

INVOLUTION

213

Expand, and test results :

28. (x + 2y)\ 33. {l-Sx'y.

29. (2x-yy.

30. (2x-5y.

31. (x'-ioy.

32. (3a + b^y. Expand :

... (!+.)'.

34. (x^-\-5yy.

35. (l + a262/.

36. (ox^-aby.

37. (2aa;-6)^

44

45.

46. f^-^Y-

55. Expand {a — b

47. 48. 49. 50.

■ cy.

Solution

38. (l-a;y.

39. {l-2xy.

40. (a2a; + 4/.

41. (x-i)«.

42. {ix-iyy.

51.

52.

53.

54.

hlj-

2x

(^--f)'

(a — 6- c) 3 =(«—&— c) 3, a binomial form,

= (a - &)3_ 3(a - 6)2c + 3(a - 6)c2 - c^

= a3-3 a26 +3 a&2_^3_3 c(rt2._2 a& +62) -|-3 ac2-3 bc^-c^

56. Expand (a-{-b — c — df.

Suggestion, (^a + b — c — dy = (a + b - c + d)^, a binomial form. Expand :

57. (a~x + yy. 60. (a + 2&-3c)'.

58. (a — x — yy, 61. (a + 6 -f- a? + 2/)^

59. (a + a; + 2)3. 62. (a-\-b-x-yy. The Binomial Theorem will be treated more fully in §§ 649-557.

EVOLUTION

I

283. The process 5^ = 5 • 5 = 25 illustrates involution. The process V25 = V5 -5 = 5 illustrates evolution, which

will be defined here as the process of finding a root of a num- ber, or as the inverse of involution.

For example, V25 = 5, for 5^ = 25;

■\/^^ = -2, for (_2)3=-8.

In general, the nth root of a is a number of which the nth power

is a. 'M

284. Since 25 = 5^ and also 25 =(- 5)(- 5) = (- 5)^, '

V25=+5or ~5.

The roots may be written together thus : ± 5, read ^plus or minus Jive.'

Or they may be written T- 5, read ' mimis or plus Jive.*

Similarly, V36= ± 6, V49 = ± 7, V| = ± f Every positive number has two square roots.

285. The square root of — 16 is not 4, for 4^ = + 16 ; nor — 4, for (— 4)2=-}-16. No number so far included in our number system can be a square root of —16 or of any other negative number.

It would be inconvenient and confusing to regard Va as a number only when a is positive. In order to preserve the generality of the discussion of number, it is necessary, there- •fore, to admit square roots of negative numbers into our num- ber system. The square roots of — 16 are written

V-16and -V-16.

214

r

EVOLUTION 215

Numbers that involve a square root of a negative number are called imaginary numbers. Numbers that do not involve a square root of a negative number are called real numbers.

Having extended the number system, we may now state the principle that every number has two square roots, one positive and the other negative.

286. Just as every number has two square roots, so every number has three cube roots, four fourth roots, etc.

For example, the cube roots of 8 are the roots of the equa- tion a;' = 8, which later will be found to be

2, _ 1 -f v^iirs, and -1- V-3. The present discussion is concerned only with real roots.

287. Since 23 = 8, ^8 = 2.

Since (-2y=-S, ■V^^S=-2.

Since 2^ = 16 and ( - 2)* = 16, ■\/l^=±2.

Since 2^ = 32, v32 = 2.

Since (_2)^=-32, V-32=-2.

A root is odd or even according as its index is odd or even.

It follows from the law of signs for involution that :

Law of Signs for Real Roots. — An odd root of a number has

same sign as the number. An even root of a positive number may have either sign. An even root of a negative number is imaginary.

288. A real root of a number, if it has the same sign as the lumber itself, is called a principal root of the number.

The principal square root of 25 is 5, but not — 5. The principal cube )otof 8 is 2; of - 8 is —2.

289. Axiom 7. — The same roots of equal numbers are equal.

Thus, if X = 16, Vx = 4 ; if x = 8,^x = 2j etc.

216 EVOLUTION J

290. Since {2^ = 2''^^ = 2% the principal cube root of 2« is

-^2^ = 2*5-3 = 21

Law of Exponents for Evolution. — The exponent of any^ root of a niimher is equal to the exponent of the nuinher divided by the index of the root.

291. 1. Since (5 ay = SW = 25 a-, the principal square root of 25 a^ is

. V25~^'= V25. V^= 5 a.

fd\^ a* a*

2. Since ( - ) = — , the principal fourth root of — is \bj b^ b^

4ja^ _ Va'* _ a

^b'~^f4~b'

Distributive Law for Evolution. — Any root of a product may

be obtained by taking the root of each of the factors and finding

the product of the results.

Any root of the quotient of two numbers is equal to the root of

the dividend divided by the root of the divisor,

292. Evolution of monomials.

EXERCISES

1. Extract the square root of 36 d%^.

Solution. — Since, in squaring a monomial, §278, the coefficient is squared and the exponents of the letters are multiplied by 2, to extract the square root, the square root of the coefficient must be found, and to it must be annexed the letters each with its exponent divided by 2.

The square root of .SO is 0, and the square root of the literal factors is a^b. Therefore, the principal square root of 36 a^h^ is 6 a^h.

The square root may also be — 6 a^6, since —Qa^bx — 6 a^b = 36 a^h^.

.'. \/36 a^&a = ± 6 a^b.

2. Extract the cube root of — 125 afy^.

Solution. y/— 125 a^y^i = «. 5 x^y~, the real root.

EVOLUTION

217

To find the root of an integral term :

EuLE. — Extract the required root of the numerical coefficient, annex to it the letters each with its exponent divided by the index of the root sought, and prefix the proper sign to the result.

Find real roots : 3. -s/'^c^W\

4. Va«6^«c'*.

5. ^Ay^

6. ^/tt^^W^:

7. Va;'*"2/^2;^'*.

8. -^ - 8 o^W\ 13. V(-m63)2.

9. v'-32a;V- 14. V{- a'bf.

10. V16V/. 15. -V^i}W^.

11. V-a''b''x'\ 16. --v/-27i)V.

12. -v/-243 2/'^ 17. -■v/-128ai^w28.

18. Extract the cube root of -^'

27 m»n*2

Solution, i/^^^ggg^ v^^ggg^ - 2 x^^ _ 2xV.

To find the root of a fractional term :

Rule. — Fi7id the required root of both numerator and denomi- itor and prefix the proper sign to the resulting fraction.

Find real roots

19

22. \l-^^€'

25.

20. A^-yr

23.

256 g;^ 6561

5/-32aV'> 3' 125 a;V

• \ 243 V' ■ ^^ 1728 c«

26. "V/ : — - — ■'

218

EVOLUTION

293, To extract the square root of a polynomial.

EXERCISES

1. Find the process for extracting the square root of

PROCESS

(

a' + 2ab + bK

2ah-{-W 2ab + b^

Trial divisor, 2 a Complete divisor, 2 a + 6

Explanation. — Since a^ + 2ab -\- b^ is the square of (a + &), we know that the square root of a^ _f. 2 a6 + 6^ is a + b.

Since the first term of the root is a, it may be found by taking the square root of a^, the first term of the power. On subtracting a^, there is a remainder of 2 a6 + b^.

The second term of the root is known to be &, and that may be found by dividing the first term of the remainder by twice the part of the root already found. This divisor is called a trial divisor.

Since 2ab + d^ is equal to &(2a + 6), the complete divisor which multiplied by b produces the remainder 2 ab + b^ is 2 a + & ; that is, the complete divisor is found by adding the second term of the root to twice the root already found.

On multiplying the complete divisor by the second term of the root and subtracting, there is no remainder ; then, a + 6 is the required root.

2. Extract the square root of 9 a^ — 30 a;?/ -f 25 y^. ^

PROCESS

I

9 a^- 30 xy + 25 ?/|3 a?-5y 9a^

Trial divisor, 6x

Complete divisor, 6x — 5y

Extract the square root of :

3. 4ar^ + 12a;4-9.

4. a^ + 2a; + l.

5. 1 — 4m + 4m^

-30xy + 25f -30xy-{-25f

6. c2-12c + 36.

7. 4a^ + 4a; + l.

8. 16 + 24 a; + 9a^.

Since, in squaring a-^-b-^-c, a + & may be represented by x, and the square of the number hj x'^-\-2 xc -h c^, the square root

EVOLUTION 219

of a number whose root consists of more tlmn two terms may be extracted in the same way as in exercise 1, by considering the terms already found as one term.

9. Find the square root of 4 ic* -f- 12 ar* — 3 ar' — 18 .r + 9.

I

PROCESS

4a;^ + 12a^-3a^-18a; + 9|2a^4-3a;

4aj2 + 3a;

12a^-3a;2 12a^ + 9a^

4aj^ + 6a; 4a^ + 6a;-3

-12«2_ig^_^9 -12a52-18a; + 9

Explanation. — Proceeding as in exercise 2, we find that the first two terms of the root are 2 x^ -{- Sx.

Considering (2 x"^ + 3 x) as the first term of the root, we find the next term of the root as we found the second term, by dividing the remainder by twice the part of the root already found. Hence, the trial divi or is 4 x2 + 6 x, and the next term of the root is — 3. Annexing this, as before, to the trial divisor already found, we find that the complete divisor is 4 a;2 -f 6 X — 3. Multiplying ihis by — 3 and subtracting the product from — 12 a;2 — 18 x + 9, we have no remainder. Hence, the square root of the number is 2 x-^ + 3 x — 3.

Rule. — Arrange the terms of the polynomial with reference to the consecutive powers of some letter.

Extract the square root of the first term, write the result as the first term of the root, and subtract its square from the given polynomial.

Divide the first term of the remainder by twice the root already found, used as a trial divisor, and the quotient will be the next term of the root. Write this result in the root, and annex it to the trial divisor to form the complete divisor.

Multiply the complete divisor by this term of the root, and sub- tract the product from the first remainder.

Find the riext term of the root by dividing the first term of the remainder by the first term of the trial divisor.

Form the complete divisor as before and continue in this man- ner until all the terms of the root have been found.

220 EVOLUTION

Extract the square root of :

10. 25a2_40a + 16. 13. 4 oj* - 52 o^ + 169.

11. 900a;2_^5Q,y^j^^ 14 ji fP _ 2 ^^^^ ^ ^ n\

12. x^-i-xy + ly\ 15. (a + bf - A (a -{- b) + 4.

16. 9x'-12a^ + 10x'-4:X-^l.

17. x^-ea^y-^-lSx^y'-Uxy^-i-Ay*.

18. x^ + 2a'x^-a'x'-2a^x' + a\

19. 25a;^+4-12a;-30a;'^ + 29a^.

20. l-2a;+3a;2_4«^ + 3aj''-2ar' + a^. ' 21. a'^-2a'b + 2a'c^-2bc^-\-b^-]-c\

22. 4 a^ - 12 a& + 16 ac + 9 6^ + 16 c^ - 24 be,

23. 9a^-\-25y^-\-9z^-S0xy + lSxz-30yz.

25. -?^^ + 15 + 9?i2. 28. a^2^2aj-l---}--- 4 n^ XX-

26. — -^ + 4. 29. a''+aJ^+-7^7- + ^ + ,T^'

16 ?-^ r 20 5 25

30. x«4-4a;^-3a;*-20a;^-2a;«4-4 + 4a;2_;j^g^_,_32aj3 „, 4:X* 4ar^ 3a;2 2aj .

y f y y

Oft ^^'^ I ^ . 4aV , 2a.r^ , «*

32. ^ + a-^.H-^ + — +-.

33. ^'_i^V^+4a^-2a5 + ^. 9 3 3 4

4771*^ 4m^ 19m^ 3m^ 73m^ 3m 9 ' 9 3 15 "^ 5 50 "^ 10 "^16*

EVOLUTION 221

36. Find four terms of the square root oil + x. Solution

1

2 + lx

2 + x-^x'^

x+ jx^

- \x^

2+x-ix^ + ^x^ 1 Ix^-^k^

Find the square root of the following to four terms :

37. 1-a. 39. x^^l, 41. / + 3.

38. a^^l. 40. 4 -a. 42. a^ + 2b.

SQUARE ROOT OF ARITHMETICAL NUMBERS

294. Compare the number of digits in the square root of each of the following numbers with the number of digits in the number itself :

MBBR	ROOT	NUMBER	BOOT	NUMBER	ROOT
1	1	I'OO	10	I'OO'OO	100
25	5	10'24	32	56'25'00	750
81	9	98'01	99	99'80'01	999

From the preceding comparison it may be observed that :

Principle. — If a number is separated into periods of tivo digits each, beginning at units, its square root will have as many digits as the number has periods.

The left-hand period may be incomplete, consisting of only one digit.

295. If the number of units expressed by the tens' digit is

represented by t and the number of units expressed by the

units' digit by u, any number consisting of tens and units will be

represented by t + u, and its square by (t + uf, ov t'- -{ 2 tu -\- u\

L Since 25 = 20 + 5, 252 = (20 + 5)2 = 202 + 2 (20 x 5) + 3^ ^ 625.

FIRST PROCESS
38'44|60 4-2 f= 36 00
2 i = 120	244
u= 2
2^ + ^=122	2 44

222 EVOLUTION

EXERCISES

296. 1. Extract the square root of 3844.

Explanation. — Separating the number into periods of two digits each (Prin., § 294), we find that the root is composed of two digits, tens and units. Since the largest square in 38 is 6, the tens of the root cannot be greater than 6 tens, or 60. Writing 6 tens in the root, squaring, and subtracting from 3844, we have a remainder of 244.

Since the square of a number composed of tens and units is equal to {the square of the tens) + (twice the product of the tens and the units) + {the square of the units), when the square of tire tens has been subtracted, the remainder, 244, is twice the product of the tens and the units, plus the square of the units, or only a little more than twice the product of the tens and the units.

Therefore, 244 divided by twice the tens is approximately equal to the units. 2x6 tens, or 120, then, is a trial, or partial, divisor. On dividing 244 by the trial divisor, the units' figure is found to be 2.

Since twice the tens are to be multiplied by the units, and the units also are to be multiplied by the units to obtain the square of the units, in order to abridge the process the tens and units are first added, forming the complete divisor 122, and then multiplied by the units. Thus, (120 + 2) multiplied by 2 = 244.

Therefore, the square root of 3844 is 62. j

SECOND PROCESS

38^44 [62

f=	36
2 ^ = 120 u= 2	2 44
2« + tt = 122	2 44

Explanation. — In practice it is usual to place the figures of the same order in the same column, and to disregard the ciphers on the right of the products. j

Since any number may be regarded as composed of tens and •anits, the foregoing processes have a general application.

Thus, 346 = 34 tens + 6 units j 2377 = 237 tens + 7 units.

EVOLUTION ^23

2. Extract the square root of 104976. Solution

10'49'76|324 9

Trial divisor = 2 x 30 = 60

Complete divisor = 60 + 2 = 62

149 124

Trial divisor = 2 x 320 = 640

Complete divisor = 640 -f- 4 = 644

25 76 25 76

Rule. — Separate the number into periods of two figures eachj beginning at units.

Find the greatest square in the. left-hand period and write its root for the first figure of the required root.

Square this root, subtract the result from the left-hand period^ and annex to the remainder the next period for a new dividend.

Double the root already found, with a cipher annexed, for a trial divisor, and by it divide the dividend. The quotient, or quotient diminished, will be the second figure of the root. Add to the trial divisor the figure last found, multijjly this complete divisor by the figure of the root last found, subtract the product from the dividend, and to the remainder annex the next period for the next dividend.

Proceed in this manner until all the periods have been used. The result will be the square root sought.

1. When the number is not a perfect square, annex periods of decimal ciphers and continue the process.

2. Decimals are pointed off from the decimal point toward the right.

3. The square root of a common fraction may be found by extracting the square root of both numerator and denominator separately or by reducing the fraction to a decimal and then extracting the root.

Extract the square root of :

3. 529. 6. 57121. 9. 2480.04.

4. 2209. 7. 42025. 10. 10.9561. 6. 4761. 8. 95481. 11. .001225.

224 EVOLUTION

12.	186624.		13.	1332.25.		14. 1
15.	m-	17.	m-	19.	289 32T-
16.	m-	18.	i\\\	20.	m-
Extract the	square	root to four deci	mal pi	aces:
23.	i-	25.	5 8-	27.	5 6"*
24.	|.	26.	.6.	28.	1-
297. To extract the cube root of a po	lynomi	al.

21. fff.

22. Iff.

29. f

30. tV

EXERCISES

1. Find the process for extracting the cube root of a^+S o?h

PROCESS

Trial divisor, 3 o? Complete divisor, 3 a^ + 3 a6 + 6^

3a26 + 3a62_f_53 i

Explanation. — Since a'* + 3 a^ft + 3 ah'^ + 6^ is the cube of (a + 6), fie, know that the cube root of a^ 4- 3 a'^h + 3 aft^ + &3 is a + 6.

Since the first term of the root is a, it may be found by taking the cube root of a^, the first term of the power. On subtracting, there is a re- mainder of 3 d^h + 3 «&2 ^ ^3^

The second term of the root is known to be &, and that may be found Dy dividing the first term of the remainder by 3 times the square of the part of the root already found. This divisor is called a tnal divisor.

Since 3 a^fo + 3 ah"- + 6'^ is equal to h (3 cfi + 3 a6 + h'^), the complete divisor, which multiplied by 6 produces the remainder 3 ol^h +3 aft^ -f h^^ Is 3 a2+3 a6 + h"^ ; that is, the complete divisor is found by adding to the trial divisor 3 times the product of the first and second terms of the root and the square of the second term of the root.

On multiplying the complete divisor by the second term of the root, and on subtracting, there is no remainder ; then, a + 6 is the required root.

Since, in cubing a-\-h-\-c, a + 6 may be expressed by ic, the cube of the number will be £c^ + 3a;^c + 3a;c^-|-c^. Hence, it is obvious that the cube root of an expression whose root consists of more than two terms may be extracted in the same way as in exercise 1, by consideHng the terms already found as one term.

Sb' Sb'-Sb^-hb'	-36^ + 56^ -3b'-^3b*-b^
3b'-6b'+3b' 3b'-6b'-{-3b + l	-3b' + 6b'- -36*4-66^-	-35-1 -36-1

EVOLUTION 225

2. Find the cube root oi b^ -3 b' + 5b^ - 3 b -1,

PROCESS

Trial divisor, Complete divisor, Trial divisor. Complete divisor,

Explanation. — The first two terms are found in the same manner as in the previous exercise. In finding the next term, b'^ — & is con- sidered as one term, which we square and multiply by 3 for a trial divisor. On dividing the remainder by this trial divisor, the next term of the root is found to be — 1. Adding to the trial divisor 3 times (62 _ 6) multiplied by — 1, and the square of — 1, we obtain the com- plete divisor. On multiplying this by — 1, and on subtracting the product from — 3 6* + 6 6^ — 36 — 1, there is no remainder. Hence, the cube root of the polynomial is b^ — b — 1.

Rule. — Arrange the polynomial with reference to the consecu- tive powers of some letter.

Extract the cube root of the first term, write the result as the first term of the root, and subtract its cube from the given polynomial.

Divide the first term of the remainder by three times the square of the root already found, used as a trial divisor, and the quotient will be the next term of the root.

Add to this' trial divisor three times the product of the first and second terms of the root, and the square of the second term. The result will be the complete divisor.

Multiply the complete divisor by the last term of the root found, and subtract this product from the dividend.

Find the next term of the root by dividing the first term of the remainder by the first term of the trial divisor.

Form the complete divisor as before, considering the part of the root already found as the first term, and continue in this manner until all the terms of the root are found. milne's stand, alq. — 15

226 EVOLUTION

Find the cube root of :

3. 63_|_g^2_|_i2 6+8.

4. a^ — 3 x^y -j- 3 xy^ — 9^.

5. m3-9m2 + 27m-27.

6. a^a;^ + 12 a^a.-^^- 48 aaj^H- 64.

7. 8 77i3-60m27i + 150mn2_125w'.

8. 2Ta^-lS9x'y-{-Ulxy^-S43f.

9. 1000p6_300p4g + 30j9V-5^.

10. 125a3 + 675a2a;-t-1215aa?2-f-729a?3.

11. 64 a^b^ - 240 aWc + 300 a&c^ - 125 c\

12. a;«-6a;5 + 15a;4-20a;3 + 15a;2-6a;4-l.

13. x^-{-Sx^ + 9x'-hlSa?-^lSx'-\-12x + S.

14. m^ + 6 m^ + 15 m^ + 20 m^ + 15 m^ + 6 m + 1.

15. 2/' + 32/^ + 122/^ 4- 19 2/3 .^362/24. 27 2/ + 27.

16. x^ + 12 a;5 + 63 x' + 184 a^ H- 315 x^ + 300 a^ -f 125.

17. a^ + 6 a;5 - 18 a^ - 1 000 + 180 «2 - 1 12 a^ + 600 a;.

18. a« + 9a5 + 21a''-9a3_42rt2_|_36a-8.

19. b^ + 3¥ + 6b'' + 7b^-\-6b' + Sb'^b\

20. 8 c« - 60 c^ 4- 198 c' - S65c^ + 396 c^ - 240 c + 64.

21. 1 - 6a + 21 a2 _ 44 a3 ^ 63 ^4 _ 54 ^5 ^ 27 a\

22. 8 ^3 + 42 w^ - 9 n^ + 36 ri^ _|_ 9 n^ _ 21 n^ - n^

23. 6 a^2/'^ - 3a;^2/ 4- a?*' — 7 a^2/^ + ^ + 6 a; V — 3 xy^

24 . c6 - 3 c^d - 3 c^(/2 + 1 1 c^rfs + 6 c^cZ^ - 12 cd^ - 8 (f .

25. aj3-12a;2 + 54x-112 + i5?_M_^i.

a; a?^ ar

EVOLUTION

227

c^ b^ b c

27. x'-{-15x''-^—-\-20 + --\-- + 6ai*. x~ x'^ x^

Qi? 2x^ 4.x 8 2

29. n«-f n5 + |7i^--i/n3_|.|y^2_3^_l_i^

30. 2V ?-« - 1 7-^ - f r^ + -2/ »^ H- V- *•' - -¥^ - 27.

31. ^x'-^-lx'y + xY-T^f-^x'y' + ^xy^-^^f.

CUBE ROOT OF ARITHMETICAL NUMBERS

298. Compare the number of digits in each number and in its cube root :

NUMBER BOOT	NUMBER	KOOT	NTTMBBB	BOOT
1 1	I'OOO	10	I'OOO'OOO	100
27 3	27'000	30	27'000'000	300
729 9	970'299	99	997'002'999	999
Observe that :

Principle. — If a number is separated into periods of three digits each, beginning at units, its cube root will have as many digits as the number has periods.

The left-hand period may be incomplete, consisting of only one or two digits.

299. If the number of units expressed by the tens' digit is represented by t, and the number of units expressd by the units' digit is represented by u, any number consisting of tens and units will be represented by ^ + u, and its cube by (t + w)', or ^3 _^ 3 fu -f 3 ^^2 ^ u\

Thus, 25 = 2 tens + 5 units, or (2'0 + 5) units,

and 255 = 203 + 3(202 x 5) + 3(20 x 5^) + 53 = 15625.

228 EVOLUTION

EXERCISES

300. 1. Extract tlie cube root of 12167.

FIRST PROCESS

12'167 120 + 3 f= 8 000

3^2 = 1200 3tu^ 180 ^2= 9	4167
= 1389	4167

Trial divisor,

Complete divisor,

Explanation. — On separating 12167 into periods of three figures each (§ 298, Prin.) there are found to be two digits in the root, that is, the root is composed of tens and units. Since the cube of tens is thousands, and the thousands of the power are less than 27, or 3^, and more than 8, or 23, the tens' figure of the root is 2. 2 tens, or 20, cubed is 8000, and 8000 subtracted from 12167 leaves 4167, which is equal to 3 times the tens^ x the units + 3 times the tens x the units"-^ + the units^.

Since 3 times the tens^ x the units is much greater than 3 times the tens X the units^ + the units^, 4167 is only a little more than 3 times the tens2 X the units. If, then, 4167 is divided by 3 times the tens^, or by 1200, the trial divisor, the quotient will be approximately equal to the units, that is, 3 will be the units of the root, provided proper allowance has been made for the additions necessary to obtain tlie complete divisor.

Since the complete divisor is found by adding to 3 times the tens^ the sura of 3 times the tens x the units and the units^, the complete divisor is 1200 + 180 + 9, or 1389. This multiplied by 3, the units, gives 4167, which, subtracted from 4167, leaves no remainder.

Therefore, the cube root of 12167 is 20 + 3, or 23.

SECOND PROCESS

12'167 I 23 Explanation. — In practice it is usual to

place figures of the same order in the same column, and to disregard the ciphers on the right of the products.

f =	8
3t^ =1200 3tu^ 180 u^= 9	4167
1389	4167

Since a root expressed by any num- ber of figures may be regarded as composed of tens and units, the pro- cesses of exercise 1 have a general application. Thus, 120 = 12 tens + 0 units ; 1203 = 120 tens + 3 units.

EVOLUTION 229

Extract the cube root of 1740992427.

Solution

1'740'992'427 { 1203 «3 = 1

^g rst^ =3(10)2 ^ 300

^■^ U^M =3(10x2) = 60 u'^ =22 =4

304 8 f2 = 3(120)2 = 43200

740

728

g p t2 = 3(1200)2 = 4320000 ll \3tu = 3(1200 X 3) = 10800 I f w2 = 32 = 9

4330809

12992

12992427

12992427

Since the third figure of the root is 0, it is not necessary to form the complete divisor, inasmuch as the product to be subtracted will be 0.

EuLE. — Separate the number into periods of three figures each, beginning at units. Find the greatest cube in the left-hand period, and write its root for the first digit of the required root.

Cube this root, subtract the result from the left-hand period, and annex to the remainder the next period for a new dividend.

Take three times the square of the root already found, annex turn ciphers for a. trial divisor, and by the result divide the divi- dend. The quotient, or quotient diminished, will be the second figure of the root.

To this trial divisor add three times the p)roduct of the first part of the root with a cipher annexed, multiplied by the second part, and also the square of the second part, Tlieir sum ivill be the complete divisor.

Multiply the complete divisor by the second part of the root, and subtract the product from the dividend.

Continue thus until all the figures of the root have been found.

1. When there is a remainder after subtracting the last product, annex decimal ciphers, and continue the process.

2. Decimals are pointed off from the decimal point toward the right.

3. The cube root of a common fraction may be found by extracting the cube root of the numerator and the denominator separately or by reduc- ing the fraction to a decimal and then extracting its root.

JO	EVOLUTION
Extract the cube root of :
3. 29791.	9. 2406104.	15.	.000024389.
4. 54872.	10. 69426531.	16.	.001906624.
5. 110592.	11. 28372625.	17.	.000912673.
6. 300763.	12. 48.228544.	18.	.259694072.
7 681472.	13. 17173.512.	19.	926.859375.
8. 941192.	14. 95.443993.	20.	514500.058197.
Extract tlie cube root to three decimal	places	j:
21. 2.	23. .8. 25.	A.	27. f ,
22. 5.	24. 16. 26.	1-	28. A. ^

ROOTS BY VARIOUS METHODS

301. By inspection and trial.

To find the cube root of a number, as 343, we estimate the root and cube it. If the cube is greater or less than the num- ber, our estimate must be modified, for the cube of the root must be the number itself.

This method, which is the general one in evolution, may be used to find any root of a polynomial.

By inspection we estimate \/x^ — 10 ic* + 40 x^ — 80 ic^ + 80 a; — 82 to be a; — 2, noting the number of terms and the first and last terms. By trial x—2 proves to be the root, for its fifth power is found to be the given polynomial.

302. By factoring.

This method consists in factoring, grouping the factors, and ■ extracting the root of each group.

Thus, v^2875 =V'5.5.6.7.7.7= V^^ = 5 . 7 = 35;

Hlso, \/iC* + 2a;8-3a;2_4a; + 4 = V{x - l)2(a; + 2)2 = (a; - !)(« + 2). = x^ + x-2.

EVOLUTION

231

303i By successive extraction of roots.

Since the fourth power is the square of the second power, the sixth i)ower the cahe of the second power, etc., any indicated root whose index is 4, 6, 8, 9, etc., may be found by extracting successively the roots corresponding to the factors of the index.

The fourth root may be obtained by extracting the square root of the square root ; the sixth root, by extracting the cube root of the square root, or the square root of the cube root ; the eighth root, by extracting the square root of the square root of the square root.

EXERCISES

304. Using any method, find the :

1. Square root of a« - 12 a^ + 36.

2. Cuberootof 125-75a; + 15a;2_^<j^

3. Fourth root of 16 - 32 a; + 24 a.-^ _ 8 x" + x\

4. Fourth root of x^ + 12 a^y + 54 a^/ + 108 xf + 81 y\

5. Fourth root of 16 m^ - 32 in^ + 24 m^ - 8 m + 1.

6. Fifthroot of 32a.'^ + 80a;^ + 80a;3 + 40a^ + 10a; + l.

7. Fifth root of a^«+ 15 a^ + 90 a« + 270 a' + 405 a^^ 243.

Find the sixth root of :

8. a;«-12a;^ + 60aj^-160a^-f240a:2_i92a; + 64.

9. 64 x^ _ 576 aj^ + 2160 x' - 4320 :^ + 4800 x^ - 2916 x + 729. 10. x^+^ ac^ + 15 a?(rx^ -f 20 aVx^ -f 15 a^cV Ji^Qa'd'x + a^c\

Find the indicated root :

11. V3375.

12. -^^1296.

13. a/50625.

14. -\/46656.

15. v/262144.

19. -v'4084101.

16. V759375.

17. v/531441.

20. V16777216.

21. -VMiSTEed.

18. V5764801.

22. V10604499373.

THEORY OF EXPONENTS

305 Thus far the exponents used have been positive integers only, and consequently the laws of exponents have been obtained in the following restricted forms :

1. a"* X a" = a"*+" when m and n are positive integers.

2. a'" -i- a" = a*^"" when m and n are positive integers and m is greater than n.

3. (a"*)"=a'"'* when m and n are positive integers.

4. -v/a"* = a"*"^" when m and w are positive integers, and m is a multiple of n.

5. (a&)" = a"6" when n is a positive integer.

If all restrictions are removed from m and n, we may then have expressions like a~^ and a*. But such expressions are as yet unintelligible, because — 2 and -| cannot show how many times a number is used as a factor.

Since, however, ^ese forms may occur in algebraic processes, it is important to discover meanings for them that will allow their use in accordance with the laws already established, for otherwise great complexity and confusion would arise in the processes involving them.

Assuming that the law of exponents for multiplication,

a"* X a" = a"'^", is true for all values of m and w, the meanings of zero, negative^ Q.nd fractional exponents may be readily discovered by substi- tuting these different kinds of exponents for m and n or both, and observing to what conclusions we are led.

232

THEORY OF EXPONENTS 233

306. Meaning of a zero exponent.

We have agreed that any new kind of exponent shall have its meaning determined in harmony with the law of exponents for multiplication, expressed by the formula,

If n = 0, oT" xa^ = a'"+^ or oT,

Dividing by a"*, Ax. 4, a^ = ^ = 1. That is,

Any number {not zero) with a zero exponent is equal to 1.

307. Meaning of a negative exponent.

Since, § 305, a"* x a" = a'""*"", is to hold true for all values of

m and n, if m = — n,

a-'* X a" = a-'*+" = aP.

But, §306, a" = l. •

Hence, Ax. 5, a"" x a" = 1.

Dividing by a", Ax. 4, «-"= — • That is,

Any number with a negative exponent is equal to the reciprocal of the same number ivith a numerically equal positive exponent.

308. By the definition of negative exponent just given,

a""* = — and &~" = — •

Therefore, ^=:^ = ix^=— . Hence,

Principle. — Any factor may be transferred from one term of a fraction to the other without changing the value of the frac- tion j provided the sign of the exponent is changed.

234 THEORY OF EXPONENTS

309. Find a simple value for :

1. 5". 3. 2-^ 5. (-3)«. 7. {a%''q)\

2. 4-2. 4. 3-3. 6. (-6)-2. 8. (-i)-'., 9. Which is the greater, (i)^ or {\f ? 0-2 or (|) "^ ? |

10. Find the value of 2^-3 • 2^ + 5 • 2^-7 • 2«4-4 • 2-1-2-2.

11. Find the value ofar^ — 3iciH-4aJ" + a;-^ — 5 aj-^ _)_ a;-* when x = ^] whena; = — |-; when a; = 1.

12. Which is the greater, (-J)-3or {\f? (_|)-4or (i)*?

Write with negative exponents :

13. 1^5. 15. 1^2". 17. c-f-a^ajs,

14. iH-al 16. a-^a?. 18. am^-i-hx\ %

19. Write 5 a;~y with positive exponents.

Solution. — By § 307, 5 x-^i'^ = ^y2l. = ^. j

Write with positive exponents :

20. 2a;-^ 23. a-^b-\ 26. 4a2c-*.

21. 5a-\ 24. x-^y-\ 27. 3 aaj-^

22. 3 6-2. 25. a-i62g-3. 28. a^^-^".

29. 4a^-2a^ + 5a;i-6a;« + 3a;-i-5a;-2.

30. 2(i3_l2a2_i6a-|-3a«4-2a-i-7a-2.

31. a^b-^ - a^b-^ + a^"^ - 1 + a-'^b - a-^b"" + a-^b\

3 (x2w 32. Write — ~ without a denominator. bar

Solution. — By § 308, §^ = 3 a^b'^x-^y.

4

Write without a denominator : ax mn

by' a2* — a-262

33. ^. 34. ^. 35. -^. 36.

THEORY OF EXPONENTS 236

37. ^^. 40. -1-. 43. -1- 46.

($)■

39. A. 42. ?-^^. 45. f-]. 48

h'^y \mj (aby

310. Meaning of a fractional exponent.

Since (§ 305) the first law of exponents is to hold true for all exponents, ^ . j .

that is, a^ is one of the two equal factors of a, or is a square root of a. The other square root of a is — a^.

Again, a^ X a^ = a^'^^ = a^ ;

that is, a^ is a square root of the cube of a. Or, similarly, a^ x a^ X a^ = a^"^^^^ = a^;

that is, d^ is the cube of a square root of a.

In general, confining the discussion to principal roots, let p and q be any two positive integers. By the first law of

€ P ? + ?+••• to « terms

exponents, § 305, a« • a* ••• to g factors = a« « = a^.

p Therefore a*, one of the q equal factors of a^, is a gth root

of the pth. power of a.

Similarly, a* is a gth root of a.

,, . ? 1 ^ ! + !+... topterm. ? ?.

Also, since a'^^a" -" top factors = a« « = a«, a* is

the pth power of a gth root of a.

The numerator of a fractional exponent with positive integral terms indicates a power and the denominator a root.

The fraction as a whole indicates a root of a power or a power of a root.

236 THEORY OF EXPONENTS

.311. Any fractional exponent that does not itself involve a root sign may be reduced to one of the forms -^ or —■^.

Thus, 8 2=8 '.

By §§ 307, 291, «"« = L =i!= f 1 K

EXERCISES

312. 1. Find the value of 16^

First Solution. 16^ = y/W^ = y/lQ • 16 . 16

= \/(2 . 2 . 2 . 2) (2 . 2 . 2 . 2) (2 . 2 . 2 . 2) = v^(2 . 2 . 2) (2 . 2 . 2) (2 . 2 . 2) (2 . 2 . 2) = 2.2.2 = 8.

Second Solution. 16^ = (16^)3 = 2^ = 8.

In numerical exercises it is usually best to extract the root first.

Simplify, taking only principal roots :

2. 8i 6. 643. 10. 64-1

3. 8l 7. 32t 11. (-8)-^

4. 8-i 8. 25t. 12. (-32)-l

5. (-8)i 9. 8lf. 13. 16"^

14. Which is the greater, 27^ or (—27)"^?

15. Which is the greater, (^)t or (i)"i?

16. Which is the greater, 64"^ or (^i^)l ? 64^ or (^)~^?

17. Find the value of x'^ — 4: x~^ + 4 when aj = — yi-g-.

18. Express -s/a^bc* with positive fractional exponents.

2 1

Solution. y/a^bc-^ = aH^c~^ = ^^^ »

THEORY OF EXPONENTS 237

Express with positive fractional exponents :

19. VoP. 22. (V«)*. 25. (■v/^)-2.

20. V^. 23. (\/^)*. 26. b^x-^y-\

21. V^. 24. {-y/aSf. 27. 2 v^ (a + 6)2.

Express roots with radical signs and powers with positive exponents:

28. ai 30. a;i 32. x^^. 34. a^-^x^.

29 a;*. 31. aW. 33. ah'^, 35. aj^-r-?/^.

.. 36. Simplify -v/^ + a;* + 8^+3 a;^- 5^^ -^27^.

37. Simplify 4.Vx + 5 a/' - 3 aj''^ + 2-\/«^ -8^-2 a;i

^ 313. Operations involving positive, negative, zero, and fractional exponents.

Since zero, negative, and fractional exponents have been defined in conformity with the law of exponents for multiplica- tion, this law hokls true for all exponents so far encountered. For the proofs of the generality of the other laws of exponents, see the author's Academic Algebra,

EXERCISES

314. Multiply:

1. a^bya-^. 3. a^ by a"''. 5. a^hy a\

2. a^ by a~\ 4. a by a~^. 6. x^ by x^.

7. ah^ by a^h\ 10. n-'^ by an^.

8. m^nbymVl 11- «"*- by a«-.

11 4 3 ^^'^ ^?^=^

9. a^6* by a~^6?. 12. a ^ by a 2 .

13. Multiply a;^2/~^ + ^^ + ^^'^ + ^^2/^ + 2/^ bj a;^2/^*

14. Multiply 2/" + a;-^/"+i + a;-y+2 ^ ^j-s^^n+s ]3y ^^z"".

I

1

I

238 THEORY OF EXPONENTS

15. Expand (aV^ + 1 + or^b^){ah-^ - 1 + a"W). First Solution ah~^-^ 1 ■\-a~h^ ah~^- 1 -\-a~h^ ah-^ + ah~^ + aW>

^ah~^- 1 -a~W

ah-^ + 1 +a~^6

Second Solution (^ah~^ + 1 + a'h^ {ah~^ - 1 + a"^6^) § 114, = (a^6"2 + a"^fti)2 _ 12

= (a^6-i + 2 a''5'^ + a"^6)- 1 = ah-'^ + 2 + a~^6 - 1 = Jb-^ + 1 + a"^6. Expand :

16. (a^ + b^)(a^ — b^). 20. (x^ — x^y~^ + y~^)(x^ -\-y~^).

17. (a;^ + 2/^)(a;^ — 2/*). 21. (a' + &"' + «V^ + l)(a^ — &"'">

18. (a;-^4-10)(a;-^-l). 22. (1 -a; + a^) (a;"' + »-' + «-') •

19. (a;'-4)(a;^ + 5). 23. (a-^ + ft'^ + c^)(a-' + &~^ + 2 c^)

Divide :

24. a* by a®. 26. a' by a"^. 28. a;^ by x^.

25. a^ by a®. 27. a;^ by a;"i 29. a;""^ by aj"-^.

30. Divide a;* + a^2/^4-2/^ by a;y,

31. Divide a-* + a-^b + 6^ by a-^ft.

32. Divide x* + 2aa?-^3 aV f a^x — a* by a^aJ*.

I

THEORY OF EXPONENTS

239

33. Divide b-^ + 3 a"^ - 10 a-^b by a V* - 2. Solution a^ft-i - 2)6-1 + 3 a~^ - 10 a-^b(^a~^ + 5 a'^b

6a~^-10a-i6 5 fl-i _ 10 a-ift

Divide :

34. a — 6 by a^ + 6^

35. a — 6 by a^ — 6^

36. a + b by a^ + fti

37. a^ 4- 62 by a' ^ ftf.

Simplify the following: 42. {a^y. 48. (-a*)l

49. (-a')*,

50. (—a^)-\

43. (a"^)«.

44. {a-y.

45

46.

Afa-l

47

. A/

a"^.

51. \a-^6-3.

52. A/a;%-3.

53. VaF¥.

Expand by the binomial formula:

60. (a^-b^y. 62. (a-^-b^y.

61. {a^ + b^y. 63. (a;-^-2/^y.

38. x—1 by a;^ + a;^ + l.

39. aj7 _ 2 + ic"^ by x^ — x~^.

40. 3 — 4 a;-^ + a?"^ by x-^ — 3.

41. a2-&3bya5 + 6i

54. (8-^)*.

55. (16-^)3.

56. (-^V«T^.

57. Q^aT^b'^yK

58. (im-'^r^)^.

59. (4a^"2/~^2^)^-

64. (a"^ + i)*.

65. (1-0;^.

Extract the square root of:

66. x^ + 2xi + Sx + 4.xi-hS-{-2x'^ + x-\

67. a^-\-y + 4:Z-^ — 2xy^-\-4:XZ-^ — 4:yh-\

68. a + 4 6^ + 9c^ — 4a^6^ + 6aM — 126^.

240 THEORY OF EXPONENTS

Extract tlie cube root of :

69. a2 + 6a^ + 12a^ + 8.

70. a-Sah^ + Sah^ — b^

71. 8x-^-12x~^y + 6x~^y^ — f.

72. x^-6x-{-15x^-20 + 15x~i — 6x-'^ + x~^.

73. Factor 4:X~^ — 9y~\ and express the result with positive exponents.

Solution. — By § 152, 4 x'^ - 9 ^/-^ = (2 x-'^ + 3 y-i) (2 x^^ - 3 ?/-i)

Factor, expressing results with positive exponents :

74. a-2-&-2 79. aj3-a;-3

75. 9-a;-2. 80. a^ + 2 + a-^

76. 16 -a-*. 81. b*-S-{-16b-*.

77. 27 — &-«. 82. 12-a;-^-a;-2. |

78. 6-3^2/"'- 83. 2-3a;-i-2a;-2.

Solve for values of x corresponding to principal roots, and test each result :

92. x'^ = 6. J

93. x'^ = U4:. ^

94. 25a;'^ = l.

95. a?^ = 243.

96. 05^ + 32 = 0.

97. x^ — a^ = 0.

98. i»^-64 = 0.

99. aj'^-27 = 0.

84.	x^ = 7.
85.	x^ = S.
86.	x^ = 9.
87.	x^ = Sl.
88.	i 0^^ = 72.
89.	x'^ = 12.
90.	\x^ = 25.
91.	2.'^ = A.

THEORY OF EXPONENTS

241

Simplify, expressing results with positive exponents :

100.

2-3

1

101. r-)

102.

103.

112.

113.

114.

115.

V4i

-I

104.

105.

\a'b-y '

16 ttT^V^

108.

-4-rK-5\4

109.

4aj-'» J

3 9^ X 32+'-

V^2-2

x-*y-^

loe. (^)

107. '''^"'^^"'

110.

111.

2-^ X 2-^^ 4-2 X 4-3 '

32^ 4- 125^ 81^ + 216^

Vb' X -^cF2

(9" X 3" X 9) - 27^

116.

27'- 3 g^ X 4 g-y

2fl7 ' X Va; x^ — y^

117.

32x33

120.

m

^ — n^

{x^ - 2/^)(a;^ + y2)

a~^ X a~^ X g^ {a + b)-'

118 ^y^ X '^^-^ X a;^ ^

Qr+l Qr+l

119. — -^—

m^ + n^

121. 2 +

6 A-i a^-CV^

V^

a; _ 3 a;^ _ 18

122. (lA^yf^A^v.

123.

^-ij * V c'x-'

124.

milne's stand, alg. — 16

r '

</>

</r

RADICALS

315. An indicated root of a number is called a radical ; the number whose root is required is called the radicand.

V5a, (a^)^, y/a'^ + 2, and (x-{- y)"^ are radicals whose radicands are, respectively, 5 a, x\ cfi + 2, and x -\- y.

316. An expression that involves a radical, in any way, is called a radical expression.

317. In the discussion and treatment of radicals only prin- cipal roots will be considered.

Thus, Vl6 will be taken to represent only the principal square root of 16, or 4. The other square root will be denoted by — ViG.

318. A number that is, or may be, expressed as an integer or as a fraction with integral terms is called a rational number.

3, i, .333, and V25 are rational numbers.

319. A number that cannot be expressed as an integer or as a fraction with integral terms is called an irrational number.

2^, v^, 1 + VS, and v 1 + VS are Irrational numbers.

320. An expression is irrational, if it contains an irrational number, otherwise it is rational.

321. When the indicated root of a rational number cannot be exactly obtained, the expression is called a surd.

V2 is a surd, since 2 is rational but has no rational square root. V 1 + V3 is not a surd, because 1 + V3 is not rational.

Radicals may be either rational or irrational, but surds are always irrational.

Both VJ and V3 are radicals but only VS is a surd.

242

RADICALS 243

322. The order of a radical or of a surd is indicated by the index of the root or by the denominator of the frac- tional exponent.

Va -{■ X and (b + xp are radicals of the second order.

323. A surd of the second order is called a quadratic surd; of the third order, a cubic surdj and of the fourth order, a biquadratic surd.

324. Graphical representation of a quadratic surd.

In geometry it is shown that the hypotenuse of a right triangle is equal to the square root of the sum of the squares of the other two sides; consequently, a quadratic surd may be rep- resented graphically by the hypotenuse of a right triangle whose other two sides are such that the sum of their squares is equal to the radicand.

Thus, to represent V5 graphically, since it may be observed that 5 = 22-1-12, draw OA 2 units in length, then draw AB 1 unit in length in a direction perpendicular to OA. Draw OB, completing the right-angled triangle OAB.

Then the length of OB represents VS in its relation to the unit length.

It will be observed that V5 can be represented graphically by a line of exact length, though it cannot be represented exactly by decimal figures, for V5 = 2.236 • • • , an endless decimal.

EXERCISES

325. Eepresent graphically :

1. V2. 3. Vi3. 5. V34. 7. V}.

2. VlO. 4. Vi7. 6. V53. 8.

326. In the following pages it will be assumed that irrational numbers obey the same laws as rational numbers. For proofs of the generality of these laws, the reader is referred to the author's Advanced Algebra.

244 RADICALS

327. A surd may contain a rational factor ^ that is, a factor whose radicand is a perfect power of a degree corresponding to the order of the surd. The rational factor may be removed and written as the coefficient of the irrational factor.

In Vs = V4 X 2 and y/bi = v^27 x 2, the rational factors are Vi and \^, respectively ; that is, V8 = 2 V2 and y/Ei = 3 v'2.

328. A surd that has a rational coefficient is called a mixedi surd.

2\/2, aVo^, and (a — 6) Va + b are mixed surds.

). A surd that has no rational coefficient except unity is called an entire surd.

V5, v^, and Va^ + x^ are entire surds.

330. A radical is in its simplest form when the index of the root is as small as possible, and when the radicand is integral and contains no factor that is a perfect power whose exponent corresponds with the index of the root.

W is in its simplest form; but Vf is not in its simplest form, because I is not integral in form ; VS is not in its simplest form, because the square root of 4, a factor of 8, may be extracted; ^^25, or 25^, is not in its simplest form, because 25* = (5"2)^ = 5^ = 6^, or v^.

REDUCTION OF RADICALS

331. To reduce a radical to its simplest form.

EXERCISES

1. Reduce V20 a^ to its simplest form.

PROCESS

V20a« = V4 a« X 5 = V4a« x V5 = 2 a^ V5

Explanation. — Since the highest factor of 20 cfi that is a perfect square is 4 a', V20 a' is separated into two factors, a rational factor vTo^, and an irrational factor Vs, that is, § 291, \/20a« = VW x VE. On extract- ing the square root of 4 a^ and prefixing the root to the irratjioEial factor as a coefficient, the result is 2 a'^Vb.

RADICALS 245

2. Eeduce V — 864 to its simplest form.

PROCESS

■^Tr864 = ^/ - 216 X 4 = -v^-216 X -^4 = - 6^/l

Rule. — Separate the radical into two factors one of which is its highest rational factor. Extract the required root of the rational factor^ multiply the result by the coefficient, if any, of the given radical, and place the product as the coefficient of the irra- tional factor.

Simplify :

3. V12. 9. Vl62. 15. V243aV«.

4. V75. 10. Vl8^. 16. ■v/l28a«6^ 6. -5^16. 11. VWb. 17. (245 aV')*-

6. Vl28. 12. -x/W?. 18. (a3-f-5a2)i

7. -^250. 13. V50a. 19. Vl8a;-9.

8. ^32. 14. \/640. 20. V^^^^2o?, 21. -y/bx^ — lOxy + byK 22. (3am2 + 6am + 3a)i

23. Reduce ^/— ^ to its simplest form.

^'2 2/3

PROCESS

a^ I a^ X2y [aF , /tt- a ,—-

2?= V27^ = W^ ^^ = 2?^23,

Explanation. — Since a radical is not in its simplest form when the expression under the radical sign is fractional, the denominator must be removed; and since the radical is of the second order, the denominator must be made a perfect square. The smallest factor that will accomplish tliis is 2 y. On multiplying the terms of the fraction by this factor, the larg-

-—. which is equal to — • Therefore, the irrational factor is V2y, and its coefficient

is-^. 2y2

246

RADICALS

Simplify
24.	v|.
25.	Vi.
26.	VI.
27.	Vf.
28.	vs
29.	V|.

30.

31.

32.

2a3

/5 a; V

83.

34.

35.

2_ 3 2/^^'

/4^

36. (a+6)\r

+ 6

37.

a-b yi(a-

I Sx

>6

50 a ^2/

(a-?,y

332. Although | = i, it does not follow without proof that 64^ = 642^, for each fractional exponent denotes a power of a root of 64, and the roots and powers taken are not the same for 642 ag for 545, j^y trial, however, it is found that each number is equal to 8 ; and in general it may be proved that

pm p

a^xi — ^q. that is,

A number having a fractional exponent is not changed in value by reducing the fractional exponent to higher or lower terms.

EXERCISES

333. 1. Reduce V9 a^ to its simplest form.

PROCESS

^9^= V(8~^'=(S a)^ = (3 a)*= ^3a* ^. Reduce V64 a^b^ to its simplest form.

PROCESS

■v/64 a^b'' = ^/2^a'b%^ = b(2ab)^ = b(2ab)^ = b Vl^b^ Simplify :

3. ^36. 5. ^1600. 7. </9^iFb¥.

4. </25. 6. ^27^8. 8. ^121 aV.

RADICALS 247

334. Simplify:

1. V600. 5. -VTS9. 9. -v/iii. 13. Vj.

2. V500. 6. V84. 10. -^. 14 JI.

3. ^/166. 7. a/72. 11. -^^43. !L

15 3/ a

4. -v^SOOO. 8. -v/192. 12. a/289. * ^Sb"'

16. V405 a-y. 18. V8-20 62. 20. A/a^6Vd«.

17. (135a;y)i 19. 5V4a2 4-4. 21. (16a;-16)i

a;-2.y\ 22/ ^Vl+.r + a^

23. V27 c2 - 36 c + 12. 26. (4: a^ - 24: a'x -{- 36 ax^K

24. ■v/a;2-2a;2/ + /. 27. (a;^?/-3a^/4-3 xy-x?/^)^

335. To reduce a mixed surd to an entire surd.

EXERCISES

1. Express 2 a^Eb as an entire surd.

PROCESS

2 a V56 = V4^2 VEl) = VUfFxEb = V 200^6

Rule. — Raise the coefficient to a power corresponding to the index of the fiven radical, and introduce the result under the radical sign as a factor.

Express as entire surds :

2. 2V2. 6. 3^/3. 10. |V2. 14. |V4f.

3. 3V5. 7. 4V5. 11. fVa?. 15. fVffa^.

4. 5V2. 8. iV8. 12. iVVc. 16. l-^iT

5. 3^/2. 9. a2-3/5^ 13. iVf. 17. fv/3|.

18. ^±yJ^EI. 19. ^±iJl §_. 20. l(a-6)*

a — y^oj-f-y a — 4^ a + 4 ab

248 RADICALS

336. To reduce radicals to the same order.

EXERCISES

1. Reduce V3, V2, and V4 to radicals of the same order.

PROCESS

■^3 = 3^ = 3r- = W«=^27 V2 = 2^ = 23^=^2^=^6i -^4 = 43=4^*^=^=^256

Rule. — Express the given radicals with fractional exponents having a common denominator.

Raise each number to the power indicated by the numerator of its fractional exponent, and indicate the root expressed by the common denominator.

Reduce to radicals of the same order:

2. V2 and v^S. 9. Vo^, -Va^, and -1/2.

3. V5 and v^6. 10. Va, -\/b, -i/x, and \/y.

4. a/7 and VlO. 11. -Va + b and ■\/x + y.

5. ^lO, V2, and ^5. 12. V|, Vj^y and 2 VS.

6. V4, V2, and V3. 13. A^'^a;, ^xy, and ^x^y\

7. v/l3, V5, and -v/4. 14. (a + b) Va^^, and a/^"T.

8. V3, -v/5, and A/^iy. 15. Va + b, -s/d^ + 6^, and Va — Z>

16. Which is greater, a/5 or V2? "v^ or a/3?

17. Which is greater, a/3 or "v^i? 3 V2 or 3 a/4?

Arrange in order of descending value :

18. a/3, V2,'and a/7. 21. V2, y/5, V2j, and v^.

19. a/2, a/4, and a/5. 22. v/7, a/48, a/4, and a/63.

20. ^2, a/3, and v^30. 23. -v/4,A/2,A/5,A/i3, and a/ISO.

RADICALS 249

ADDITION AND SUBTRACTION OF RADICALS

337. Radicals that in their simplest form are of the same order and have the same radicand are called similar radicals.

Thus, 2\/3, a a/3, and TV'S are similar radicals.

338. Principle. — Only similar radicals can be united into one term by addition or subtraction.

EXERCISES

339. 1. Find the sum of V50, 2^8, and 6 VJ.

PROCESS Explanation. — To ascertain whether the given

expressions are similar radicals, each may be re-

VoO = 5 V2 duced to its simplest form. Since, in their simplest

2-v/8 = 2 V2 form, they are of the same order and have the same

o /J Q /o radicand, they are similar, and their sum is obtained

2 _ by prefixing the sum of the coefficients to the common

Sum = 10 V2 radical factor.

Find the sum of: [For other exercises, see page 477.]

2. V50, Vl8, and V98. 5. V28, V63, and VTOO.

3. V27, Vi2, and V75. 6. ^250, "v^, and v/54.

4. V20, V80, and V45. 7. ^I28, ^/686, and ^/J.

8. a/135, v/320, and -\/625.

9. V500, V108, and -\/-32.

10. Vi v/l2i, VJ, and VlJ.

11. VJ, V75, |V3, and Vl2.

12. Vi, iV3, 1^/9, and V147.

13. a/40, V28, -s/2E, and VlTd.

14. a/375, V44, -^192, and V99.

250 RADICALS

Simplify :

15. V245-V405 + V45.

16. VI2 + 3V75-2V27.

17. 5V72+3V18- V50.

18. -5/128 + -^086 --v^M.

23. ■\/l2Sx-i-^/375x-V54.x ^ an. \ ryT)

26. ^/^ +

3. be

19. VI12-V343+V448.

20. -\/l35- -^625+ -5^320.

21. -^| + -^T+^5|.

22. Sy864-^4()00 + v'32. 27. V (a + bfc - -V (a - bfc.

28. 6^ + 4-^I|:-8^||f.

29. ^'_96»^ + 2^3^--\/5^ + ^40^.

30. ^/a6^--v^a^6^ + v'8^^6V.

31. V3a^ + 30a:2^753._^3^_6^2_j_3^^

32. V5a« + 30a^ + 45a-^-V5a^-40a^ + 8()a3.

33. V50 + ^''9-4V| + -v/^=^ + -v/27-a/64.

34. V}+(5 v/|-iVl8 4-^/36 --V/JI + -^125-2 VS

35. (f)^-(fr^ + VT^p+Vi:35--</(T|p.

36. 5.2-^4-2-^+3.2-^+3.5-i.2^H--v/^||^.

«

MULTIPLICATION OF RADICALS

340.

That is,

a^ X a^

,M-nM

Va X Va = Va* X Va^ =^ Vo*

\

Since fractional exponents to be united by addition must be expressed with a common denominator, radicals to be united by multiplication must be expressed with a common root index.

RADICALS 251

EXERCISES

341. PROCESSES. — 1. V7xV5=V35.

2. 5V3x2Vl5 = 10V45 = 10x3V5 = 30V5.

3. 2V3 X 3^2 = 2a/27x 3^4 = 6-^108.

Rule. — If the radicals are not of the same order ^ reduce them to the same order.

Multiply the coefficients for the coefficient of the product and the radicands for the radical factor of the product ; simplify the re- sidt, if necessary.

Multiply : [For other exercises, see page 477.]

4. V2by V8. 11. 2\/6by3V6.

5.	V2by V6.	12.	3 V3 by 2-y/K
6.	V3 by Vi5.	13.	^5by ^WO.
7.	2 V5 by 3ViO.	14.	2^250 by V2.
8.	3V20 by 2V2.	15.	2^24 by -v^iS.
9.	V2 by 3^'3.	16.	2^2 by ^512.
10.	2</3 by 3^45.	17.	V2 xy by S^/x'f.

Find the value of :

18. Vmn X ^mhi X ^\fmr?.

I

19. V2 axy x -y/xy x -s/a^xy.

20. V^~V X Vx-'^y"^ X Va? y.

21. VcT^ X \/ci2p X </{a - h)-\

22. V| X V| X V|. 25. -^1- X ^1 X V|.

23. VJ X V| X V|. 26. 16^ X 2^ X 32i

24. ^|x^fxV|. 27. 27^x9^x81^

252 RADICALS

28. Multiply 2 V2 + 3 V3 by 5 V2 - 2 V3. Solution

2\/2 + 3V3 5V2-2V3 20 + 15V6

- 4V6-18

20 +11V6-18 = 2+ IIV6. Multiply :

29. V5 + V3 by V5 - V3.

30. V7 + V2 by V7 - V2.

31. V6-V5by V6- Vs.

32. 5 - V5 by 1 4- V5. . J

33. 4V7 + 1 by 4 V7 - 1.

34. 2V2 + V3 by 4 V2 + V3.

35. 2 V3 + 3V5 by 3V3 + 2V5.

36. 3 a + V5 by 2 a — V5.

37. 2 V6 - 3 V5 by 4V3 - ViO. J

38. a"- - abV2 + b^ by a" + «& V2 + b\

39. a; — -\Jxyz + 2/2; by V^ + V2/2;.

40. icVa; — xs/y + 2/ Va; — 2/ V^ by Va; + Vy- Expand :

41. (>i3+V5)(Al3-V5). 43. (M6+vn)(V6-vn).

42. (^l9TVi)(V9-V6). 44. (\5a+aV5)(A/5a-aV5).

45. (a/7 c + V5 c=^)(Af7 c - V5 c2).

46. (Ml4 ar + ajV27)(All4 a; - a; V27).

RADICALS 253

DIVISION OF RADICALS

342. a* -i- a^ = a^"* = a^~^ = aK

That is, Va^Va = </a'^</^= ^^^T^= ^a.

In division, when one fractional exponent is subtracted from another, the exponents must be expressed with a common de- nominator. When one radical is divided by another, the radicals must be expressed with a common root index.

EXERCISES

343. PROCESSES. — 1. V6()--Vl2=V5.

(2/)^ (^)tV \fj "^f y

Rule. ^- If necessary, reduce the radicals to the same order.

To the quotient of the coefficients annex the quotient of the radicands written under the common radical sign, and reduce the result to its simplest form.

Fin	d quotients : [For other exercises, see page 477.]
4.	Vso-Vs.	12.	2^12 -V8.
5.	V72-2V6.	13.	■wax -j- Vic?/.
6.	4 V5 - ViO.	14.	V2ab'-^</a'b\
7.	6 V7 ^ V126.	15.	■Va'x^^-\/2ax.
8.	^4-V2.	16.	</9a''b^^-VSab.
9.	7^/135 --^/9.	17.	^4.xY^'\/2xy.
10.	7V75-f-5V28. •	18.	Va — b-7--Va + b.
11.	■v/16^a/32.	19.	3\/i-V|.

254 RADICALS

20. Divide Vl5 - V3 by V3.

21. Divide V6 - 2 V3 + 4 by V2.

22. Divide V2 + 2 + i V42 by i V6.

23. Divide 5 V2 -f 5 V3 by VIO + Vl5.

24. Divide5 + 5V30 + 36by V5 + 2V6.

INVOLUTION AND EVOLUTION OF RADICALS

344. In finding powers and roots of radicals, it is frequently convenient to use fractional exponents.

EXERCISES

345. 1. Find the cube of 2Va^.

Solution. (2y/ax^y = 2\a^x^y = 8 aV^^ = 8 Vo^is = 8 ax^Vax. '

2. Find the square of 3Va^. Solution. (3 Vx^y = 9(x^y = 9 x^ = 9 v^ = 9 xVsfi,

3. Obtain by involution the cube of V2 + 1.

Solution (\/2 + 1)3 = ( v^)3 + 3( V2)2 . 1 + 3 V2 . 12 + 13 = 2\/2 + 6 + 3v^ + 1 = 7 + 5V2. In such cases expand by the binomial formula.

Square :

4. 3Va6.

5. 2a/3¥.

6. x-V2^.

7. w^yTft.

8. aVa^,

1 I

Cube:	Involve as indicated :
9. 2V5.	14. (-2V2a6/.
10. 3V2.	15. (-V2's/x)\ 1
11. 2^/^.	16. (- V2A/aa!2/. 1
12. -\/a'b\	• 17. (-2Vx\'yy. 1
13 ^4w3.	18. (-3a^a;3)^ 1

RADICALS

256

25. (Vx±iy.

26. (Va-V6y

27. (■Vx±iy.

Expand :

19. (2 + V6)2. 22. (2-V3)3.

20. (2 + V2)2. 23. (V7-v6}2-

21. (2 + V5)^ 24. (2V2-V3)2.

28. What is the fourth root of V2x ? Solution. "V^v^ = [(2a;)^]^ = (2x)^ = \^.

Find the square root of : Find the cube root of :

29. V2. 32. -V^. 35. V2a;. 38. -27V^.

64 V ay.

30. ^5. 33. -^/x^^ 36. V7tt^ 39. — Va"6".

31- a/«^. 34. VaV. 37. -v^SmV. 40.

Simplify the following indicated roots :

41- ^ll/I^. 43. (V8^^)*-

42. -^Va^V.

44. (Vx'^y"*y Rationalization

// x%^ \l

346. Suppose that it is required to find the approximate "value of -—^} having given V3 = 1.732 •••.

V o

1.732... 1 1.000000 1 .577 8660

3)1.732... .577...

We may obtain a decimal approximately equal to — -^ as in

V3

le first process (incomplete), by dividing 1 by 1.732..-; but

great saving of labor may be effected by first changing the

fraction to an equivalent fraction having a rational denomi-

nator, thus

1. V3 V3

V3 V3.V3 3' and employing the second process.

256 RADICALS

347. The process of multiplying a surd expression by any number that will make the product rational is called rationali- zation.

348. The factor by which a surd expression is multiplied to render the product rational is called the rationalizing factor.

349. The process of reducing a fraction having an irrational denominator to an equal fraction having a rational denomi- nator is called rationalizing the denominator.

EXERCISES

350. Eind the value of each of the following to the near- est fourth decimal place, taking V2 = 1.41421, V3 = 1.73205, and V5 = 2.23607 :

1. A. 3. -^. 5. ^^

V2 V8 V50

2. A. 4. 1^. 6. -.1

V5 V45 V125

\

Rationalize the denominator of each of the following, using the smallest, or lowest, rationalizing factor possible :

7. J_. 9 jV^, j^ -Va + b

Va;^ ■ Vl2 * Va^=^'

^ ax Va

8. ■-' 10. -: ■- 12

_ rzn. i

V2a»aj -s/ax" ' \ x-\-2 ^

351. A binomial, one or both of whose terms are surds, is ( ailed a binomial surd.

V2 + VS, 2 + V5, \/2 + 1, and VS - v^ are binomial surds.

352. A binomial surd whose surd or surds are of the secon order is called a binomial quadratic surd.

\/2 + VS and 2 + VS are binomial quadratic surds.

353. Two binomial quadratic surds that differ only in the sign of one of the terms are called conjugate surds.

3+ VS and 3— VS are conjugate surds ; also V'3+ V2 and V3— \/2.

)

RADICALS 257

354. The product ofmiy ttvo conjugate surds is rational. For, by § 114, ( Va + \/6) ( \/a - V&) = (Va)^ - {y/ly^ = a - h.

Hence, a binomial quadratic surd may he rationalized by multi- plying it by its conjugate.

EXERCISES

2

355. 1. Rationalize the denominator of

3-V5

Solution

2(8 + VS) _^ 2(3 +V5) ^ 3+\/5

3_V5 (3-V5)(3+V5) 9-5 2

2. Rationalize the denominator of _ ~ _»

V7 + V3

Solution

V7 - V3 ^ ( V7 - V3) ( V7 - V8) ^ 7 -2V21 +3^5- v^ V7+V3 (V7+V3)(V7-V8) 7-3 2 *

Rationalize the denominator of :

3 3 g V3 + V2 ^ a-2V6

2 + V3 * V3-V2 * a + 2-\/h'

4. _^ 6. ^-^^^. 8. ^ + '^,

V5-V3 ' 2-V2 * V^-V^/^

g 4\^+6V3 j^^ Va^ + g + l-l.

3V3 -2V2 ' Va^ + a + 1 + 1

j^^ a; — Vo;^ — 1 ^^ Va; + y — Va; — ?/

a; + Va;^ — 1 \'x -\-y -f Va — ^

Reduce to a decimal, to the nearest thousandth :

13. ^=1^3. ^^ _^_. ,g V3 + V2

2-V3 34-V5 V3-V2

milne's stand, alg. — 17

258 RADICALS

16. Kationalize the denominator of _ ~ "^ — IlJX^.

V2+V3 + V5 Solution V2 _ V3 - V5 ^ (y/2 - VS) - V3 ^ ( V2 - V5) + V3 V5 + V3 + \/6 ( V2 + V3) + V6 ( V2 + \/3) - VS

_ 2 - 2 VlO + 5-3^4-2 VlO 2 + 2 V6 + 3-5 2\/6

_ 2- VlO ^ 2\/6-2\/l5 _ VQ- VTE V6 6 ~ 3 '

Rationalize the denominator of:

17. V2-V5-V7. ,3. 1

V2 + V5 + V7 ' V2+-V3 + V5

18. V3 + V2 2^j 2V2-3V3 + 4V5^

' V3 + V2-V6 * V^+V3-V5

21. Rationalize the denominator of - — r;=> or

Va + W a* + 6t Solution. — By § 134, Va+ y/b^, or a^ + b^, is exactly contained in the sum of any hke odd powers of a^ and &3, and also in the difference of any Hke even powers of a^ and 6^. The lowest like powers of a^ and b^ that are rational numbers are the sixth powers, w-hich are even powers. Hence, the rational expression of lowest degree in which a^ + 63 is exactly contained is {a^y — (bfy, or a^ — 6*. M

Dividing a^ — &* by a^ + &f , the rationalizing factor for the denomina- tor is found to be ai - a%^ + ah^ - ab^ + ah^ -b^^. Multiplying both terms of the given fraction by this factor,

a Qj. a _ a(a^ - a^b^ + «^&^ - «&^ + «^&^ - &^)

Va+sy6^' aU&^ «'-^*

Rationalize the denominator of :

■Vab «, V^ o. V.

22. „ . _ ' 24. „ V"-- _^ 26.

aa?

</a-</b Va'-^¥ Va-</x

? 25. ^ + ^ . 27. 5^_.

^^+.V^ </a-V6 V«+</2/«

RADICALS 259

Square Root of a Binomial Quadratic Surd

356. To find the square root by inspection.

The square of a binomial may be written in the form

(a + 6)' = («' + &') + 2 a6. Thus, (V2 + V6)2=(2 + 6)+2Vi2 = 8-f2Vl2.

Therefore, the terms of the square root of 8 + 2 Vl2 may be obtained by separating Vl2 into two factors such that the sum of their squares is 8. They are V2 and V6.

V8+2Vl2= V2+ V6.

Principle. — The terms of the square root of a binomial quadratic surd that is a perfect square may be obtained by divid- ing the irrational term by 2 and then separating the quotient into two factors, the- sum of whose squares is the rational term.

EXERCISES

357. 1. Find the square root of 14 + 8 V3. Solution

14 + 8 \/3 = 14 + 2 (4 V3) = 14 + 2 V48. Since \/48 = V6 x VS and 14 = 6 + 8,

Vl4 + 8 V3 = V6 + V8 = V6 + 2 V2.

2. Find the square root of 11 — 6 V2.

Solution

Vll - 6 \/2 = Vll - 2 VT8 = V9 - V2 = 3 - V2.

Find the square root of :

3. 12 + 2Va5. 7. 11 + 2V30. 11. 12+4V5.

4. 16-2V60. 8. 7-2VI0. 12. 11+4V7.

5. 15 4-2V26. 9. 12-6V3. 13. 15-6V6.

6. 16-2V55. 10. 17 + 12V2. 14. 18 + 6V5.

260 RADICALS

Find the square root of :

15. 3-2V2. 17. a^-\-b-{-2aVb.

16. 6 + 2 V5. 18. 2a-2Va^-b\

358. To find the square root by using conjugate relations.

This method is useful in the more difficult cases. It depends upon the following properties of quadratic surds.

1

359. Principle 1. — The square root of a rational number cannot be partly rational and partly a quadratic surd.

For, if possible, let Vy =:Vb±m, V^.and Vb being sards.

By squaring, y = b±2mVb + m^j

and VE=±y-'^"-\

2 m

which is impossible, because (§ 321) a surd cannot be equal to a rational number.

Therefore, y/y cannot be equal to Vb ± m.

360. Principle 2. — In ayiy equation containing rational numbers and quadratic surds, as a-\- -y/b = x-\- Vy, the rational parts are equal, and also the irrational parts.

Given a + Vi = x + Vy. (1)

Since a and x are both rational, if possible, let

a=x±m. (2)

Then, a: ± m + >/6 = z + Vy, (3)

and Vy = y/b± m. (4)

Since, § 359, equation (4) is impossible, a = a: ± wi is impossible ; that is, a is neither greater nor less than x.

Therefore, a =^ x, and from (1), y/h = Vy.

Hence, if a + V6 = a; + Vy, a = x and Vb = Vy.

RADICALS 261

361. Principle 3. — If a-\-^b and a — Vb are binomial quadratic surds and '\a-\--\/b = \^x-\-^y, then yja —-y/b =

■y/x — V?/.

To exclude imaginary numbers from the discussion, suppose that a — Vb'is, positive.

Given v a + V6 = Va; + y/y.

Squaring, § 277, a + V6 = x + 2 Vxy + y.

Therefore, § 360, a = x -{■ y and \/6 = 2 y/xy ;

whence, Ax. 2, a — Vh= x + y — 2 Vxy.

Hence, § 289, Va-Vb=Vx - Vy.

EXERCISES

362. 1. Find the square root of 21 + 6 VlO.

Solution

Let Vx + Vy = V2I + 6 VlO. (1)

Then, § 361, \^ _ Vy = V2I - 6 VIO. (2)

Multiplying (1) by (2), x-y= V441 - 360 = Vsl,

or X -y = 9. (3)

Squaring (1), § 277, x + 2 Vxy + y = 21 + 6 VlO. Therefore, § 360, x-{- y = 21. (4)

Solving (4) and (3), x = l5,y = 6.

.-. Vx = VT5, Vy = VQ. Hence, from (1), V2I + 6 VlO = Vl5 + V6.

Find the square root of :

2. 25 + 10V6. 8. 16+6V7. 14. 2 4-V3.

3. I9 + 6V2. 9. 21-8V5. 15. 6+V35.

4. 45 + 30V2. 10. 47-12vTl. 16. l+iV2.

5. 35-14V6. 11. 56H-32V3. 17. 2 + fV6.

6. II + 6V2. 12. 35-I2V6. 18. 3O + 20V2.

7. 24-8V5. 13. 56- 12 V3. 19. 18-6V5.

262 RADICALS

RADICAL EQUATIONS

363. An equation involving an irrational root of an un known number is called an irrational, or radical, equation.

o;^ = 3, Vx + 1 = Vx -4+1, and Vx — 1 = 2 are radical equations.

364. A radical equation may be freed of radicals, wholly or in part, by raising both members, suitably prepared, to the same power. If the given equation contains more than one radical, involution may have to be repeated.

When the following equations have been freed of radicals, the resulting equations will be found to be simple equations. Other varieties of radical equations are treated subsequently.

EXERCISES

365. 1. Given V2^ + 4 = 10, to find the value of x.

Solution V2x + 4 = 10. Transposing, y/2x = 6.

Squaring, 2 a; = 36.

.-.a; = 18.

Verification. — Substituting 18 for x in the given equation and (§ 317) considering only the positive value of V2 x, we have V30 + 4 = 10 ; that is, 10 j= 10, an identity ; hence, the equation is satisfied for x = 18.

_ i

2. Given -y/x — 7 -f Vx = 7, to find the value of a;. '

Solution Vx-7 +Vx = 7. Transposing, y/x — 7=7 — Vx.

Squaring, x — 7 = 49 - liVx + x.

Transposing and combining, 14Vx = 66. Dividing by 14, Vx = 4.

Squaring, x = 16.

Verification. VlO - 7 + vl6 = V9+ \/i6= 3 + 4=7; that is, 7=7.

RADICALS 263

3. Given Vl4+Vl + Vx+S = 4, to find the value of a?. Solution

\ 14 + VT+Vx+l = 4.

Squaring, 14 + Vl + Vx + 8 = 16.

Transposing, etc., Vl + Vx + 8 = 16 - 14 = :

Squaring, 1 + vx 4- 8 = 4.

Transposing, etc., Vx+ 8 = 4 — 1=3.

Squaring, a + 8 = 9.

.-. x = 9-8 = l.

Verification. \ 14 + Vl + Vl + 8 = Vl4 + Vl + 3

= V14+2 = 4 ; that is, 4=4.

General Directions. — Transpose so that the radical term, if there is but one, or the most complex radical term, if there is more than one, may constitute one member of the equation.

Then raise each member to a power corresponding to the order of that radical and simplify.

If the equation is not freed of radicals by the first involution, proceed again as at first.

Solve, and verify each result : 4. V^+li=4. 11. 1 + 2V^ = 7-V^.

5. Va; + o = 31. 12. Vi» + 16 — Va; = 2.

6. Va; — a2=&. 13. V2 ic — V2 a; — 15 = 1,

7. Va;-1 = 2. 14. ■\/x' -\-x + l = 2 -x.

I

8. -y/x-a^^a. 15. 3Va;2-9=3a;-3.

9. -Vx + b^a. 16. Va; + 2 = Va;+32.

10. l-f-V^ = 5. 17. 5 — Va; + 5 = V«.

264 RADICALS

Solve, and verify each result :

18. Va^-5a; + 7 + 2 = a;. 19. V9^+~8 + V9» - 4 = 0.

20. 4- V4-8a; + 9a^ = 3a;.

21. V 2 (1 - aj) (3 - 2 ic) - 1 = 2 a?.

22. V2a;-1 + V2a;H-4 = 5.

23. V3a;-5 + V3i» + 7 = 6.

24. Vl6« + 34- Vl6a;H-8 = 5.

25. A/i -|-W^Tl2 = l + a;.

26. V7 4-3V5^^I§-4 = 0.

27. 2a;-\4i«2- V16a;2-7 = 1.

28. 2Va;- V4ic-22-V2 = 0.

29. V2(a; + 1) + V2 x-l = VS x-\-l.

30. V3ic + 7+ V4a;-3= V4a; + 4 + V3a;.

31. ^1^|V2^H^56 = 2.

33. Solve the equation

. \7 + \l+V4+A/l+2Va; = 3 5

V3a; + 2 Suggestion. — Clear the equation of fractions.

= V3a; + 2 + V3a;-l.

34. Solve the equation

VSx-^15 V3a; + 6

^Sx-\- 5 V3a; + 1 Suggestion. — Some labor may be saved by reducing each fraction to a mixed number and simplifying before clearing of fractions. 10 , . 5

Thus,

1 +

1 +

V3x + 5 VSx+l

Canceling, and dividing both members by 5, 2^1

VSx + 6 VSx + 1

RADICALS 266

_ -V'2x-\-9 V2a; + 20 ■ ^ V27- + 6 _ V27 + 2 35. — — = — • *v.

Solve and verify :

V2 X + 9 _ -y

V2^-7~V2^-12

V^ + 18^ 32

V^+2 V^ + 6

36. V^ + 18^_32_^l. 41.

37. V^-l^V^-^. 42.

Vs + 5 Vs-1

Vv — 1 Vv — 5

38. V^^^Vi^. 43.

39. V£^=VFEi. 44.

V2i	r+4 V2r+1
Vll	.n + V2n + 3_8^
Vli 2V2 2V2	. n - V2 71 + 3 ^ a; + 4_3Va; + l4-9 a;-4 3Va; + l-9
Vm	-|-l_Vm-l_l.
Vm	4.1 + Vm-l ^
V4.	5_^.34.2V2-l_g

Vi + 1 ■Vt-2 V40 + 3-2V2-1

^^ AfV5a;-9 'VV5aj-21 45. = — — •

Ws^+ll VV5^-16

Suggestion. — First square both members.

46. ^-3 3^y^+v3^^^^

V^-V3 2

SuGGESTiox. — Begin by simplifying the first member.

3

47. V2a;-V2i»-7

V2a;-7

48. Solve V^ + g + V^-^ ^ 2 + ^^-'-^^ for ^. ■y/x -\-a— -y/x — a ^

Suggestion. — Rationalize the denominator of the first fraction.

Solve for x, and verify :

49. ^x-{-Vx — {a — by = a + b,

50. a-\/x — b\^x = a^-\-b^ — 2 ah.

51. V5 ax — 9a^-{-a= VS ax,

52. ■Vx-\-3a = — V^.

Va; + 3 a

266 RADICALS

53. Solve V^H- V2~x-\- V3»= Va for x, •> Solution

Vx + V2x + VSx = Va. (1)

Factoring, Vi(l + V2 + V3) = Va. _ (2)

.-. ^/^=. ^^i 1

1 + v'2 + V3 /'

_ _ i

„ Va(l4-V2-V3) 1

(1 + V2 + V3)(l + V2 - V3)

^ Vq(l + V2 - V3) ,3.

2V2 j

1

3

Squaring, a; = « (1 + V2 - V3)2. (4)

8

Solve for x:

54. V2^"+ V3¥+ V5aj = Vm.

55. V2^ + V3~x— -VBx = Vc.

56. Va; — a + V2 (a; — a) = V3 a; + a V^.

366. Erom §§ 364, 365, the student will have observed thai radical equations are freed of radicals either by rationalization or by involution.

Thus, V2^-6 = 0 (1) V2^ + 6 = 0 (2)

Multiplying by V2^ + 6 V2^— 6

2aj-36 = 0 2a;-36 = 0

/. a; = 18 .-.a; = 18

If the positive, or principal, square root of 2 a; is taken, a; = 18 satisfies (1) but not (2) ; if the negative square root of 2 a? is taken, a; = 18 satisfies (2) but not (1).

It has been agreed, however, that the sign ■\/~ shall denote only principal roots in this chapter, and because of this arbi- trary convention, our conclusion must be that (1) has the root a; = 18 and that (2) has no root, or is impossible.

RADICALS 267

According to this view, when both members of (1) are mul- tiplied by V2 a; + 6, no root is introduced because V2 a; -j- 6 = 0 has no root; but when both members of (2), which has no

root, are multiplied by V2^ — 6, the root of \/2^ — 6 = 0, which is x= 18, is introduced (§ 230).

A root may be introduced in this way by rationalization^ or by the equivalent process of sqiuxring.

Thus, V2» + 6 = 0. (2)

Transposing, V2^ = — 6.

Squaring, § 277, 2 a; = 36.

..a; = 18. Verifying, V2TI8 +6=6 + 6^^0.

The symbol ^t is read ' is not equal to.'

EXERCISES

367. 1. Solve, if possible, the equation Va;— 7— -s/x— 7. Solution. — Transposing, squaring, simplifying, etc.,

Vx=-4. Squaring, x = \Q.

Verification. VI6 - 7 — VlG = \/9 - Vl6 = 3 - 4 ijfc 7. Hence, the equation has no root, or is impossible.

Solve, and verify to discover which of the following equa- tions are impossible ; then change these to true equations :

2. V2a;+ V2a;-3 = 1. 5. V4a; + 0 -2Va;- 1 =9.

3. V3a; + 7 + V3a; = 7. 6. V4a; — V^ = V9a; — 32.

4. 2Va;+V4a;-ll = l. 7. V5 a; - 1 - 1 = V5 a; + 16.

8. Va; + 1 + Va; -I- 2 - V4 a; -I- 5 = 0.

9. V2(a^ + 3a;-5)=(a; + 2)V2.

10.

■\/x-b Va; + 1^Q 11. Vl9^ + V2a;+11^2i VaT^ V^T8 ' Vi9«-V2a; + ll ^'

IMAGINARY NUMBERS

368. Our number system now comprises natural numbers, 1, 2, 3, ... ; fractions, arising from the indicated division of one natural number by another ; negative numbers (denoting oppo- sition to positive numbers), arising from the subtraction of a number from a less number; surds, arising from the attempt to extract a root of a number that is not a perfect power; and finally imaginary numbers, arising from the attempt to extract an even root of a negative number (§ 285).

In this chapter only imaginary numbers of the second order will be treated. i

Before the introduction of imaginary numbers, the only numbers known were those icliose squares are positive, now called real numbers to distinguish them from imaginary num- bers, whose squares are negative. • j

369. Since the square of an imaginary number is negative, imaginary numbers present an apparent exception, in regard to signs, to the distributive law for evolution. Apparently

sf^^ X V^^ would equal V(- 1)(- 1) = V+T = ± :

But by the definition of a root, the square of the square root of a number is the number itself.

Hence, V^^ X V^^ =r(V^ri)2= _ 1, not + 1. {A)

In this chapter it will be assumed that imaginary numbers obey the same laws as real numbers, the signs being deter- mined by {A), which we call the fundamental property of imaginaries.

IMAGINARY NUMBERS

370. Powers of V— 1.

(V=l) ' ,

(V^)^ = (V^)(V^l)

(V^^f == (VZijy ^zrj =(-i)V'

269

= -1;

(V- 1)* = ( V- 1)X V- 1/ = (- 1)(- !)=+!; and so on. Hence, if n = 0 or a positive integer,

(^)

Hence, any even power of ^ — 1 is real and any odd power is imaginary.

For brevity V— 1 is often written i.

371. Operations involving imaginary numbers.

EXERCISES

Find the value of :

1. (V^^ 3. (V^^y. 5. (V^^«.

2. (V^^. 4. (V^^^ 6. (V^^^

7. (-0^

8. (-*)«

9. Add V- a* and V- 16 a\ Solution V^^ + V- 16 a* = aV^=3 + 4 a^y/^H. = 5 a2 V^^.

Simplify :

10. V-4+V^^="49l

11. V'^+V^64.

12. 2 V^Tl _|_ 3 V^Ti.

13. V-12 + 4V-3.

14. 5V^=^18-V^=^^72.

15. 3V^^20-V^^80.

270

IMAGINARY NUMBERS

16. V-16aV4- V-aV-V-9a2a^.

17. (V^^ + 3V^^) + (V^a-3V^.

18. (V— 9 a??/ — V— xy) — (V— 4 a;?/ -f V— «?/).

19. V^^^+V-4aj^-V^^^ + 3a;V^^.

20. V-16-3V^4 + V^^l84- V^504- V^^25.

21. V^ + aV^2-V^^^98-5V-

22. Vr^=^-3Vl-10 4-2V5^^^30.

2a2

23. Multiply 3 V^IO by 2 V-8.

PROCESS

3V^=n^ X 2 V^^ = 3 ViO V

lx2V8V-l

= 6Vl0xSx(-l)

= -6V80"=-24V5

Explanation. — To determine the sign of the product, each imaginary number is reduced to the form 6 V— 1. The numbers are then multiplied as ordinary radicals, subject to (^), § 369, that V— 1 X V— 1= — 1.

24. Multiply V'-2 + 3 V^ by 4 V^^^ - v' - 3.

First Solution

VZ^ + 3\/^^ = (\/2 + 3\/3) V^n:, 4V"Z^- ^/:^3 = (4V2- \/3)V^=T; ... (v^Zr2 + 3V^^)(4A/"^-\/^^) = (\/2 + 3V3)(4V2 - V3)(V^)2 = (8 + 12a/6 - V6 - 9)(- 1) = l-ll\/6.

Multiply :

Second Solution

y/zr2^sy/zrz 4v/z:2- V^^

-4\/4-12\/6

+ 3\/9 + V6

1

11 V6

25. 3 V^=r5 by 2V^^l5. 28. 8V^n:by V-6'.

26. 4V^^^byV^n^. 29. V^=^125 by V-108.

27. 2 V^^ by 5 V^^TS. 30. V-100 by V-30.

31. V^=^+ V^^ by V^=^ - v'^^.

IMAGINARY NUMBERS

271

32. V— a6 + V— a by -V—ab — V— a.

33. V—xy-{- V— ic by V— a;i/+ V — «.

34. V^=^- V^^12 by V^^-V^^75.

35. V— a+ V— b-\- V— c by V— a+ V— 6 — V^.

36. Divide V— 12 by V^^.

Solution

V^=r3 V3V^ V3

37. Divide Vi2 by V^^.

Solution

Vl2 Vl2 Vi 2

V- 3 V3 V- 1 \/^n V- 1

.2\/^rT - 1

= _2A/:rT.

38. Divide 5 by (V- 1)^

Solution

(V-l)3

Divide :

39. V^^38 by V^^.

40. V27 by V^3.

41. 14 V"=^ by 2V^^.

42. -V^^^by V^^.

43. 1 by V^^.

44. V8 + 3 Vll by V^^.

45. Vl2 + V3 by V^3.

63. V^^ by A/

46. - 2 by V^nr.

47. (V^^byiV^^.

48. (V^^f by (V^n:)!^

49. V4a6 by V— 6c.

50. (V"^^)"by -|V^=T.

51. (V^^« by (V^^)-'.

52. V — a-^4-&V — 1 by ^ —ab.

272 REVIEW

REVIEW i

372. 1. Distinguish between an equation and an identity ; between an integral and a fractional equation. Illustrate.

2. When is a literal equation an identity ? 1

3. State what is meant by a graph. Of what practical use are graphs ?

4. Define abscissa; ordinate; coordinates. Interpret the equation ^ = (— 4, 3).

5. Tell how to determine where a graph crosses the avaxis ; the ?/-axis.

6. Construct the graph of 2y = 3x — 4:.

7. Why are simple equations sometimes called linear equa- tions ?

8. State the law of signs for involution; the law of exponents. J

9. How may the involution of a trinomial be performed by the use of the binomial theorem ?

10. Tell how the 12th root of an expression may be obtained.

11. Define root of an equation ; equivalent equations ; si- multaneous equations; independent equations; indeterminate equations ; elimination of an unknown number.

12. Upon what axiom is elimination by addition based ? elimination by comparison ?

13. Define radical ; radicand ; surd ; conjugate surds. Illus- trate each. Is \2-|-V4 a surd? State reasons for your answer.

14. What are similar radicals ? Illustrate.

15. Tell what is meant by the principal root of a number. What is the principal square root of 4? the principal cube root of -8?

REVIEW 273

16. Represent VlO inches exactly by a line.

17. Show the difference in meaning between (a^y and a* x a".

18. What does the numerator of a fractional exponent indi- cate ? the denominator ?

19. Show that aj" = 1 ; that — , = a'b\

20. Factor a~^ + 2 a'^b'^ -f b-% giving reasons for each step.

21. Define and illustrate mixed surd ; entire surd. Tell how to reduce a mixed surd to an entire surd.

22. What is a radical equation? Give the steps in the solution of such an equation.

23. Define real number ; imaginary number ; rational num- ber ; irrational number.

24. Classify the following numbers as real or imaginary ; as rational or irrational :

2, Vi", V2, V5, V^ V=^, V^^ <^al </^ a being a positive number.

25. Illustrate how, in finding the value of an expression with an irrational denominator, it is advantageous to ration- alize the denominator first.

26. Find the value of i^, i*, i^, ^^ What are the values of even powers of i ? of odd powers of i ?

27. Solve graphically the simultaneous equations

{2x-Sy = 10,

28. If a system of two linear equations is indeterminate, how will the fact be shown by the graphs of the equations, referred to the same axes ? how, if they are inconsistent ?

29. When a is positive, is V — « real or imaginary ? When a is negative, is V— a real or imaginary ?

30. Find the sum and the product of 2V— 4 and 3V— 9.

31. Subtract V-9 from V-81; divide V-81 by V-9.

MILNE'S STAND. ALG. 18

274 REVIEW

EXERCISES

373. Reduce to simplest form :

6g^ — 7a^ — 5a; a: — y y + x 4:a^y^

9ar^ — 25 a; ' x-\-y y — x x'^ — y*

2 8a;^ + 18a;-5, V2 - V3 V2 + V3

12a;^ + 5a;-2 * V2 + V3 V2-V3*

4.

5.

10. .- .--1 +

a-Vb ' Va+V6 Va-Vft'

V2-V3-V5 _^^ VT+^-vri:^

V2+V3+V5 * vr+^+vir^^'

2- V5 , 2V3 1.1.1

+ ——=' 12.

2+V5 V243 * 1-V2a; 1 + V2a? l-2a;

- „ x-{- ^x^ — a^ X — Va;^

JLo. ^

14.

X — -y/o? — a? x-\- Va;^ — o?

Va + 1 + Va-1 Va + l - Va-1

Va + 1 — Va — 1 Va + 1 4- Va— 1

15 a^-6 ^^a^-4«V6 + 46

a2-2aV6 + & a2^2aV6 + b

l + Va + a Wa" « Av« «/

17. -^ ;;= • 19-

1 — ^a-\-a

1-Va

f_a V^Y a ■\/x\ m

Wx « AVx a J I

REVIEW 276

Expand :

20. (a' -by, 24. (a-2 + a-i)2. ^q (a-VSy.

21. (2 a -3 by, ''• ^^" + ')' 31. (V^+V^)«.

26. (J-b^y. 32. (V2-V3y.

1^3 ~ 2; 27. (a*-6-2)^ 33. (V5-2y.

28. (a-^- 6-^)6. 34. (^_^)3.

29. (a^4-6^)«. 35. (V2--</2)«.

22.

23.

I

Extract the square root of :

36. ^^VsoT^-o^-^ + l

37. ^ + 4.f + l-2xy + '^-yz.

38. a^-^12ah^ + 54.ab + 10Salb'+SlbK

39. l+2V^-a;-2a;V^+ar'.

40. aH-46 + 9c-4Va6-h6Vac-12V6c.

41. 0^ — 4a;Va;2/ + 6a^ — 42/Va;^ + 2/^. Find the square root of:

42. 81234169. 46. 56 + 14 Vl5.

43. 64064016. ^ 47. 47-12Vi5.

44. .00022801. 48. 62 4-20V6.

45. .1 to four places. 49. 51 - 36 V2.

Extract the cube root ot :

50. a^-9x+27x-^-27x-\

1 1

51. 27^ + 27.^-5 + ^^ ^^^

52. x^-{-Sc(^Vx-5xVx-i-SVx—l,

276 REVIEW

53. Find the cube root of 2V2 - 6^/2 + 3V2\/4 - 2.

54. Extract the cube root of 405,224.

55. Extract the cube root of 510,082,399.

56. Extract the cube root of 2 to three decimal places.

57. Find the first four terms of VI + x — x^.

58. Find the first three terms of "v^l + a^.

59. Find the fourth root of

a6 _ 4 aWab^'^ + 6 a?h-^ - 4 ab-Walr^ + b-\

60. Find the sixth root of

8 - 48 Va + 120 a - 160 a Va + 120 a" - 48 aWa + Sa\ If a"* X a" = a'"+'* for all values of m and n, show that :

64. {aby = l.

65. (abcf = a^b^c^.

66. f^)' = < \bj b'

70. (aV)i 73. (A%)~^-

71. (6V)-i 74. (-}f)-i

72. (a-b^yl 75. (-2^)"^.

76. For what values of n is(a — by = {b — a)" ?

Simplify, expressing results with positive exponents :

77. (36a-3--25a-2)-i. 80. (Vo^^h- ^/«V^)^-

78. (8 a^a^e X 64 a-^'a;-^)-* 81. (Va^^^ ^ V«^ri

79. (aV)^-4-(aV)2. 82. ( Va ^ \/a) ^ a/^.

61.	a-2 =	1 = . a2
62.	«l =	V^ = (Va)
63.	2a 3= . a
Find the value of :
67.	16l
68.	27l
69.	8-t.

REVIEW

277

a-b

a-\-b . 2 a^b -r

a^ — b^ a' + 53 a^ — b^

84 ^ + ft~^& ^ /I + ab-'' + ft'^>"^ X 1 + a-«63\ 1 - a-15 ' VI - ab-\ + a'b-' 1 - a'^^V

Solve the following equations :

7-2a; 7a;-4 1 - 6 a;

85.

86.

10 15 30

ic+l_a;— 1_3 — 5a; » — 1 a; + 1 ~" 1 — a;2 *

4^-17_3i-22

88. .

89.

9 33

Sx — 5y 4:X — 3y_7

'—X'-£)-

3 ~ 12 "12'

i(a.- + y + 3)-|(a^ + y) = 0.

3x+l=22/,

(aj + 5)(7/ + 7) = (a^ + l){y - 9) + 112.

94.

a; ^2/ -1 -1 La; ^2/

2„2 -1-3

Simplify, expressing results with positive exponents :

90- —1 ^-125-1

25^ X 64^

91. \a-\J{a^fy'\i

92. 5«- -5/^=32+^/256-8-1.

o-^l

95. \(ah'^)^-i-(a-h)-'\K a-{-b a — b

93.

\yyj_

6

96.

97.

a* - 6^ a* + 6^

(,27 • TJ "^

3 rt-i 4- 2 .T

278 REVIEW

Exercises on this page are from recent examination papers.

98. Write the first five powers of V— 1.

i i JL

99. Prove that (cry — a*"^ ; that a"^ x a"" = a*"-".

100. Which is the greater, 2^^ or 5 ? Prove it.

101. Find the value of to two decimal places.

2-f-V5-V2

102. Extract the square root of

a;(a; - V2)(a; - V8)(a; - VT8) + 4.

103. Find a factor that will rationalize x^ + y'^.

Simplify :

104. 16* . 2^ . 32l 107. ^ . ^8 . 3v^.

105. V38-12V10. 108. ^^:^ + (V^« + 8-i

106. ^^-^^. 109. J^±5!. Va;-f-Vic-2 >'V23-Vr

110. 120 + 4^-9-^4- ^ 4-271 V-64

111.

112.

113.

V3 + V2 . 7 + 4V3 2-V3 * V3-V2

^'3 + V5+V^-V5

2V15 + 8 . 8\/3-6V5 5+V15 ' 5V3-3V5

114. J^xV^X^^-^^

115 l-\-x a;^ + 3a; + 2

(a; + 2)-^-(a^ + l)-^(2 + a;)-i'

QUADRATIC EQUATIONS

374. The equation a; — 2 = 0 is of the first degree and has one root, x = 2. Similarly, ir — 3 = 0 is of the first degree and has one root, a; = 3. Consequently, the product of these two simple equations, which is

(a;-2)(a;-3)=0, or ar'-5a; + 6 = 0,

is of the second degree and has two roots, 2 and 3.

375. An equation that, when simplified, contains the square of the unknown number, but no higher power, is called an equation of the second degree, or a quadratic equation.

It is evident, therefore, that quadratic equations may be of two kinds — those which contain only the second power of the unknown number, and those which contain both the second and first powers.

ic2 = 15 and ax'^ -\-hx — c are quadratic equations.

PURE QUADRATIC EQUATIONS

376. An equation that contains only the second power of the unknown number is called a pure quadratic.

ax2 = h and ax^ — cx'^ = be are pure quadratics.

Pure quadratics are called also incomplete quadratics, because they lack the first power of the unknown number.

377. Since pure quadratics contain only the second power of the unknown number, they may be reduced to the general form aa^ = b, in which a represents the coefficient of x^, and b the sum of the terms that* do not involve a^.

279

280 QUADRATIC EQUATIONS

378. The equation 3a^= 300 has two roots, for it may be reduced to the form {x— 10)(a; + 10) = 0, which is equivalent to the two simple equations,

a; - 10 = 0 and « + 10 = 0,

each of which has one root.

The roots, + 10 and — 10, are numerically equal but opposite in sign.

Principle. — Every pure quadratic equation has two roots, numerically equal hut opposite in sign.

It is proved in § 436 that every quadratic equation has two roots an(^ only two roots.

EXERCISES

379. 1. Given 10 a^ = 99 - a;^^ to find the value of x.

Solution 10 a;2 = 99 - x'^. Transposing, etc., 11 x^ = 99.

Dividing by 11, a;^ = 9.

Extracting the square root of each member, § 289,

x=±3.

Note. — Strictly speaking, the last equation should be =b a; = ± 3, which stands for the equations, 4-0;=+ 3, +x = — 3, and — re = — 3, and — x = + 3. But since the last two equations may be derived from the first two, by changing signs, the first two express all the values of x. For convenience, the two expressions, x = + 3 and x = — 3, are written x=± 3.

Consequently, in extracting the square roots of the members of an equation, it will be sufficient to write the double sign before the root of one member.

2. Find the roots of the equation 3 aj^= — 15.

Solution

3x2 =-15. Dividing by 3, x^ = — 6.

Extracting the square root, x = ± V— 5. ,

Verification. — The given equation becomes — 15 = — 15 and is there- fore satisfied when either + V— 5 or — V— 6 is substituted for x.

QUADRATIC EQUATIONS 281

Solve for x, and verify each root :

3. 3x'-5 = 22.. 9. 7a^-25 = 5ar^4-73.

4. 2x' + Sx' = S0. 10. (x + 4)2 = 8a; + 25.

5- 4i»2 = i 11. (a_a;)2=(3a;4-a)(aJ-a).

6. |a^-5 = 22. 12. aa^ = (a - 6) (a^ - 6^) _ 5^^.

7. a;2_5^0. 13. aV + 2aa^ = (a^- l)2_a^.

8. 6aa;2-54a^ = 0. 14. (a; + 2)2-4(a;4- 2) =4.

a; — 8 6 _, a,a;a6

15. ^ = — -^' 21. - + - = — . o x-\-o X a X

16. J_ + ^ = §. 22. -^ i^ = 0.

1 — X 1+x 3 a-\-b X

X x'^-15^x x-2 x + 2^ 40

* 12"^ 5« 5* '^ • a; + 2 2-a; a^-4*

18 ^^+^ + ^Zl? = 4. 24 V^M^-V^^^^^l.

x-3 x-h3 ' ^ ' v^qpi_^V^^^'=n[ 2

19. ^::l? + ^±^ = -1. ' 25.:^+^ + ^:^ = -^^. x-{-l x — 1 x—a x-\-a 1 — a

x — 3 , x-\-3_^rj x-\-a , x — a_a^-\-b'^

2^- ^i:2^^T2~ «• ^^' ^T"6^^^"^^^^^'

27. V(aj + 3)(a; - 5) = V49 - 2 a;.

28. V25 - 6 a;+ V25i:6lt' =8.

29 a; + 7 a;-7 ^ 7 _

x'-lx x^-\-7x 0)2-73

Va; + 2a — ^x — 2a x

30. , =0 —

2 2

31. -I — =0?.

a? + V2-ic2 a;_v'2-a;2

282 QUADRATIC EQUATIONS

Problems [Additional problems are given on page 478.] 380. 1. What negative number is equal to its reciprocal ? J

2. If 25 is added to the square of a certain number, the sum is equal to the square of 13. What is the number ?

3. What number is that whose square is equal to the dif- ference of the squares of 25 and 20 ?

4. When 5 is taken from a certain number, and also added to it, the product of these results is 75. Find the number.

5. How many rods of fence will inclose a square garden whose area is 21 acres ?

6. A certain number multiplied by ^ of itself is equal to 16. Find the number.

7. The sum of two numbers is 10, and their product is 21. What are the numbers ?

Suggestion. — Represent the numbers by 5 + a: and 6 — a;.

8. The sum of two numbers is 16, and their product is 55. What are the numbers ?

9. Factor a^ -{- 17 a -\- 60 by the method suggested in the preceding problems.

Suggestion. — We need the two factors of 60 whose sum is 17. Rep- resent them by -^ + x and J/ _ x. Then, (\^ + x) (V- - x) = 60.

10. Separate a^-\-2a — 2 into two factors.

11. Separate ic^ — 2 a; — 1 into two factors.

12. Separate 24 into two parts the product of which is 143.

13. The sum of the squares of two numbers is 394, and the difference of their squares is 56. What are the numbers ?

14. The difference between two numbers is 4, and the sum of their squares is 208. Find the numbers.

Suggestion. — Represent the numbers by x + 2 and x — 2.

QUADRATIC EQtTATIONS 283

15. The length of a 10-acre field is 4 times its width. What are its dimensions in rods ?

16. The area of a sheet of mica is 48 square inches and its length is 1-J times its width. Find its length and its width.

17. The perimeter of a rug is 30 feet and its area is 54 square feet. Find its length and its breadth.

18. The length of a sheet of paper is 14 inches more than its width and its area is 912 square inches. Find its length.

19. At 75 cents per square yard, enough linoleum was pur- chased for $ 36 to cover a rectangular floor whose length was 3 times its breadth. Find the dimensions of the floor.

20. The lock of the St. Mary's Canal, Michigan, is 8 times as long as it is wide ; the surface of the water it contains is 80,000 square feet in area. Find the dimensions of the lock.

21. A man had a rectangular field the width of which was | of its length. He built a fence across it so that one of the two parts formed a square containing 10 acres. Find the dimensions of the original field in rods.

22. Two square fields together contain 51^ acres. If the side of one is as much longer than 50 rods as that of the other is shorter than 50 rods, what are the dimensions of each field?

Formulae 381. Solve the following formulae from physics :

1. s = i^^^for^. ^ F=—,forv.

2. E = iMv% for V. ^

3. P = rE,iovI. 5. G:=^,iovd.

' d^

6. When g = 32.16, formula 1 gives the number of feet (s) through which a body will fall in t seconds, starting from rest. How long will it take a brick to fall to the sidewalk from the top of a building 100.5 feet high ?

284

QUADRA^riC EQUATIONS

7. To lighten a balloon at the height of 2500 feet, a bag of sand was let fall. Find the time, to the nearest tenth of a second, required for it to reach the earth. 1

Solve the following geometrical formulae :

8. c^ = a^ + h\ for h. 10. A = .7854 (f , for d.

9. 4m2 = 2(a2 + &2)_c2,form. 11. F= Jtt^-Vi, for r.

12. Using formula 8, find the hypotenuse (c) of a right tri- angle whose other two sides are a = 8 and 6 = 6.

13. Solve formula 8 for b, and find the side (b) of a right triangle whose hypotenuse (c) is 5 and whose side (a) is 3.

14. From formula 8 and the accompanying figure find, to the nearest tenth, the side (a) of a square inscribed in a circle whose diameter (d) is 10.

15. Find, to the nearest tenth of an inch, the dimensions of the largest square timber that can be cut from a log 12 feet long and 18 inches in diameter.

16. Using formula 8 and the accompany- ing figure, deduce a formula for the alti- tude (h) of an equilateral triangle in terms of its side (c). j

17. From formula 9, find the length of the median (m) to the side (c) of the triangle in the accompanying figure, if a = 11, 6 = 8, and c=9.

18. Substituting in formula 10, find, to the nearest tenth of a foot, the diameter (c?) of a circle whose area (A) is 1000 square feet.

19. Using formula 11, find, to the nearest tenth of a centi- meter, the radius (r) of the base of a conical vessel 20 centi- meters high (h = 20) that will hold a liter of water (y= 1 liter = 1000 cu. cm. j ir = 3.1416).

QUADRATIC P:QLATI()NS 285

AFFECTED QUADRATIC EQUATIONS

382. A quadratic equation that contains both the second and the tii'st powers of one unknown number is called an affected quadratic, or a complete quadratic.

x^-\-Sx = 10, i x^ — X = S, and ax^ + bx + c = 0 are affected quadratics.

383. Since affected quadratic equations contain both the second and the first powers of the unknown number, they may always be reduced to the general form of aa^ -\-bx-\- c= 0, in which a, b, and c may represent any numbers whatever, and X, the unknown number.

The term c is called the absolute term.

384. To solve affected quadratics by factoring.

Reduce the equation to the form aa^ + &» -f- c = 0, factor the first member, and equate each factor to zero, as in § 172, thus obtaining two simple equations together equivalent to the given quadratic, subject to the exceptions given in § 230 as to equivalence.

Thus, 3 a;2 = 10 X - 3.

Transposing, 3 x^ _ lo x + 3 = 0.

Factoring, (x - 3) (3 x - 1) = 0.

.•.x-3 = 0or3x-l = 0;

whence, x = 3 or |,

EXERCISES

[Additional exercises are given on page 470.]

385. Solve by factoring, and verify results :

1. a^_5a.4.6 = 0. 7. 20^- 7 x-{-S = 0.

2. ie2_|_i0a; + 21 = 0. 8. 2z''-z-S = 0.

3. .^2 + 120^-28 = 0. 9. 3v2_2v-8 = 0.

4. x2-20.« + 51=0. 10. 10?-2-27r + 5 = 0.

5. x^-ox=24:. 11. 6(s2-|-l)=13s.

6. x^-l = 3(x-^l). 12. 2a;2 + 7a;=4.

286 QUADRATIC EQUATIONS

386. First method of completing the square.

Since (x + ay=:x^-^2ax-^ a^, J

the general form of the perfect square of a binomial is

a^ + 2ax-\-a\ j

Consequently, an expression like a^ -^ 2ax may be made a" perfect square by adding the term a^, which it will be observed is the square of half the coefficient ofx.

Thus, to solve x^+6x= —5

by the method of extracting the square root of both members (the method used in solving pure quadratics), we must complete the square in the first member.

The number to be added is the square of half the coefficient of x; that is, (f)^, or 9. The same number must be added to the second member to preserve the equality.

Therefore, Ax. 1, a^_j_6a; + 9=— 5 + 9;

that is, a;2 -h 6 a; + 9 = 4. m

Extracting the square root, § 289, a; + 3 = ± 2 ; 1

whence, a; = —3-1-2 or —3 — 2.

.-. a; = — 1 or — 5.

EXERCISES

[Additional exercises are given on page 479.]

387. 1. Solve the equation ar» - 5 a; - 14 = 0.

Solution. • a;^ — 5 a; - 14 = 0.

Transposing, x^ — 6 x = H.

Completing the square, oj^ — 5 x -|- ^ = 14 +^^ = ^^.

Extracting the square root, x — | = ± f ;

whence, a: = f4-for^-|.

.•. X = 7 or — 2.

Verification. — Either 7 or — 2 substituted for x in the given equa- tion reduces it to 0 = 0, an identity ; that is, the given equation is satis- fied by these values of x.

QUADRATIC EQUATIONS 287

2. Solve the equation 4a^ + 4a; + 6 = 0. Solution

Transposing, 4a:2 + 4ic = — 6.

Dividing by 4, x^ + x = — f .

Completing the square, «''^ + aj + J = — f + i = — f. Extracting the square root, x -{■ i = ±iv'— 6-

•'• « = -i + W^ or - ^ - iV^ which would usually be written, x = l{— 1 ± V— 6).

Steps in the solution of an affected quadratic equation by the first method of completing the square are :

1. Transpose so that the terms • coiitaining x^ and x are in one piember and the known terms in the other.

2. Make the coefficient of x^ positive unity by dividing both mbers by the coefficient of a^.

3. Complete the square by adding to each member the square 9/ half the coefficient ofx.

4. Extract the square root of both members. 6. Solve the two simple equations thus obtained.

Solve, and verify all results :

3. x^-2x = U3. 12. y^ = 10-3y.

4. a^ + 2a; = 168. 13. z'-lSO = Sz.

5. a^- 4 a; = 117. 14. v^ -\- 15 v = 54..

6. a;2-6a; = 160. 15. v^ + 21v = -54..

7. 8aj = a^-180. 16. n(7i- 1) = 930.

8. x'-\-2x = 120. 17. r2 + 27r + 140 = 0.

9. a^ + 22a; = -120. 18. V -111 + 2^ = 0.

10. a;2^28a;-187. 19. 5a^-3a;-2 = 0.

11. a;2_i2x = 189. 20. Qx^-5x-% = 0.

288 QUADRATIC EQUATIONS

21. .2x'-\-.9x = 3k ^^ 1,3 10 •

24. \-

22. .03af-.07x = .l.

x-\-l x-1 3

25 ^ 3x — 5_ x+^ 23. 2ar'-Y-aj = |. * a;-2 2 ~ 5

388. Other methods of completing the square. To apply the first method of completing the square, the coef- ficient of a^ must be -}-l or be made -|-1.

Other methods of completing the square are based on mak- ing the coefficient of o?^ a perfect square, if it is not already one, by multiplying or dividing both members of the equation by some number.

Thus, given 3 a.-^ + 10 a; = - 3.

Multiplying by 3, 9 a^ + 30 a; = - 9.

When the square of the first member is completed, 30 a; will be twice the product of the square roots of the terms that are squares. Hence, the square root of the term to be added is 16x-T- V9 ay^, or 5 ; and the number to be added is 5^, or 25.

Completing the square,

9a^ + 30a; + 25=-9-h25 = 16.

When the coefficient of a? has been made a perfect square, the number to be added to complete the trinoynial square is obtained by dividing half the term containing x by the square root of the term containing x^, and squaring the quotient.

EXERCISES

389. 1. Solve the equation 8 a^ - 10 a; = 3.

Solution

8 a;2 - 10 a; = 3. Multiplying by 2, 16 a;2 - 20 a; = 6.

Completing the square, 16 x2 - 20 a; + ^ = 6 + ^/ = ^.

Extracting the square root, 4 a; — | = ± |.

4a; = f±| = 6or-l. '.x=\or -\.

QUADRATIC EQUATIONS

Solve, and verify :

289

4. 4.x'-12x = 27.

5. lSa^ + 6x = 4..

6. 2a^-lla; + 12 = 0.

8. 7^2^2^ = 32.

9. Sa^-lSx = 5.

10. 6m2 + 5m = 4.

11. 5n^-Un = -S.

12. Solve the general quadratic equation ax^-\-bx + c = 0.

Solution

ax^ + bx + c = 0. Transposing c, ax^ +bx= — c.

Multiplying by a, c^^x^ + a&a: = — ac.

Completing the square,

Multiplying by 4, 4 a%2 + 4 aba; + 6^ = 6^ - 4 ac.

Extracting the square root,

a2a;2 4. aft^ + — = — - ac. 4 4

2 ax + 6 = ± V 6-^ — 4 ac.

_-6±v62-4ac 2a

(1) (2) (3)

(4)

(6)

(6)

(7)

It is evident that (5) can be obtained by multiplying (2) by 4 a and add- ig 52 to both members. Hence, when a quadratic has the general form

(1), if the absolute term is transposed to the second member, as in (2), ^he square may be completed and fractions avoided by

Multiplying by 4 times the coefficieM of x^ and adding to each member ie square of the coefficient of x in the given equation.

This is called the Hindoo method of completing the square -

Solve by the Hindoo method, and verify results:

13. 2a^ + 3a; = 27.

14. 2aj2 + 5a; = 7.

15. 2ic2 + 7a; = -6.

16. 3 a;- — 7 a; = — 2.

17. 4a^-17ar = -4.

milnk's stand, alg.

18. 4ar'-a;-3 = 0.

19. 5a^-2x-16 = 0.

20. 3ar^ + 7a;-110 = 0.

21. 2ar^-oa;-150 = 0.

22. 3a^4-a;-200 = 0.

19

290 QUADRATIC EQUATIONS

23. 5a^-7x=-2. 25. ISaj^- 7a;-2 = 0.

24. 6a^-f5a; = -l. 26. 7ar^-20a:- 32 = 0.

390. To solve quadratics by a formula.

The general quadratic

ax^-{-bx-\-c = 0 (1)

has been solved in exercise 12, § 389. Its roots are - 1

''- 2^ ^^^

Since (1) represents an?/ quadratic equation, the student is

now prepared to solve any quadratic equation whatever, that

contains one unknown number.

The roots of any quadratic equation, then, may be obtained by

reducing it to the general form and employing (2) as a formula.

EXERCISES [Additional exercises are given on page 479.]

391. 1. Solve the equation Ga;^ = aj + 15. Solution. — Writing the equation in the general form

find that a = Q, h =- 1, and c =- 15.
.. by (2), 1 390, . = 1 ±^(-1);-/ x«(- 1"^) =»±,'« = | or	3 2

Solve by the above formula, and verify results :

2. 2a;2 + 5a; + 2 = 0. 11. l-3a; = 2iB2.

3. 3a;24.11a;+6 = 0. 12. 4 = a;(3a; + 2).

4. 6a^4-2 = 7£c. 13. a?-5x = -S.

5. 4«2_|_4a; = l5. 14. Sa^ — 6x = -2.

6. 2x2^9_3a;. 15. 4iB2-3aj-2 = 0.

7. a;(2a; + 3) = -l. 16. x^ + 10 = 6x.

8. 13a; = 3aj2-10. 17. x''= -4(x-\-S).

9. 7a;2-|-9a; = 10. 18. 4(2a;-5)=ar^. 10. 5x2-18a; = 72. 19. a;(3x + 4) = -2.

QUADRATIC EQUATIONS 291

392. Miscellaneous equations to be solved by any method.

EXERCISES

1. Solve the equation Ssc^ -\-2x = 0.

Remark. — Dividing by x removes the root x= 0 and reduces the equation to the simple equation 3 x + 2 = 0, whose root is a; = — |.

If the given equation is solved by quadratic methods, the roots are found to be the same, namely, 0 and— § ; consequently, it is important to account for roots that may be removed (§ 230) by dividing by an ex- pression that involves the unknown number. The root removed is the root of the equation formed by equating the divisor to 0.

2. Solve the equation -i^ - 6ijtlf == 4.

X— 1 a^

Solution. JL^ _ ?!_±1^ = 4.

x-1 x^

First reducing the second fraction to its lowest terms, then multiplying

)th members by the L.C.D., x (x — 1), simplifying, etc., we have

x^-2x-S=0.

Factoring, (x - 3)(« + 1) = 0.

.-. a; = 3or-l.

Verification. — When a; = 3, each member = 4 ; when x = — I, each

iember = 4 ; that is, both x = 3 and x = — 1 are found to be roots of the

given equation.

Note. — If the second fraction is not reduced to lowest terms before

clearing the equation of fractions, the multiplier is x^ (x — 1 ) instead of

«(x— 1), and the root x= 0 so introduced must be rejected.

In general, no root is introduced by clearing an equation of fractions,

provided that : fractions having a common denominator are combined ;

each fraction is expressed in its lowest terms ; and both members are then

multiplied by the lowest common denominator.

General Directions. — 1. Reduce the equation to the general form ax"' + 6a; + c = 0.

2. If the factors are readily seen, solve by factoring.

3. If the factors are not readily seen, solve by completing the square or by formula.

4. Verify all results, reject roots introduced in the process of reducing the equation to the general form, and account for roots that have been removed.

292 QUADRATIC EQUATIONS

Solve according to the general directions just given : 3. a^-6x'-f5 = 0. jQ 9a; ■ 3 ^^

4. 2x^-5x = 0.

5. 7a;2 + 2a; = 32. 20.

7. a^-30 = 13a;.

12. a;2-4.3a; = 27.3.

9 3a;	-6	_35 4
	X	a;	-2
• 9(a	-1)		6
	4		1
• x'-	-2a; + l	4
4	2a;_ 3 ■	= 28

2a;2_^^ a;-3

1 H-a; x — 1 _4 a;-3 a;-2~5'

6. a;^ = 3a; + 10. ^^ ^ ^_5 3

x — 5 X 2

8 a-2-12a; = 28. 22. ^^±1 + iH±l? = 7.

a; 4- 5 a; + 6 9. ar^-12a; = 0.

23 ^' + 4 o_(a; + 3)\

10. 18ar^ + 6a; = 0. ^^- ^^ + '^- a;2_9

11. 4a;2-12a; = 0. ' a «^ 4 ^.

24. = -f- o.

x-2 x-2

X ,1 x+2

13. a;^ + .25a; = .015. 25. -_^ + - = -^_.

14. a;+i-| = 0. 26. J^ + ^+^ = 3.

a? 2 a; + 7 a; + 3

27. ^±2_^±J? = 1. a; — 7 a;— 5

28 ^7-3 a; + 2^23

a;4-4 a;-2 10*

2a;4-l 5^a;-8 ■ l-2a; 7 2

30. 1^^ = 2- ^

ar*-3a; x'-Sx

Find roots to the neai'est thousandth :

31. ar^-4a;-l=0. 33. u^ + 5u-^5.5 = 0.

32. ^2 + 6^ + 7 = 0. 34. «2_i2i + 16.5 = 0.

»

QUADRATIC EQUATIONS

Literal Equations

293

393. The methods of sohition for literal quadratic equations are the same as for numerical quadratics. The method by factoring (§ 384) is recommended when the factors can be seen readily. If it is necessary to complete the square, the first method (§ 386) is usually more advantageous, provided the coefficient of ar' is +1, otherwise the Hindoo method (§ 389). 'is better, because by its use fractions are avoided. Results •may be tested by substituting simple numerical values for the literal known numbers.

EXERCISES

394. Solve for x by the method best adapted : 1. a^ — ax=ah — bx. 4. 5x — 2ax = oi^ — 10 a.

2. X- -\- ax = ac -{- ex. 5. a?^ -|- 3 bx = 5cx-^ 15 be

3. X' = (m — n) X -\- mn. 6. 6x^ -{-Sax = 2bx-\- ah

7. acx^ — bcx —bd-\- adx = 0.

8. 0^ + 4 mx + 3 wa; 4- 12 mn = 0. 9. x^ = 4:ax—2a\

10. x^ — ax-^a^ = 0.

11.	4.ax-x' = 2>aK
12.	^ax-[-&a^ = ^x\
13.	21b^-4:bx = x'.
14.	^-^ mx = ^\ 12 3
15.	x^ 5x b 36 4 3
^R	X X

1 x-\-l

m.

17. x-{- — = — -\-b. X b

18. 2a;-^^ = a-2a;.

19.

aa;H-4

= 1-

ax

16

20. a^^^x = ^±^. b b

21. ^ + 2 = (2a^)

X.

22. ^._2j;^4(a6-l). ab ab

294 QUADRATIC EQUATIONS

23. a^-2(a-6)a; = 4a6.

24. x^ — 2 x{m — n) = 2 mn.

25. oc^ ^2{a-\- S)x = -32 a.

26. oc^ -\- X -^ bx + b = a(x -\- 1).

27. a(2 ic - 1) + 2 6a; - 6 = a;(2 a; - 1).

28. 0(^-{-A(a-l)x = 8a-4:aK

29. _1_ = 1 + 1 + 1.

a + 5 + ic a b x

30. »' + ! 1 ^

n^a; — 2 n 2 — na; n

_ 2a4-a; a — 2a;_8 2 a - a; a + 2 a; ~ 3*

32. -1 i- = %+^.

a — a; a -[- x or — x^

33. a(a5 — 2 a + &) + a(a; + a — 6) = a;2 — (a — 6)2.

34. ^_fl+J-^.+i+l=o.

a + 6 V a6y a 6

35.

ar^ + 1 a + 6

X c a 4- b

36. ^^-^ + 3 ^^

2 a;

37.

6a;

^ ^^a(a: + 2 6)

a —X a H- 6

3a; -1-6

a; + 6 2a; — a 14.^

39. ^^ + f. + «^Y_('^ + ^)=a.. a^ \ xj \a x)

a + 6

QUADRATIC EQUATIONS 295

Radical Equations

395. In §§ 364, 365, the student learned how to free radical

equations of radicals, the cases treated there being such as lead to simple equations. The radical equations in this chapter lead to quadratic equations, but the methods of freeing them of radicals are the same as in the cases already discussed.

396. The principles of § 230 in regard to the equivalence of equations have been illustrated in § 392, exercise 1, showing the removal of a root by dividing by an unknown expression, and exercise 2, showing the introduction of a root by clearing of fractions unless certain precautions are taken. In the dis- cussion of § 366 it was shown that the processes of rationali- zation and involution, used in freeing radical equations of radicals, are likely to introduce roots according to the con- vention adopted there as to how roots shall be verified.

Hence, it is important in the solution of equations that roots be tested not only to determine the accuracy of the work, but to discover whether the solutions obtained are really roots of the given equation, and also to examine the processes em- ployed in reducing equations to see whether any roots have been removed.

EXERCISES

397. 1. Solve the equation 2^x — x — x — 8 V^.

Solution

2Vx-x = x-^Vx.
Dividing by Vx, 2 - Vx = Vx - 8.
Transposing, etc., Vx = 5.
Squaring, x = 25.
Verification. — When x = 25,
1st member = 2 V25 - 25 = 10 - 25 =	-15;
2d member = 25 - 8V25 = 25 — 40 =	-15.

Hence, x = 25 is a root of the equation ; x = 0, the root of the equation Vx = 0, also is a root of the given equation, removed by dividing both members by Vx.

296 QUADRATIC EQUATIONS

2. Solve and verify V^ + l + ■Vx — 2 — V2 a; — 5 = 0. Solution

Vx+ 1 + Vx-2 - V2x-b = 0.

Transposing, Vx + 1 + Vx - 2 = \/2 a; — 6.

Squaring, x + 1 + 2 Vx^ -x — 2+ x-2 = 2x-5. Simplifying, Vx^ — x - 2 = — 2.

Squaring, x^ — x — 2 = 4.

Solving, X = — 2 or 3.

Verification. — Substituting — 2 for x in the given equation,

that is, V^ + 2 V^^ - 3 V^^= 0.

Therefore, — 2 is a root of the given equation. Substituting 3 for x in the given equation,

Vi -f Vi - vT = 0,

which is not true according to the convention adopted in the discussion in J § 366. Hence, 3 is not to be regarded as a root of the given equation.

Note. — The equation could be verified for x = 3 if the negative square root of 1 were taken in the second term and the positive square root in \ the third, thus : |

Vi + VT - Vi = 2 + (- 1) -(+ 1) = 0. J

i

This is an improper method of verification, however, for it has been agreed previously that the square root sign shall denote only the positive square root.

Solve and verify, rejecting roots that do not satisfy the given equation, and accounting for roots that otherwise might be lost :

3. 8V«-8a; = |. 5. a;-l + Va; + 5==0.

4. 3a;4- V^ = 5V4^. 6. a;-5- Va;- 3 = 0.

7. V4 a; + 17 4- Va; + 1 — 4 = 0.

8. l + V(3-5a;)2 + 16 = 2(3-a;).

QUADRATIC EQUATIONS

9. -^1 -\-xVx^ + 12 = 1 + a7.

10. Va'-1 + V2x-1-V5a; = 0.

11. ■V2X — 7 — V2x + V'i» — 7 = 0.

12. Va; + 3 + V4 a; + 1 — VlO a; + 4 = 0.

13. Va 4- ic — Va-^ = V2 «.

14. Va; — a + V6 — a; = V& — a.

15. vV— 6^= Va; + 6Va + 6.

16. ^2^+ VlOa; + l= V2^4-l.

17. VO + a; 4- Va; - VlO — 4 a; = 0.

18. V4a;-3- V2a; + 2 = Va;-6.

19. V2 a; + 3 — Va; + 1 = V5 a; - 14

20. V3a; — 5 + Va; — 9= V4a; — 4. 6

Vx'-hS

= x.

22. a; + Va;- 4- ^'^-^

m'

^x' +

mr

23. X -\- ^x^ — a-

V^^^^

24 2a;+V4a^-l^^^ 2a;-V4a^-l

_^ x — a, x-\-a 2

25. A — — +a/— ^— =:a2.

^x + a ^a; — a

26. ■Va — x + -Vb — x=Va-^b — 2x,

27. V^T^ — V a; — 2 a- = V2 a; — 5 a^.

28. Vm^i — x— -y/x ^mii — 1 = Vwn Vl

297

298 QUADRATIC EQUATIONS

Problems

[Additional problems are given on page 480. ]

398. 1. The sum of two numbers is 8, and their product is 15. Find the numbers.

Solution Let X = one number.

Then, 8 - a; = the other.

Since their product is 15, (8 — a;)x = 15. Solving, a; = 5 or 3,

and 8 - X = 3 or 5.

Therefore, the numbers are 5 and 3.

2. Separate 20 into two parts whose product is 96.

3. Separate 14 into two parts whose product is 45.

4. Find two consecutive integers the sum of whose squares is 61. • I

5. A rectangular garden is 12 rods longer than it is wide and it contains 1 acre. What are its dimensions?

6. A plumber received $ 24 for some work. The number of hours that he worked was 20 less than the number of cents per hour that he earned. Find his hourly wage.

7. The 1860 bunches of asparagus from an acre of land were sold in boxes each holding 1 less than i as many bunches as there were boxes. Find the number of bunches in a box.

8. The area of a tablet is 2838 square inches. If its length exceeds its width by 23 inches, what are its dimensions?

9. An ice bill for a month was $4,80. If the number of cakes used was 4 less than the number of cents paid per cake, how many cakes were used ?

10. The height of a box is 5 feet less than its length and 2 inches more than its width. If the area of the bottom is 8J square feet, what are the dimensions of the box ?

r

QUADRATIC EQUATIONS 299

11. A party hired a coach for $12. Since 3 of them failed to pay, each of the others had to pay 20 cents more. How many persons were there in the party ?

Solution

Let X = the number of persons.

Then, a; — 3 = the number that paid,

12

— = the number of dollars each should have paid,

X

1 2 — = — = the number of dollars each paid. ic-3

Therefore,

12 1^12 -3 6 X

Solving, a; = 16 or — 12.

The second value of x is evidently inadmissible, for there could not be a negative number of persons. Hence, the number in the party was 16.

12. A club had a dinner that cost $ 60.* If there had been

5 persons more, the share of each would have been $1 less. How many persons were there in the club ?

13. A party of young. people agreed to pay $8 for a sleigh ride. As 4 were obliged to be absent, the cost for each of the rest was 10^ greater. How many went on the ride ?

14. The sum of the three dimensions of a block is 35 feet and its width and height are equal. The area of the top ex- ceeds that of the end by 50 square feet. Find its dimensions.

15. The sum of the three dimensions of a box is 58 inches and its length and width are equal. The area of the bottom exceeds that of one end by 176 square inches. Find its height.

16. A bale of cotton contains 21 cubic feet. Its length is 41 feet, and its width is ^ of a foot less than its thickness. Find its width ; its thickness.

17. A man sold raisins for $480. If he had sold 2 tons more and had received $ 20 less per ton, he would have re- ceived the same amount. How many tons of raisins did he sell?

300 QUADRATIC EQUATIONS

18. A tub of dairy butter weighed 20 pounds less than a tub of creamery butter, and 360 pounds of dairy butter re- quired 3 more tubs than the same amount of creamery butter. What weight of butter was there in a tub of each kind ? J

19. A moving picture film 150 feet long is made up of a certain number of individual pictures. If these pictures were J of an inch longer, there would be 600 less for the same length of film. How long is each separate picture ? i

20. A purchase of 80 four-inch spikes weighed 3 pounds less than one of 80 five-inch spikes. If 1 pound of the former con- tained 6 spikes more than 1 pound of the latter, how many of each kind weighed 1 pound ? J

21. Mr. Eield paid $8.00 for one mile of No. 9 steel wire and $2.88 for one mile of No. 14 wire. The No. 9 wire weighed 224 pounds more, and cost i ^ per pound less, than the No. 14 wire. Find the cost of each per pound. J

22. A train started 16 minutes late, but finished its run of ' 120 miles on time by going 5 miles per hour faster than usual. What was the usual rate per hour ?

23. To run around a track 1320 feet in circumference took one man 5 seconds less time than it took another who ran 2 feet per second slower. How long did it take each man ?

24. Two automobiles went a distance of 60 miles, one going 6 miles per hour faster than the other and completing the journey | of an hour sooner. How long was each on the way ?

25. If the rate of a sailing vessel was 1^ knots more per j hour, it would take ^ an hour less time to travel 150 knots, i Pind the rate of the vessel per hour.

26. A man rode 90 miles. If he had traveled ^^ of a mile more per hour, he would have made the journey in 10 minutes less time. How long did the journey last ?

t

QUADRATIC EQUATIONS 301

27. A cistern can be filled by two pipes in 24 minntes. If

it takes the smaller pipe 20 minutes longer to fill the cistern

than it does the larger pipe, in what time can the cistern be

filled by each pipe ?

Solution

Let jc = the number of minutes required by the larger pipe.

Then, x + 20 = the number of minutes required by the smaller.

Since - = the part that the larger pipe fills in one minute,

X

= the part that the smaller pipe fills in one minute,

rK+20 and 2^5 = the part that both pipes fill in one minute,

M 1,11

' X x + 20 24

Solving, a; = 40 or — 12.

I Hence, the larger pipe can fill the cistern in 40 minutes. 28. A city reservoir can be filled by two of its pumps in days. The larger pump alone would take If days less time than the smaller. In what time can each fill the reservoir ?

29. A tank can be emptied by two pipes in 3^ hours. The larger pipe alone can empty it in 1^ hours less time than the smaller pipe. In what time can each pipe empty the tank ?

30. A company owned two plants that together made 25,200 concrete building blocks in 12 days. Working alone, one plant would have required 7 days more time than the other. What was the daily capacity of each plant ?

31. The sum of the reciprocals of two consecutive integers is /q. Find the integers.

32. Find the price of eggs per dozen, when 2 less for 30 cents raises the price 2 cents.

ft 33. A merchant sold a hunting coat for $ 11, and gained a per cent equal to the number of dollars the coat cost him. What was his per cent of gain ?

34. By receiving two successive discounts, a dealer bought for $9 silverware that was listed at $20. What were the dis- counts in per cent, if the first was 5 times the second ?

302

QUADRATIC EQUATIONS

Formulae f

399. 1. In any right-angled triangle (Fig. 1), c^ = a'^-\-W Find all the sides when a = c — 2 and 6 = c — 4.

Fig. 1.

Fig. 3.

2. The area {A) of a triangle (Fig. 2) is expressed by the formula A = ^ ah. If the altitude Qi) of a triangle is 2 inches greater than the base (a) and the area is 60 square inches, what is the length of the base ?

3. If two chords intersect in a circle, as shown in Fig. 3, a • 6 is always equal to c- d. Compute a and h when c = 4, d = 6, and h = a -{-^.

4. The formula h = a -^vt — l^f gives, approximately, the height (h) of a body at the end of t seconds, if it is thrown vertically upicard, starting with a velocity of v feet per second from a position a feet high.

Solve for t, and find how long it will take a skyrocket to reach a height of 796 feet, if it starts from a platform 12 feet high with an initial velocity of 224 feet per second.

5. How long will it take a bullet to reach a height of 25,600 feet, if it is fired vertically upward from the level of the ground with an initial velocity of 1280 feet per second ?

6. When a body is thrown vertically downward, an approxi- mate formula for its height \%h — a — vt — \^ f, in which 7i, a, V, and t stand for the same elements as in exercise 4.

Solve for t, and find when a ball thrown vertically downward from the Eiffel tower, height 984 feet, with an initial velocity of 24 feet per second, will be 368 feet above ground.

Find, to the nearest second, when it will reach the ground.

QUADRATIC EQUATIONS

303

EQUATIONS IN THE QUADRATIC FORM

400. An equation that contains but two powers of an un- :nown number or expression, the exponent of one power being

twice that of the other, as ao^" + 6x** + c = 0, in which n repre- sents any number, is in the quadratic form.

EXERCISES

401. 1. Given a;* + 6 a:^ — 40 = 0, to find the values of x. Solution

x* + 6 a;2 - 40 = 0. Factoring, (x^ - 4) (a;2 + lO) = 0.

.-. a;2_4 =oora;2 + 10 = 0, a; = ± 2 or ± V- 10.

2. Given x^ — x^ = 6, to find the values of x. First Solution

x^ - x^ = 6. Completing the square, x^ — x* + ( J)^ = ^5^-. Extracting the square root, x? — ^ = ± f •

.-. xi = 3 or-2. Raising to the fourth power, x = 81 or 16.

Second Solution Let X? = i), then, x2 = p^^ and p^ — p = 6.

/. p2 _ p _ 6 = 0. Factoring, (l> - 3) (p + 2) = 0.

.-. ^ = 3 or — 2 ; that is, x^ = 3 or — 2.

Whence, x = 81 or 16.

Since x = 16 does not verify, 16 is not a root and should be rejected.

I

304 QUADRATIC EQUATIONS

Solve the following equations :

3. aj*-13ar^ + 36 = 0. 11. x^-Sx^ = -2.

4. a;^ - 25 oj^H- 144 = 0. 12. x^-x^ = 6.

5. a:*-18a^ + 32 = 0. 13. aj+2Va- = 3.

6. 3x'-^5x'-S = 0. 14. a;*-2a;^ = 3.

7. 5a;^+6a^-ll=0. 15. x^ = 10x^-9.

8. 2a;*-8a.'2-90 = 0. 16. (a; -3)^ + 2 (a; -3) = 3.

9. x^-5x^ + 6 = 0, 17. {x'+iy-\-4.(x''-\-l)=4.5. 10. a;^ + 3aj^-28 = 0. 18. (x^-4.y -3{x^-4:)=10.

19. Solve the equation a; — 4 a;^ + 3 a;^ = 0. 4

Solution Let x^ =p, then, x'^ =P% and x =pK

Then, p^ - ip"^ + Sp = 0.

Factoring, p(p2 _ 4 ^ _|. 3) _ q.

Whence, p = 0

or p2_4p^3 _0.

Factoring, (^ _ 1 ) (j, _ 3) = 0.

Whence, p = 1 orp = 3.

That is, a;^ = 0, 1, or 3.

.-. a; = 0, 1, or 27. Solve:

20. aj^ — 4 a; — 5a;^ =0. 23. 5x = xVx + 6Vx.

21. a;-3a;^ + 2a;'=0. 24. Sx = x^x-^2y/x^.

22. a; + 2a;^ — 3a;^ = 0. 25. 2a; + v^=15a?Va;. '

QUADRATIC EQUATIONS

305

26. Given x^ -7 x + Vx'-l x-\-l^ = 24, to find the value

^of X,

Solution

		x'2-lx + Vx^-7x + lS =	24. 42.	(1)
Adding 18,	a;2 _ 7 a; + 18 + \/a;--2 - 7 a; + 18 =	(2)
Put;)for (a;2-	-7x	+ 18)^ and p'^ for (x^ - 7 x + 18)	.	(3)
Then,		p^+p-42 =0.		(4)
Solving,		p = 6 or - 7.		(6)
That is,	y/x^-1x+lSz=:6	(6)
p	Vx2-7x+ 18=-7.	(7)
Squaring (6),		»2 _ 7 X + 18 = 36.
Solving,		a; = 9 or - 2.
Since (§ 366)	the	radical in (7) cannot equal a	negative	number,

^x2 — 7 x + 18 = — 7 is an impossible equation. Hence, the roots of (1) are x = 9, - 2.

27. Solve the equation x-^2 Vic + S = 21.

28. Solve a^-3a; + 2Va^-3a?H-6 = 18.

29. Solve the equation afi-9x^-\-S =0,

Solution

a^_9x3 + 8 = 0. (1)

Factoring, (x^ - l)(x3 _ 8) = 0. (2)

Therefore, x^ - 1 = 0. (8)

x3 - 8 = 0. (4)

If the values of x are found by transposing the know^n terms in (3) and /(4) and then extracting the cube root of each member, only one value of X will be obtained from each equation. But if the equations are factored, '\.three values of x are obtained for each.

^ Factoring (3), (x - 1) (x2 + x + 1) = 0, (6)

and (4), (a; _ 2) (x2 + 2 x + 4) = 0. (6)

Writing each factor equal to zero, and solving, we have

From (5), x = 1, i (- 1 +V^), I (- 1 - V^l). From (6), X = 2, - 1 + V^^, - 1 - V^^.

(7) (8)

MILNE'S STAND. ALG.

20

B06 QUADRATIC EQUATIONS

Note. — Since the values of x in (7) are obtained by factoring jc^ — 1 = 0, they may be regarded as the three cube roots of the number 1. Also, the values of x in (8) may be regarded as the three cube roots of the number 8 (§ 286).

Solve;

30. a^-28aj»-H27 = 0. 31. a?<-16 = 0.

32. Find the three cube roots of — 1.

33. Find the three cube roots of — 8.

34. Solve the equation a;*-f-4a^ — 8a; + 3 = 0.

First Solution Extracting the square root of the first member as far as possible, x* + 4 a;3 - 8 a; + 3|a;'-^ + 2x - 2 a:4

2 ic2 + 2 x	4a;8
	4 x3 + 4 a;2
2 a;2 + 4 X	-2	- 4 aj2 - 8 X + 3 - 4 x2 - 8 x + 4

-1

Since the first member lacks 1 of being a perfect square, the square may be completed by adding 1 to. each member, which gives the follow- ingequation: ^ + i^-i^ + i=l.

;Extractiiig the square root, x^ -\-2x — 2 = ±\.

.-. a;2 + 2 a; - 3 = 0, and x2 + 2 X - 1 = 0. Solving, a; = 1, - 3, - 1 i \^.

Second Solution

If 4 x2 is added to the first two terms of the given equation, the re- sulting trinomial, x* + 4 x'^^- 4 x^, will be a perfect square. Adding 4 x^ and subtracting 4 x^ does not change the first member.

Then, a* + 4x3 + 4x2- 4x2- 8x + 3 = 0;

whence, (x2 + 2 x)2 - 4 (x2 + 2 x) + 3 = 0.

Factoring, (x2 + 2 x - 3) (x2 + 2 x - 1) = 0.

Solving, « = 1, - 3, - 1 ± V^.

QUADRATIC EQUATIONS

307

Third Solution

By applying the factor theorem (§ 164), the factors of the first mem,' ber are found to be a; — 1, x + 3, and x'-^ + 2 x — 1 ; that is,

(X - 1) (x + 3) (x2 + 2 X - 1 ) = 0.

Solvmg, X = 1, - 3, - 1 ± V2.

Solve :

35. x* + 2a^-x = S0. 37. «* - 2 a;^ + a? = 132.

36. x*-4:a^ + Sx=-3, 38. a;* - 6 ar^ + 27 a; = 10.

39. x* + 2x^ + 5x'--\-4.x-60 = 0.

40. x* + 6a^-{-7x^-6x-S = 0.

41. a;^-6a^ + 15a;'^-18a; + 8 = 0.

x-tl x" 12

Suggestion. — Since the second term is the reciprocal of the first, put

for the first term and - for the second. P

Then,

43. to +

P +

a;^ + a;

= 2.

1^25^ p 12*

45 ^ + 2 2(0^ + 4)^51 a;2_^4^ ^_l_2 5

44. ^ +

a.-^ + l 2

46 a^'^+l 4(a;-l)_^21 a;-l a^ + 1 5 *

Solve the following miscellaneous equations :

47. a;t — 6a; + 8a;i = 0. 52. a; V^ + 20 V^ = 9 ».

48. 2 a; — 3a;? = — a;2.

49. ■</a; + 3V^ = 30.

50. aa^^n _|_ 5^« ^ g ^ Q^

61. a;-7a;l + 10a;i = 0.

53. a; — 5 + 2 Va; — 5 = 8.

54. aj + 10 = 2V^TlO + 5.

55. a;-3 = 21-4Va;-3.

56. 2aj-3V2a; + 5 = -5.

308 QUADRATIC EQUATIONS

57. 2x-6V2x-l = S. 59. x-^ - 5 x-l -]- 6 = 0.

58. a; = 11 —3Vx-\-J. 60. ic~^ — 5 a;"^ + 4 = 0.

61. a^-5x + 2Va^-5x-2 = 10.

62. x^-x- ■VxF -ic + 4-8 = 0.

63. o^ — 5x-^o Vxr — 5 ic + 1 = 49.

64. (x'-2xy-2(x'-2x)=S.

65. {jf-xy-ix'-x)- 132 = 0.

66. ri?-lY + Sr-^-lV33. a;

67. (. + lY-2(x + l) = f.

68. (l±£.y + 2(l±^ = 8.

70. rc^-10a.'3 + 35a^-50a; + 24 = 0.

71. 16x^-8 a;^-31a;2 + 8a; + 15 = 0.

72. 4a;*-4a^-7ar + 4a; + 3 = 0.

73. a^ + flj + l

74. x'-2x-\-- — =4.

x'-2x-\-l

75. aj2-3a;+- — ? =1.

1		8 = 3-
a^ + ic	+ 1
76.	a^-	" + a^-	2 aj-	-4	=7
77.	X	1^'-	-1	= -	13
a^-	1 ' a;		6

78. i + ^ 3^Q,

l + aj + a^ Vl+aJ-ha;'

QUADRATIC EQUATIONS 309

SIMULTANEOUS EQUATIONS INVOLVING QUADRATICS

402. Two simultaneous quadratic equations involving two unknown numbers generally lead to equations of the fourth degree, and therefore they cannot be solved usually by quad- ratic methods.

However, there are some simultaneous equations involving quadratics that may be solved by quadratic methods, as shown in the following cases.

403. When one equation is simple and the other of higher degree.

Equations of this class may be solved by finding the value of one unknown number in terms of the other in the simple equation, and then substituting that value in the other equation.

EXERCISES

\x + y — l,

404. 1. Solve the equations <^

Solution

x-^y = l. (1)

3a;2 + «/2^43. (2)

From (1), y = 1-x. (3)

Substituting in (2), 3 «2 + (7 - xY = 43. (4)

Solving, X = 3 or J. (5)

Substituting 3 for x in (3), y = 4. (6)

Substituting ^ for a; in (3), y = Y- C^)

r when a; = 3, y = 4, That is, X and y each have two values \

I when x = ^^y = ^.

Solve the following equations :

2.

3.

x = 2y. \x — y = 2.

10x-^y = Sxy, ^ fm'-3n' = lS,

y-x = 2.

(;

Im — 271 = 1.

810 QUADRATIC EQUATIONS

^ (x = e-y, ^ (Sx(y + l) = 12,

[a^ + f = 72, ' [Sx = 2y.

(xy(x-2y) = 10, (Srs-10r = s,

[xy = 10. ' [2-8= -r.

405. An equation that is not affected by interchanging the unknown numbers involved is called a symmetrical equation.

2 aj2 + xy + 2 2/2 = 4 and x^ -i- y"^ = 9 are symmetrical equations.

406. When both equations are symmetrical.

Though equations of this class may be solved usually by substitution, as in §§ 403, 404, it is preferable to find values for x-{-y and x — y and then solve these simple equations for x and y,

EXERCISES

{ic -f w = 7, xy=:10.

Solution

« + y = 7. (1)

xy = 10. (2)

Squaring (1), x^ + 2xy + y^z= 49. (3)

Multiplying (2) by 4, 4xy = 40. (4)

Subtracting (4) from (3), x^-2xy + y^ = 9. (5)

Extracting the square root, x — y=±S. (6)

From (1) + (6), a; = 5 or 2.

From (1)- (6), y = 2 or 6.

2. Solve the equations \

Suggestion. — From the square of the second equation subtract the first equation ; then subtract this result from the first equation and pro- ceed as in exercise 1.

QUADRATIC EQUATIONS

311

3. Solve the equations

(x' + y' = { [x + y = l.

Solution

97,

jc* + y4 = 97.

x+y = l. Raising (2) to the fourth power,

x^ + 4 a;8y + 6 x'^y^ -{■ixy'^ + y^ = 1.

Subtracting (1) from (3), 4x^y + 6 x'^y'^ + ixy^ = - 96.

Dividing by 2, 2 x^y + 3 xV + 2 x?/^ = _ 48.

2xy X square of (2), 2x^y + i xV + 2 xy^ = 2 xy.

Subtracting (5) from (6), x'^y^ -2xy = 48.

Solving for xy, xy = — 6 or 8.

Equations (2) and (8) give tvv^o pairs of simultaneous equations,

1

\xy=:-G \.

Solving as in exercise 1, the corresponding values of x and y are

x= 3; ,y=-2;

2; 3;

i(l+V_31); Kl-V-31);

Solve the following equations : .xy = l.

4.

5.

6.

I

8.

aj2 4-2/2 = 34.

x + y=-.^, a^-h2/' = 243.

x-\-y = ^,

x^ -\- xy -{- y^ = 49.

x^-\-xy-\-y' = Sl, a^ + 2/'=26.

9.

10.

11.

12.

13.

(1)

(2)

(3)

(4) (5) (6) (7) (8)

i(i_V-3i); i(i+v:r3r).

I ar* — ic?/ + 2/^ = 4.

^x-^y + xy = 11.

x' + Sxy + y' = 31, xy = 6.

(x' [(x

x^-i-y^-. (x-{-yy

100, = 196.

x-\-xy-{-y = 19, x^y^ = 144.

312 QUADRATIC EQUATIONS

^^ |«' + /=17, ^^ (x* + x^f + y* = 21, I

lfl; + ^ = 3. la^ + iC^ + 2/" = 7. I

408. An equation all of whose terms are of the same degree with respect to the unknown numbers is called a homogeneous equation.

x'^ — xy = y^ and Sx^ + y^ = 0 are homogeneous equations. Jm

An equation like a^ — xy -{- y^ = 21 is said to be homogeneous in the unknown terms.

409. When both equations are quadratic, one being homogeneous.

In this case elimination may always be effected by substitu- tion, for by dividing the homogeneous equation through by y^, it

becomes a quadratic in -• The two values of - obtained from

y y

this equation give two simple equations in x and y, each of which may be combined with the remaining quadratic equa- tion as in §§ 403, 404.

Thus, a2(? -\- hxy -\- cy^ = 0 is the general form of the homo- geneous equation in which a, b, and c are known numbers.

Dividing by /, we have af - J -|- 6i - J + c = 0, a quadratic in - •

EXERCISES

410. 1. Solve the equations ] + ^~2/— >

[5a^-^4:xy-y' = 0.

[5x'-}-4:xy-f = 0.
Solution
«2 + 3 a; - 2/ = 6.	0)
5 a;2 -1- 4 xy - 2/2 = 0.	(2)

Dividing (2) by y^, 5( - J + 4(- J — 1 = 0, a quadratic in - which may

be solved by factoring or by completing the square.

To avoid fractions, however, (2) may be factored at once ; thus,

(x + y)(^x-y) = 0. .'. y = — X or 6 aj.

QUADRATIC EQUATIONS

313

Substituting — a; for y in (1), simplifying, etc.,

a;2 + 4 X - 5. Solving, x = l or — 6.

:.y z= — x=— 1 or 6.

(8) (4)

Substituting 5a; for y in (1), simplifying, etc.,

x^-2x = 5. Solving, a; = 1 + V6 or 1 - V6. (5)

.•.y = 5a; = 5(l+\/6)or5(l-V6). (6)

Hence, from (3), (4), (5), and (6) the roots of the given equation are U = -l; 5; 6(1+V6); 6(1-V6)

Solve the following equations :

5.

5a^ + 8a;2/-42/' = 0, xy + 2y^ = 60.

' 2x^ — xy — y^ = 0, .4:X^-\-4:xy + y^ = 36,

6x'-{-xy-12y' = 0, [x^-{-xy — y = l.

7.

Sx^-7xy-4:0y^ = 0y x^-xy-12y^ = S.

x^ — xy — y^ = 20,

3x'-13xy + 12y' = 0. ^ (3x^-7xy + 4.f = 0,

g (a^-^y'-\-x-y = 12, \3x'-\-2xy-y^ = 0.

411. When both equations are quadratic and homogeneous in the unknown terms.

Either of the following methods may be employed in this case :

Substitute vy for x, solve for y^ in each equation, and com- pare the values of y^ thus found, forming a quadratic in v.

Or, eliminate the absolute term, forming a homogeneous equation; then proceed as in §§ 409, 410.

314

QUADRATIC EQUATIONS

	EXERCISES
412. 1. Solve the	equations	■x'~xy-\-y' = 21, y-2xy=-W,
	First Solution
	x'^-xy + y'^ = 21.
	y^-2xy= -15.
Assume	x = vy.
Substituting in (1),	vY - 'oy^ + y'^ = 21.
Substituting in (2),	y2 _ 2 vy^ = - 15.
Solving (4) for y"-,		«2_ 21
^ t?2 - t; + 1
Solving (5) for y2,	^ 2v-l
Comparing the values of y'^^ 2	15 21
V - 1 v^—v-\-l
Clearing, etc.,	5 w2 _ 19 u + 12 = 0.
Factoring,	(v-^)Coi	- 4) = 0.

I

(1)

(2) (3) (4) (5) (6)

(7)

(8)

(9) (10)

(11)

(12)

(13)

.-. V = S or f. Substituting 3 for v in (7) or in (6), y = ±VS 1 and since x = vy, x— ±3V3. j

Substituting \ for ?) in (7) or in (6), y = ± 5 1 and since x = vy, a; = ± 4. J

When the double sign is used, as in (12) and in (13), it is understood that the roots shall be associated by taking the upper signs together and the lower signs together.

3\/3; -3\/3; 4; -4

\/3 ; - \/3 ; 5 ; - 5,

Suggestion for Second Solution. — Multiplying the first equation by

6 and the second by 7, and adding the results, we eliminate the absolute

term and obtain the homogeneous equation

5a;2_19xy+ 12^/2 =0, which may be solved with either of the given equations, as in exercise 1, §410.

Solve the following equations :

Hence,

I

0-2/ + 3 2/' = 20,

'o?-{-xy = 12^ .xy + 2y^ = 5.

QUADRATIC EQUATIONS

316

4.

5.

rar' + 2 2/2= 44, U2/-2/^ = 8.

p(aj-2/)=6,

7.

8.

'x^ — xy + y^ = 21y 2a^-3icy + 2/ = 100,

rzx"-

lar-2/'

75.

6.

Qi? — xy — y^ = Zxy^2f

20, ra^-5a;2/ + 32/' = 8,

= 8. ^' |3ar' + a;2/ + 2^=24.

413. Special devices.

Many systems of simultaneous equations that belong to one )r more of the preceding classes, and many that belong to none of them, may be solved readily by special devices. It is impos- sible to lay down any fixed line of procedure, but thp object often aimed at is to find values for any two of the expressions, x-f-y, x — y, and xy^ from which the values of x and y may be obtained. Various manipulations are resorted to in attaining this object, according to the forms of the given equations.

EXERCISES

414. 1.

12,

Solve the equations ]

la^-l- 2/^ = 4.

	Solution
	a;2 + a;y = 12.	(1)
	xy + y^ = A,	(2)
Adding (1) and (2), x'^ -{■ 2 xy ^ y'' = 16.	(3)
	.'. ic + y = + 4 or — 4.	(4)
Subtracting (2) from ^1),	a;2-2/2 = 8.	(5)
Dividing (5) by (4),	a;-?/=+2or-2.	(6)
Combining (4) and (6),	a; = 3or— 3; ?/ = lor— 1.

Note. — The first value of x — y corresponds only to the first value of X + y, and the second value only to the second value.

Consequently, there are only two pairs of values ofx and y.

Observe that the given equations belong to the class treated in § 411. The special device adopted here, however, gives a much neater and simpler •olution than either of the methods presented in that case.

316 QUADRATIC EQUATIONS

2. Solve the equations I ^ ^ ^ ^ ^ > m

[xy = 3. 1

Solution *

x^ + y'^-\-x + y = U, (1)

xy = 3. (2)

Adding twice the second equation to the first,

x^ -\- 2xy + y'^ + X + y = 20. Completing the square, (a: + y)^ + (x -\- y) + (|)2 = 20^. Extracting the square root, x -}- y + ^ = ±|.

.'. X + y = 4 or - 5. (3)^

Equations (2) and (3) give two pairs of simultaneous equations,

y = -5 3

[xy = S [xy =

Solving, the corresponding values of x and y are found to be

(x = S; 1; i(_5+Vl3); |(-.5-Vl3); ^ ^

[y = l; 3; |(_5-Vl3); ^(-5 + >/l3). '|

Symmetrical except as to sign. — When one of the equations is symmetrical and the other would be symmetrical if one or more of its signs were changed, or when both equations are of the latter type, the system may be solved by the methods used for symmetrical equations (§ 406).

3. Solve the equations	■a^ + y' = 53, Solution
	a;2 + y2 = 68.	0)
	ic — y = 5.	(2)
Squaring (2), x	2-2a;y + y2 = 25.	(3)
Subtracting (3) from (1),	2xy = 28.	(4)
Adding (4) and (1), x	2 + 2a;y + y2 = 81.	(5)
Extracting the square root.	x + y=±9.	(6)
From (6) and (2),	ic = 7 or - 2 ;
and	2/ = 2 or - 7.

QUADRATIC EQUATIONS 317

4. Solve the equations .

1-1=2.

\.x y Suggestion. — Square the second equation and proceed exactly as in

exercise 3. Solving at first for - and - instead of for x and y, the result is

X y

- = 7 or - 6, (1)

X

and i = 5 or - 7. (2)

y

Solving (1), x = \oT-\.

Solving (2), y = jor-|.

Note. — It is sometimes convenient to begin by solving for expressions ler than x and y, as Vxy, \/x+ y, etc.

Whether the equations are symmetrical or symmetrical ex- cept for the sign, it is often advantageous to substitute u-\-v for X and u — v for y.

2.

5. Solve the equations \ ^

\x-y

Solution

a^ + 1/4 = 82. (1)

x-y = 2. (2)

Assume x = u + v^ (8)

and y = u — V. (4)

Substituting these values in (1),

w* + 4 uH + 6 u'^v^ + 4 uv^ 4- ^

+ w* - 4m5o + 6mV^- 4m?j*J + d*=82, (5)

and in (2), 2v = 2. (6)

Dividing (5) by 2, # + 6 mV + v* = 41. • (7)

Dividing (6) by 2, v=\. (8)

Substituting 1 for v in (7) and solving, m = ± 2 or ± V— 10. (9)

P Hence, substituting (8) and (9) in (3) and (4), the corresponding Talues of X and y are found to be

J x = 3; -1; 1+V^^^; 1- V-10;

1^ = 1; - 3 ; - 1+ y/~~\(i ; - 1 - V^lO.

Note. — The given system of equations may ^e so\ye<i also by the method of exercise 3, § 407.

318 QUADRATIC EQUATIONS

Division of one equation by the other. — The reduction of equa- tions of higher degree to quadratics is often effected by divid ing one of the given equations by the other, member by member.

6. Solve the equations | "^^ "^ ^'^' + ^' " ^^' [a:r — xy-{-y'^ = 7.

	Solution
	X4 + a;2y2 + y4 = 91.		(1)
	x^-xy -{-y^ = 7.		(2)
Dividing (1) by (2),	aj2 + xy 4- y2 =: 13,		(3)
Subtracting (2) from (3),	2xy = 6;
whence,	xy = S.		(4)
Adding (4) and (3),	x^-\-2xy + y'^ = 16.		(5)
Subtracting (4) from (2),	x'^-2xy -\-y^ =4.		(6)
Extracting the square root of	(5), a; + y = 4or	-4.	(7)
Extracting the square root of	(6), x-y = 2oT	-2.	(8)

Solving these simultaneous equations in (7) and (8), fx=S; 1; -1; -3; U = l;3;-3;-l.

Note, — Since (7) and (8) have been derived independently, with the first value of x-\-y we associate each value of x — y in succession, and with the second value ot x-{- y each value of x— y in succession, in the same order. •

Consequently, there are four pairs of values of x and y.

7. Solve the equations

ra^ -2/3 = 26, 1

U-2/ = 2. \

Suggestion. — Dividing the first equation by the second,

x2 + xy 4- y2 = 13.

Therefore, solve the system,

fa;2 + xy + 2/2 = 13,

U-2/=2, instead of the given system.

I

Note. — It is sometimes the case that a root is removed when one equation is divided by the other, member by member. j

Observe that the given system may be solved by the method used in exercise 6, but the solution suggested here is briefer and simpler.

I

QUADRATIC EQUATIONS 319

Elimination of similar terms. — When the equations are quad- tic and each is homogeneous except for one term, if these xcepted terms are similar in the two equations, they may be

eliminated and the solution of the system be made to depend

on the case of § 409.

Some equations belonging to this class, namely, those that are homo- geneous except for the absolute term, have been treated in § 411.

8. Solve the equations ] ^V—^Vy

\2x'-xy + f = 2y.

Suggestion. — Multiplying the first equation by 4 and the second by 5, 4x2 + 8xy = 10y, (a)

id . 10a;2-5a;y + 5y2 = lOy. (6)

Subtracting (a) from (&),

6 aj2 — 13 X2/ + 5 ^2 _ 0^ a, homogeneous equation.

Therefore, solve the system,

r 6x2 -13a;y + 5^2 = 0, l2x2-xy + 2/2 = 2y,

stead of the given system, using the method of § 409.

Instead of eliminating terms below the second degree, as in exercise 8, in certain systems it is advantageous to eliminate similar terms of the second degree.

9. Solve the equations J ^2/ + ^ — »

I a;?/ 4-2/ = 32.

Solution

xy^x = 36. (1)

xy + y = 32. (2)

Subtracting (2) from (1), x — y = 3;

rhence, y = x — 3. (3)

Substituting (3) in (1), x(x-3)-\-x = 35,

x2-2x = 35. Solving, X = 7 or — 5. (4)

Substituting (4) in (3), 2/ = 4 or — 8.

320

QUADRATIC EQUATIONS

Solve, using the methods illustrated in exercises 1-9 ^ + mn = 2,

-I

11.

12.

13.

14.

mn-\-n^= —1.

pq= — 15.

= 130, 2.

\a-b =

of y

1.

15.

16.

17.

18.

jc^ + c^d^-f ^4 = 3, \c'-cd + d- = 3.

s3 = 54,

r — s^iiQ.

\2:^-xy-^f = ^y.

{xy + x = 32, \xy + y=.21.

fx* + 2/^ = 17, .X -y =1.

19 (2^-32/^ = 5, 130^-2^2^30.

415. All the solutions in §§ 403-414 are but illustrations of methods that are important because they are often applicable. The student is urged to use his ingenuity in devising other methods or modifications of these whenever the given system does not yield readily to the devices illustrated, or whenever a simpler solution would result.

EXERCISES

416. Solve the following miscellaneous systems of equations :

I

x + y = 3, xy^2.

5.

2.

3.

5 a;^ — 4 y^ = 44, .4if2-5/ = 19.

a-fa& + 28 = 0, 6 + a6 4-40 = 0.

3 xy = 8 X, xy -\-y^=z% X.

I2x2_

1 0^+1/2 = 61.

4 {^-xy \xy-y-

7.

3(? — xy = 48, 12.

x'-\-x^f^y':=21, x^ — xy -i-y^ = 7.

ar^ + / + a; + 2/ = 26, xy

(x' + f + x- \xv=- 12.

10.

11.

x^ + xy = —Qy [0^2/ + / = 15.

^ + f = 2^, a; + 2/=4.

a; + 2/ = 4.

x' + y' = ll, x — y=—S.

iCy -J^ X^ = 4:4:,

xy + f==-2S.

x'^4:X + Sy= -1, 2o^ + 5xy-\-2f- = 0.

^i + 1-1

r - — o'

m n Z

J— i=o.

mu 18

x^ — xy — 6, [a^ + / = 61.

xy — y^ = 12.

2a;-2/ = 2, 2a:2 + / = f.

a^ + a:2/ + / = 19,

ra^ + 3^2/ = / + 23,

milne's stand, alg. — 21

QUADRATIC EQUATIONS 321

t xy = 96 — .'^y,

22. ] ^

V — a;2/ = 8,

23.

xy^f = 12.

25.

a;(a; + y) = a;, y{x-y) = -

a^ + 3a.'2/-?/2 = 43,

l.y(^-2/) = -l.

2y = 10.

2g r2a^ + 3a;2/+2/' = 20, l5cc2 + 42/'^ = 41.

27.

28.

2xy-y'=12, 3xy + 6x' = 104..

x'^xy + y^^lbl, ic2 + 2/2 = 106.

l + a; = y,

30.

31.

a;<-?/* = 369, .x2-2/'=9.

r

32

33.

■I

x' + xy^y'' = U, ■Vxy + y=6.

4:x'-2xy + y^ = 13, 6x' + 6y^ = lSxy,

34_ fa^ + y^-3(a. + 2/) = 8, U + y + a;2/ = ll.

322

35.

36.

37.

38.

QUADRATIC EQUATIONS

[xy(y — x) = -12, (x-^y = 25,

ra^ + f = 225

'aP + f = 225y,

a^+2/^ = 3a^j + 5, 2/* = 2.

39.

40.

41.

42.

3a^ + 2a; + 2/ = 25,

ix? -Ixy + 12 f- = 0, xy-\-^y = 2x-{-21,

(x + y)(x^ + f) = %^,

(x — y)(x'^ — y'') = 5.

fa^ + y = x^y^ + 42, {xy==20.

43.

x + y + 2Vx + y = 24., ■y-{-SVx — y = 10.

r

la;

a^ + 2,24-6V^+7 = 55,

44 ra^ + 2^« + 6

45.

Jo^— 6 «?/ + 9/ + 2 a; — 6 y — 8 = 0, [a^ + 4.xy + 4:y^- Ax -Sy -21 = 0.

Suggestion. — The equations may be written in the quadratic form Thus |(x-3^)2 + 2(x-3i/)-8=0,

oi? — xy = a^-\-h'^'

46.

Solve j I for a; and y,

[xy — y^ = 2ab J

48. Solve K + ^ = ^^^^^ + ^^n for. and,.

Wy + xy' = 2a(a^-b'))

49. Solve I "" ^ I for a and t,

[v = at J

60. Solve J ^ I for V and «.

U = a« J

QUADRATIC EQUATIONS 323

Problems [Additional problems are given on page 482.]

417. 1. The sura of two numbers is 12, and their product is 32. What are the numbers ?

2. The difference between two numbers is 4 and their prod- uct is 77. Find the numbers.

3. The product of two numbers is 108 and their quotient is 1^. Find the numbers.

4. The sum of two numbers is 17, and the sum of their squares is 157. What are the numbers ?

5. The difference between two numbers is 1, and the dif- ference between their cubes is 91. What are the numbers ?

6. The sum of two numbers is 82, and the sum of their square roots is 10. What are the numbers ?

7. The perimeter of a rectangle is 20 inches and its area is 24 square inches. Find its dimensions.

8. It takes 52 rods of fence to inclose a rectangular garden containing 1 acre. How long and how wide is the garden ?

9. The area of a rectangular field is 3 acres and its length is 4 rods more than its width. Find its dimensions.

10. An Indian blanket has an area of 35 square feet. If its width were 1 foot less and its length 1 foot more, the former dimension would be ^ of the latter. Find its dimensions.

11. The product of two numbers is 18 less than 10 times the larger number and 8 less than 10 times the smaller number. Find the numbers.

12. If a two-digit number is multiplied by its units' digit, the result is 24. If the sum of the digits is added to the num- ber, the result is 15. What is the number ?

13. If 63 is subtracted from a two-digit number, its digits will be transposed; and if the number is multiplied by the sum of its digits, the product will be 729. What is the num- ber?

324 QUADRATIC EQUATIONS J

14. The difference between two numbers is 2 a and their product is b. Find the numbers.

15. A certain door mat has an area of 882 square inches. If its length had been 6 inches less and its width 5^ inches more, the mat would have been square. Find its dimensions.

16. The course for a 36-mile yacht race is the perimeter of a right triangle, one leg of which is 3 miles longer than the other. How long is each side of the course ?

17. I paid 75 ^ for ribbon. If it had cost 10 ^ less per yard, I should have received 2 yards more for the same money. How many yards did I buy and what was the price per yard ?

18. A man. expended $6.00 for canvas. Had it cost 4 cents less per yard, he would have received 5 yards more. How many yards did he buy, and at what price per yard ?

19. Mr. Fuller paid $ 2.25 for some Italian olive oil, and $ 2.00 for ^ gallon less of French olive oil, which cost $ .50 more per gallon. How much of each kind did he buy and at what price ?

20. In papering a room, 18 yards of border were required, while 40 yards of paper i yard wide were needed to cover the ceiling exactly. Find the length and the breadth of the room.

21. A grocer sold carrots for $4.40. If the number of bunches had been 4 less and the price per bunch 1 ^ more, he would have received the same amount. Find the price per bunch.

22. An Illinois farmer raised broom corn and pressed the 6120 pounds of brush into bales. If he had made each bale 20 pounds heavier, he would have had 1 bale less. How many bales did he press and what was the weight of each ?

23. A ship was loaded with 2000 tons of coal. If 50 tons more had been put on per hour, it would have taken 1 hour 20 minutes less time to load the whole amount. How long did jt take to load the coal ?

QUADRATIC EQUATIONS 325

24. A boy has a large blotter, 4 inches longer than it is wide, and 480 square inches in area. He wishes to cut away enough to leave a square 256 square inches in area. How many inches must he cut from the length and from the width ?

25. The total area of a window screen whose length is 4 inches greater than its width is 10 square feet. The area inside the frame is 8 square feet. Find the width of the frame.

26. The total area of a rug whose length is 3 feet more than its width is 108 square feet. The area of the rug exclusive of the border is 54 square feet. Find the width of the border.

27. A rectangular skating rink together with a platform around it 25 feet wide covered 37,500 square feet of ground. The area of the platform was J the area of the rink. What were the dimensions of the rink ?

28. After a mowing machine had made the circuit of a 7-acre rectangular hay field 11 times, cutting a swath 6 feet wide each time, 4 acres of grass were still standing. Find the dimensions of the field in rods.

29. Two men working together can complete a piece of work in 6| days. If it would take one man 3 days longer than the other to do the work alone, in how many days can each man do the work alone ?

30. My annual income from an investment is $ 60. If the principal were $ 500 less and the rate of interest 1 % more, my income would be the same. Find the principal and the rate.

31. A sum of money on interest for one year at a certain per cent amounted to $ 11,130. If the rate had been 1 % less and the principal $ 100 more, the amount would have been the same. Find the principal and the rate.

32. The fore wheel of a carriage makes 12 revolutions more than the hind wheel in going 240 yards. If the circumference of each wheel were 1 yard greater, the fore wheel would make 8 revolutions more than the hind wheel in going 240 yards. What is the circumference of each wheel ?

GRAPHIC SOLUTIONS

QUADRATIC EQUATIONS

418. Graphic solution of quadratic equations in jr.

Let it be required to solve graphically, a^ — 6flJ + 5 = 0.

To solve the equation graphically, we must first draw the^ graph of a^ — 6 a; + 5. To do this, let y = of — 6x -{- 5.

The graph of y = a^ — 6x -{- 5 will represent all the corre- sponding real values of x and of a^ — 6 a; + 5, and among them will be the values of x that make x^ — 6x -\- 5 equal to zero, that is, the roots of the equation a^ — 6a; + 5 = 0.

In substituting values of x, when the coefficient of a^ is + 1, as in this instance, it is convenient to take for the first value of a; a number equal to half the coefficient of x with its sign changed. Next, values of x differing from this value hy equal amounts may be taken.

Thus, fii-st substituting a; = 3, it is found that y = — 4, locating the point ^ = (3, — 4) . Next give values to x differing from 3 by equal amounts, as 2\ and 3^, 2 and 4, 1 and 6, 0 and 6. It will be found that y has the same value for a; = 3^ as for x = 2|, for a; = 4 as for a; = 2,

etc. The table below gives a record of the points and their coordinates :

■■

I

•o

^

lil

0

D

i,M P 1

^ 1

D

Q

^

y^^

J

w-

_^

X	y	To I NTS
3	-4	A
2i,3i	-33-	B, B>
2,4	-3	C, C
1,5	0	2>, D'
0,6	6	E,E'

GRAPHIC SOLUTIONS

327

1 Plotting the points A; B, B' ; (7, C" ; etc., whose coordinates are

I given in the preceding table, and drawing a smooth curve through them, we obtain the graph ofy = x^ — Qx-\-bBS shown in the figure.

It will be observed that : I When a; = 3, a^— 6a;+5=— 4, which is represented by the j negative ordinate PA.

When x = 2 and also when a; = 4, ar* — 6 a; -f- 5 = - 3, I which is represented by the equal negative ordinates MO I and JSfC.

When x = 0 and also when x = 6, a;^ — 6a? + 5 = 5, repre- sented by the equal positive ordinates OE and QE'. I Thus, it is seen that the ordinates change sign as the curve ! crosses the a;-axis.

t At D and at D', therefore, where the ordinates are equal to 0, I (he value of a^ — 6 a; + 5 is 0, and the abscissas are x = l and

x = 5. '\ Hence, the roots of the given equation are 1 and 5.

Note. — Half the coefficient of x with its sign changed, the number

'' first substituted for x, is half the sum of the roots, or their mean valuer

I when the coefficient of x^ is + 1. This will be shown in § 433. I

I The curve obtained by plotting the graph of a;^ — 6 a; -f- 5, or

I of any quadratic expression of the form ax^ +bx -\-c, is a

I parabola.

I 419. Let it be required to solve each of the equations I aj2_8ar + 14 = 0, (1)

W x'-Sx + 16=0, (2) mk a^- 8a; + 18 = 0. (3)

SF The graphs corresponding to equations (1), (2), and (3), found as in § 418, are marked I, II, and III, respectively.

The roots of (1) are seen to be OF = 2.6 and OW = 5.4, ap- proximately.

—

\\\

/

/

w

///

\\

7

0

A

K

i^

\'^

1

328

GRAPHIC SOLUTIONS

Since graph II has only one point, K, in common with the ic-axis, equation (2) appears to have, only one root, OK = 4. 1

But it will be observed that if graph I, which represents two un- equal real roots, 0 V and 0 W, were moved upward 2 units, it would coincide with graph II.

During this process the unequal roots of (1), OF and OW, would approach the value OK, which repre- sents the roots of (2). Consequently, the roots of (2) are regarded as two in number. They are real and equal, or coincident.

The movement of the graph of (1) upward the distance JK, or 2 units, corresponds to completing the square in (1) by adding 2 to each member. Since the roots of the resulting equation, ic^ — 8 ic + 16 = 2, differ from those of (2) or from the mean value OK = 4, by ± V2, or ± VjK, it is evident that the roots of (1) are represented graphically by

~~

—

—

\\

\\\

w

Y

y

\

^c

]/

L^

k^A

0

^

A

J .

/w

VnT>

3]

and

OK^ -JJK^ 4 + V2 = 5.414+, OK- VJK= 4 - \/2 = 2.586-

Since graph III has no point on the a>axis, there are no real values of x for which a^ — 8 ic -f 18 is equal to zero ; that is, (3) has no real roots. Consequently, the roots are imaginary.

If graph III were moved downward 2 units, it would coincide with graph II. If the square in (3) were completed by subtracting 2 from each member, the roots of the resulting equation, o;"^ — 8 x + 16 = — 2, would differ from the mean value by ± V— 2, or ± VLK.

Hence, it is evident that the roots of (3) are represented graphically by

OK-h VLK=^+ V-2,

and

OK-

^LK=4:- V-2.

The points /, K, and L, whose ordinates are the least alge- braically that any points in the respective graphs can have, are called minimum points.

GRAPHIC SOLUTIONS 329

420. When the coefficient of ic^ is + 1, it is evident from the preceding discussion that :

Principles. — 1. The roots of a quadratic in x are equal to the abscissa of the minimum point, plus or minus the square root of the ordinate with its sign changed.

2. If the minimum point lies on the x-axis, the roots are real and equal.

3. If the minimum point lies below the x-axis, the roots are real and unequal.

4. If the minimum point lies above the x-axis, the roots are imaginary.

EXERCISES

421. Solve graphically, giving real roots to the nearest tenth :

1. ic2_4a; + 3 = 0. 8. ar'=6a;-9.

2. a^-6a; + 7 = 0. 9. a? -\- 4.x -^-2 = 0.

3. ic^ - 4 a; = — 2. 10. a^ + 3 a; + 4 = 0.

4. ic2 = 2(a;+ 1). 11. a;2_5^_j_-^3^0.

5. a;2-f 2(i» + l) = 0. 12. r'-2a; + 6 = 0.

6. aj2-4aj-f6 = 0. 13. a^-4a;-l=0.

■ 7. a^ - 2 a; - 2 = 0. 14. ar^ + 7 a; + 14 = 0.

15. Solve graphically 4a;— 2a^ + l = 0.

Suggestion. — On dividing both members of the given equation by - 2, the coefficient of x^, the equation becomes a;2 _ 2 a: - i = 0.

The roots may be found by plotting the graph oiy — x"^ — Ix — \.

Solve graphically, giving real roots it, the nearest tenth :

16. 2ar^ + 8a; + 7 = 0. 18. 12 a; - 4a^ - 1 = 0.

17. 2 a;2 _ 12 a; + 15 = 0. 19. 11 -f- 8 a; + 2 a;^ = 0.

Note. — Another method of solving quadratic equations graphically is Iven in § 426.

330

GRAPHIC SOLUTIONS

422. Graphs of quadratic equations in x and /.

EXERCISES

1. Construct the graph of the equation a? -\- y^ = 25.

Solution. — Solving for y, y = ± V25 — x^.

Since any value numerically greater than 6 substituted for x will make the value of y imaginary, we substitute only values of x between and in- cluding — 5 and + 5. The corresponding values of y, or of ± \/25 — x"^. are recorded in the table below.

It will be observed that each value substituted for a;, except ± 5, gives two values of ?/, and thac values of x numerically equal give the same values of y ; thus, when x = 2, y = ± 4.6, and also when x = — 2, Ir = ± 4.6.

X

y

K*V

^

?r-<

S

s'

0 ± 1 ±2 ± 3 ±4 ± 5

± 5 ±4.9 ± 4.6 ±4 ±3 0

/

K

^

/

K

(

\

'

\

1

\

/

\

A

1

^

A

M

~

~

~

~

The values given in the table serve to locate twenty points of the graph of a;2 + y2 _ 25. Plotting these points and drawing a smooth curve through them, the graph is apparently a circle. It may be proved by geometry that this graph is a circle whose radius is 5. I

The graph of any equation of the form jr^ + /^ = r* is a circle whose radius is r and whose center is at the origin.

2. Construct the graph of the equation {x — 2)^+ {y — 3)^ =9.

The graph of any equation of the form (x — ay+(j — by = r^ is a circle whose radius is r and whose center is at the point (a, b).

3. Construct the graph of the equation y^ = 3x-\-9. Solution. — Solving for y, y = ± V3 a; + 9.

GRAPHIC SOLUTIONS

331

It will be observed that any value smallei* than — 3 substituted for x will make y imaginary ; consequently, no point of the graph lies to the left of re = — 3. Beginning with x = — 3, we substitute values for x and determine the corresponding values of ?/, as recorded in the table :

X	y
-3	0
-2	±1.7
-1	±2.4
0	±3
1	±3.5
2	±3.9
3	±4.2

^^^"

f%rf

-p^

A- J

7^

i

^

^

Plotting these points and drawing a smooth curve through them, the graph obtained is apparently & parabola (§418).

The graph of any equation of the form /^ = ax-^c is a parabola.

4. Construct the graph of the equation 9 a?^ + 25 y^ = 225.

Solution. — Solving for y, y = ± | V25 — x^.

Since any value numerically greater than 5 substituted for x will make the value of y imaginary, no point of the graph lies farther to the right or to the left of the origin than 6 units ; consequently, we substitute for x only values between and including — 5 and + 5.

Corresponding values of x and y are given in the table :

X	y
0	±3
±1	±2.9
±2	±2.7
±3	±2.4
±4	±1.8
±5	0

Plotting these twenty points and drawing a smooth curve through them. we have the graph of 9 ar^ + 25 y'^ = 225, which is called an

332

GRAPHIC SOLUTIONS

The graph of any equation of the form 6V -\- a^y^ — a^b^ is an ellipse.

5. Construct the graph of the equation 4 a?^ — 9 2/'' = 36.

Solution I

Solving for y, y = ± | y/x^ — 9.

Since any value numerically less than 3 substituted for x will make the value of y imaginary, no point of the graph lies between a; = + 3 and X = — 3 ; consequently, we substitute for x only ± 3 and values numer- ically greater than 3.

Corresponding values of x and y are given in the table :

'

X

y

^

k

J

1^4

rv

Sk

/

^

±3 ±4 ±5 ±6

0 ±1.8 ±2.7 ±3.5

5

N.

A"

j

'

'

\

/

p

\

i^r'

N

k

}

V

^s

K

_^

-i

Plotting these fourteen points, it is found that half of them are on one side of the 2/-axis and half on the other side, and since there are no points of the curve between a; = + 3 and x = — S, the graph has two separate branches, that is, it is discontinuous.

Drawing a smooth curve through each group of points, the two branches thus constructed constitute the graph of the equation ix^-9y^ = 36, which is an hyperbola.

The graph of any equation of the form 6V — o^y^=a^b^ is an hyperbola. An hyperbola has two branches and is called a discontinuous curve.

6. Construct the graph of the equation aJ2/ = 10.

Solution ^

Substituting values for x and solving for y, the corresponding values found are as given iu the table on the next page.

GRAPHIC SOLUTIONS

333

X

y

%

y

—

—

—

—

—

m

—

—

—

'

\

1 2 3 4 5 6 7 8 9 10

10 5 2^ 2 If 1^ li 1

-1 -2 -3 -4 — 5 -G -7 -8 -9 -10

-10 — 5 -31 -2| -2 -If -If -u -1

\

\

%

^

A,

^

C

/o

y.

s

H

Y^

Hi

M

H

»=^

n

^

?>w

f>.

K

^

'?\

'

_

_

_

_

_

_

_

1

_

_

_

_

_

_

Plotting these points and drawing a smooth curve through each group of points, the two branches of the curve found constitute the graph of the equation xy — 10, which is an hyperbola.

The graph of any equation of the form jr^ = c is an hyperbola.

Construct the graph of :

10. 9a;2_i6 2/2 = 144.

0^2 _j_ 2/2 = 9.

9. 9£c2 + 16 2/2 = 144.

11. xy = 12.

12. {x-lf + {y-2y = lQ.

423. Summary. — The types of equations and their respec- tive graphs, here summarized, will aid the student in plotting graphs, but he will meet other forms of equations that will have some of the same kinds of graphs, the varieties in equa- tions giving rise to varieties in form, size, or location of the graphs.

For example, § 422, exercises 1 and 2, are both equations of the circle, the first having its center at the origin and the second at the point (2, 3).

It is possible to determine many characteristics of the vari- ous graphs from their equations alone, but a discussion of this is beyond the province of algebra. In the study of graphs, therefore, the student will rely principally on plotting a sufficient number of points to determine their forni accurately.

334

GRAPHIC SOLUTIONS

The following types have been studied : I. ax + by = c (§ 267) II. x^^y'' = r^

III. {x-ay + {y-by = r'

IV. y = ax'^ + bx^c V. y^ = ax-\-c

VI. b'^x^ + ay =a''b^ VII. b'x'-ay=a'b^ VIII. xy = c

Straight line

Circle

Circle

Parabola

Parabola

Ellipse

Hyperbola

Hyperbola

424. Graphic solution of simultaneous equations involving quadratics.

The graphic method of solving simultaneous equations that involve quadratics is precisely the same as for simultaneous linear equations (§ 271). Construct the graph of each equation, both being referred to the same axes, and determine the coordi- nates of the points where the graphs intersect. If they do not intersect, interpret this fact.

The student should construct the following graphs for him- self. Roots are expected to the nearest tenth of a unit. To obtain this degree of accuracy, numerous points should be plotted and a scale of about i inch to 1 unit should be used.

EXERCISES

425. 1

. Solve graphically \

x'-^y' = 25,

y =

1.

Solution. — Constructing the graphs of these equations, we find the first, as in § 422, exercise 1, to be a circle ; and the second, as in § 267, a straight line.

The straight line intersects the circle in two points, (— 4, — 3) and (3, 4). Hence, there are two solutions,

X =— 4, y =—S; and a; = 3, y = 4.

Test, — The student may test the roots found graphically by performing the numerical solution.

c

. -^ z

^4^-^ ^^

UF ^

V / \

t 7 A

Z ±

W- T

v^ 3^

\t t-

^^^^ ^^

z ^^ ^^

P'

GRAPHIC SOLUTIONS

335

2. Solve graphically \

9a;2 + 25/ = 225,

2.

Solution. — On constructing the graphs (for the first, see exercise 4, §422), it is found that they intersect at the points a; = 3.7, y=2, a; = — 3.7, y = 2.

Since the graphs have these two points in common, and no others, their coordinates are the only values of x and y that satisfy both equations, and are the roots sought.

Observe that the pairs of values x = 3.7, y = 2 and x = — 3.7, y = 2, are real, and different, or unequal.

Note. — The roots are estimated to the nearest tenth ; their accuracy may be tested by performing the numerical solution.

9a;2 + 25/=225,

'

V

a4

V

= 3

.^

r*

■~~"

y

«2

•

\

/

I

\

4.

v^

^

— -

^

a*

opc

•m-

3. Solve graphically

[y

3.

4. Find the nature of the roots

•'(

Solution. — Imagine the straight line y = 2 in the figure for exercise 2 to move upward until it coincides with the line y = S. The real unequal roots represented by the coordinates of the points of intersection approach equality, and when the line becomes the tangent line ?/ = 3, they coincide. Hence, the given system of equations has two real equal roots, a; = 0, y = S, and x = 0, y =3.

9a;2 + 25/ = 225,

2/ = 4.

Solution. — Imagine the straight line y = 2 in the figure for exercise 2 to move upward until it coincides with the line y = i. The graphs will cease to have any points in common, showing that the given equations have no common real values of x and y.

It is shown by the numerical solution of the equations that there are two roots and that both are imaginary.

A system of ttvo independent simultaneous equations in x and y, one simple and the other quadratic, has two roots.

TJie roots are real and unequal if the graphs intersect, real and equal if the graphs are tangent to each other, and imaginary if the graphs have no points in common.

336

GRAPHIC SOLUTIONS

^

-

- .«3.^

.^>r

H

■N

V

y

r

9

\

L

^^v^"

->>

V

^^

XT''

/A'

(^

\

i i

^^

1

\

jf

^

■^.

.-

^

-

J

\

/

^

4'

•^

^

X

^

5. Solve graphically

la^ + / = 25.

Solution. — The graphs (see ex- ercises 5 and 1, § 422) show that both of the given equations are satis- fied by four different pairs of real values of x and y :

I a; = 4.5; 4.5; -4.5; -4.5;

12^ = 2.2; -2.2; -2.2; 2.2.

6. What would be the nature of the roots in exercise 5, if the second equation were x^-{-y^ = 9?

A system of two independent simultaneous quadratic equations in X and y has four roots.

An intersection of the graphs represents a real root, and a point of tangency, a pair of equal real roots. If there are less than four real roots, the other roots are imaginary.

Find by graphic methods, to the nearest tenth, the real roots of the following, and the number of imaginary roots, if there are any. Discuss the graphs and the roots.

7.

8.

9.

la?

10.

11.

12.

«2_ 92,2 = 36,

-32/ = l. Ux'-9y' = 36, l4a^4-92/' = 36. r9a^ + 162/2=r=144,

a^-h/ = 4, y-5. |a^_4/ = 4, Lic2 + 2/2 = 4.

x-y = 2, [xy = — l.

la;

13.

14.

15.

16.

17.

18.

Ax'-9y' = 36, 42/ = aj2-16.

|9ic2 + 16/ = 144, \Sx + 4:y = 12. (x^i.f=:9, [y=:a^-5x-\-6.

(a^ + f = 9,

[x = y'-\-5y + 6.

(y = x^-4,

U=:(3/ + l)(y + 4).

( y=zx^ — 5x-\-4:.

GRAPHIC SOLUTIONS 337

19

' y" -\- x^ -\- 3 y — 4: X + 3 = 0.

It is not possible to solve any two simultaneous equations in X and y, that involve quadratics, by quadratic metliods, but approximate values of the real roots may always be found by the graphic method.

Solve the following by both methods, if you can :

20.

[x^y + y = 2Q. ^ • {y' + x^ll.

* 426. Another graphic method of solving quadratic equations

in X (§ 418).

It has been seen that the real roots of simidtaneous equor tions are the coordinates of the points where their graphs inter- sect or are tangent to each other, and that when there is no point in common, the roots are imaginary.

In §§ 418-421, it was found that the real roots of a quadratic equation were the abscissas of the points where the graph of the quadratic expression crossed or touched the avaxis, and that when it had no point in common with the a^axis, the roots were imaginary.

In other words the solution of a quadratic equation in x was made to depend upon the solution of the simultaneous system,

' y = aoi? + 6a; 4- c, (a parabola)

2/ = 0, (a straight line)

the second being the equation of the »-axis.

In the following method, by substituting y for ni? in the given equation,

a^ + 6a; + c = 0,

ii^the equation is divided into the simultaneous system,

y, f ay + 6a; -h c = 0, (a straight line)

yy=zo(?. (a parabola)

The solution of this system for x gives the required roots of aa;^ -f 6a; + c = 0.

milne's stand, alg. — 22

I.

338

GRAPHIC SOLUTIONS

It will be observed that whether system I or II is used, the solution requires the construction of a parabola and a straight line, but the advantage of using II instead of I lies in the fact that the parabola y = oi^ is the same for all quadratic equations in x and when once constructed can be used for solv- ing any number of equations, while with I a different parabola must be constructed for each equation solved.

EXERCISES

427. 1. Solve graphically the equation a^ — 2a; — 8 = 0. Solution

Substituting y for x^, we have
.^ - 2 a; - 8 = 0. '
Consequently, the values of x
at satisfy the system,
	y-2x-S=0,
	[y = x^

are the same as those that satisfy the given equation.

Constructing the graph of y = aj2, v^e have the parabola shown in the figure.

Constructing the graph of y — 2a; — 8 = 0, a straight line, we find that it intersects the parabola at a; = — 2 and a; = 4.

Hence, the roots of the equation x2 _ 2 x — 8 = 0 are - 2 and 4.

Solve graphically, giving roots to the nearest tenth

^ h

T-± -r —

= -=+ = -t - -

- "f " 5r

I -S^i

4 -^ t

A -^ t

i 7 jt

i- ^ c

t 7 S

X ^ t

a7 ^

- - it- —

A-^ I

7 V ^

: zzzl~ S2: :== =

/

2. a^ + x-2 = 0.

3. a^-x-6 = 0.

4. x^-Sx-4: = 0.

5. a.-2-2a;-15 = 0.

6. ar' + 5a; + 14 = 0.

7. ar'-7a; + 18 = 0.

8. 2ar'-a; = 6.

9. 2ar^-a;-15 = 0.

10. 3ar^ + 5a;-28 = 0.

11. 6x^-7x-20^0.

12. Sa^ + 14:X-W = 0.

13. 15ar» + 2a;-20 = 0.

PROPERTIES OF QUADRATIC EQUATIONS

428. Nature of the roots.

In the following discussion the student should keep in mind the distinctions between rational and irrational, real and imaginary.

For example, 2 and \/4 are rational and also real; \/2 and VH are irrational^ but real; V— 2 and V— 5 are irrational and also imaginary.

429. Every quadratic equation may be reduced to the form

aic^ + 6«; + c = 0,

in which a is positive and & and c are positive or negative. Denote the roots by rj and rg. Then, § 390,

^ and 7-2 = 7.

* 2a 2a

An examination of the above values of r^ and rg will show that the nature of the roots, as real or imaginary, rational or 1 irrational, may be determined by observing whether V^^ — 4 ac is real or imaginary, rational or irrational. Hence,

Principles. — In any quadratic equation, ay? -|- 6a; + c = 0, when a, 6, and c represent real and rational numbers :

1. Ifb'^ — 4:acis positive, the roots are real and unequal.

2. If b^ — 4: OX) equals zero, the roots are real and equal.

3. Ifb^ — Aac IS negative, the roots are imaginary.

4. Ifb^—Aac is a perfect square or equals zero, the roots are r irational; otherwise, they are irrational,

430. The expression 6^ — 4 ac is called the discriminant of the quadratic equation aa? 4 6aj + c = 0.

339

340 PROPERTIES OF QUADRATIC EQUATIONS

431. If a is positive and h and c are positive or negative, the signs of the roots of ax^ -\- bx -{- c = 0, that is, the signs of

- 6 + V6' - 4 ac •, -b- V&'-4ac

'^ = ra ^''^ "^ = 2^r — '

may be determined from the signs of b and c.

Thus, if G is positive, — 6 is numerically greater than ± V^^ — 4 ac, whence both roots have the sign of — 6 ; if c is negative, — 6 is numerically less than ± V&^ — 4 ac, whence Vi is positive and ?'2 is negative. The root having the sign opposite to that of b is the greater numerically. Hence,

Principle. — If c is positive, both roots have the sign opposite to that ofb; ifc is negative, the roots have opposite signs, and the numerically greater root has the sign opposite to that of b.

Note. — If 6 =0, the roots have opposite signs. (See also § 378.)

EXERCISES ^

432. 1. What is the nature of the roots of a;^ _ 7 a; - 8 = 0 ?

Solution. — Since &2 _ 4 (35c = 49 + 32 = 81 = 9^, a positive number and a perfect square, by § 429, Prin. 1, the roots are real and unequal ; and by Prin. 4, rational.

Since c is negative, by § 431, Prin., the roots have opposite signs and, h being negative, the positive root is the greater numerically.

2. What is the nature of the roots of3a^ + 5aj + 3 = 0?

Solution. — Since 62 _ 4 ^c = 25 — 36 = — 11, a negative number, by § 429, Prin. 3, both roots are imaginary.

Find, without solving, the nature of the roots of :

3. a;2_5a,_75^0 g. 4^^^- 4a? + 1 = 0. ^. a? + 5x-\-& = 0. 9. 4a;2_f_ga._4^0.

5. »2 + 7a;-30=:0. 10. a?^ 4- a; + 2 = 0.

6. a^_3a; + 5 = 0. 11. 4 a;^ + 16 a; + 7 = 0.

7. a^ 4-3 a; -5 = 0. 12. 9a;2^12a;4-4 = 0.

PROPERTIES OF QUADRATIC EQUATIONS 341

13. For what values of m will the equation

have equal roots ? imaginary roots ?

P Solution

The roots will be equal, if the discriminant equals zero (§ 429, Prin. 2) ; that is, if (3 m)-2 - 4 . 2 . 2 = 0,

or, solving, if i^ = | or — |.

The roots will be imaginary, if the discriminant is negative (§ 429, Prin. 3) ; that is, if (3 m)2 - 4 • 2 • 2 is negative,

which will be true when m is numerically less than |.

14. For what values of m will 9a?^ — 5 ma? + 25 = 0 have equal roots ? real roots ? imaginary roots ?

15. For what values of a will the roots of the equation

4a^_2(a-3> + l = 0 be real and equal? real and unequal? imaginary?

16. Find the values of m for which the roots of the equation

4 «^ + mx 4- a? + 1 = 0 are equal. What are the corresponding values of a;?

17. For what values of n are the roots of the equation

3 a^ 4- 1 = n(4 x — 2x^ — 1) real and equal ?

18. For what value of a are the roots of the equation

ax^ - (a - l)a; + 1 = 0 numerically equal but opposite in sign ? Find the roots for this value of a.

19. For what values of d has ar^ + (2 - c«)a; = 3 d^ - 27 a zero root ? Find both roots for each of these values of d.

20. For what values of m will the roots of the equation

(m -f |)a^ - 2 (m + l)a; + 2 = 0 be equal?

21. Solve the simultaneous equations for x and y

r3a:2-4/ = 8, |5(a;-A:)-4 2/ = 0. For what values of k are the roots real ? imaginary ? equal ?

342 PROPERTIES OF QUADRATIC EQUATIO^^S

433. Relation of roots and coefficients.

Any quadratic equation, as ax^ -\-hx -\- c = 0, may be reduced, by dividing both members by the coefl&cient of ic^, to the forn? x^ -\- px -\- q = 0, whose roots by actual solution are found to be

Adding the roots, ri + rg = — = — P-

Multiplying the roots, r^^ = -^ -~ \r — 22 = g.

Hence, we have the following:

Principle. — The sum of the roots of a quadratic equation having the form x^ -\- px -{• q =^ 0 is equal to the coefficient of x with its sign changed, and their product is equal to the absolute term.

434. Formation of quadratic equations.

Substituting — (rj + r.^) for p, and rirg for q (§ 433) in the equation a^ + pa; + g = 0, we have

^^ — (n + ^2)^ + n^2 = 0.

Expanding, a^ — riX— r^ + r^r^ = 0.

Factoring, (x — ri)(x — rg) = 0.

Hence, to form a quadratic equation whose roots are given :

Subtract each root from x and place the product of the remain' ders equal to zero. J

EXERCISES

435. 1. Form an equation whose roots are — 5 and 2. Solution. (« + 5) (a - 2) = 0, or x^ + 3 a; — 10 = 0.

Or, since the sum of the roots with their signs changed is +5 — 2, or 3, and the product of the roots is — 10, (§ 433) the equation is x2 + 3 x - 10 = 0

PROPERTIES OF QUADRATIC EQUATIONS 343

Form the equation whose roots are :

2.	6,4.		8.	a, —3 a.	14.	34-V2, 3-V2.
3.	5,-3.		9.	a + 2, a-2.	15.	2-V5, 2+V5.
4.	3, -i.		10.	6 + 1, 6 - 1.	16.	2± V3.
5.	hi-		11.	a -{-by a — b.	17	-i(3±V6).
6.	-2, -	h	12.	Va — V6, V6.	18.	K-i±V2).
7,	-i, -	f.	13.	i(a±V6).	19.	a(2±2V5).

20. What is the sum of the roots of 2 mV — (5m— l)a;=6? For what values of m is the sum equal to 2 ?

21. When one of the roots of ax^ -^ bx -{- c = 0 is twice the other, what is the relation of b^ to a and c?

I

Solution Writing ax^ + &a; + c = 0 in the form

+ ^x + ^ = 0, (1)

a a

and representing the roots by r and 2 r, we have fl| r + 2r=3r = -^, (2>

and r'2r = 2r^ = -. (3)

a

On substituting the value of r obtained from (2) in (3) and reducing,

22. Obtain an equation expressing the condition that one t of 4a:2 — 3 aa; + 6 = 3 is twice the other.

23. Find the condition that one root of ax^ -\-bx-\-c = 0 shall be greater than the other by 3.

24. When one root of the general quadratic ax^ -\-bx-\-c = 0 is the reciprocal of the other, what is the relation between a and c?

25. If the roots of aa^ -}- bx -{- c = 0 are rj and ra, write an equation whose roots are — Vi and — r^

i

344 PROPERTIES OF QUADRATIC EQUATIONS

1

26. Obtain the sum of the squares of the roots of

2a^ — 12x-\-3 = 0, without solving the equation.

Solution

Sum of roots = ri + r^ = 6. (1)

Product of roots = riVo = f . (2)

Squaring (1), rt^ + ra^ + 2 nra = 36. (3)

(2) X 2, • 2 rira = 3. (4)

(3) -(4), ri2 + r22=33.

Find, without solving the equation:

27. The sum of the squares of the roots of «^ — 5 cc — 6 = 0.

28. The sum of the cubes of the roots of 2 o;^ — 3 a; + 1 = 0.

29. The difference between the roots of 12 a:^ -)- a; — 1 = 0.

30. The square root of the sum of the squares of the roots of a^2_7a,^_i2 = o.

31. The sum of the reciprocals of the roots of ax^-\-bx-\-c=0. Suggestion. L^±= ?!l±J!2 .

32. The difference between the reciprocals of the roots of 8fl^-10a; + 3 = 0.

436. The number of roots of a quadratic equation.

It has been seen (§ 433) that any quadratic equation may- be reduced to the form x^ -\-px-\- q=:0, which has two roots, as ?'i and rg. To show that the equation cannot have more than two roots, write it in the form given in § 434, namely,

(a;-ri)(a;-?-2) = 0. (1)

If the equation has a third root, suppose it is rg.

Substituting r^ for x in (1), we have

which is impossible, if i\ differs from both rj and rg. Hence,

I

Principle. — A quadratic equation has tivo and only tivo roots.

PROPERTIES OF QUADRATIC EQUATIONS 345

437. Factoring by completing the square.

The method of factoring is useful in solving quadratic equa- tions when the factors are rational and readily seen. In more difficult cases we complete the square. This more powerful method is useful also in factoring quadratic expressions the factors of which are irrational or otherwise diffiicult to obtain.

EXERCISES

438. 1. Factor 2 a^ + 5 a; -3.

Solution Let 2a;2 + 5a;_3 = o.

Dividing by 2, etc., x2 + fx = |.

Completing the square, a;^ + f a; -f f| = f|. Solving, X = ^ or - 3.

Forming an equation having these roots, § 434,

(x-|)(x + 3)=0. Multiplying by 2 because we divided by 2,

(2a;- l)(a; + 3) = 2x2 + 5a;-3 = 0. Hence, the factors of 2 x^ + 5 x — 3 are 2 x — 1 and x + 3.

Factor :

2. 4.x' - 4.x -S. 6. 7a^4-13£c-2.

3. 5a^ + 3a;-2. 7. 24. x" - 10 x - 25.

4. Sx' + Ux-5. 8. 10a^ + 21x-10.

5. Sa^-Ux-\-3. 9. 15a^-5.5a;-l. 10. Factor a^ + 2 » - 4.

Solution Let x2 +. 2 X - 4 = 0.

Completing the square, x^ + 2 x + 1 = 5.

Solving, x = -\ + VloT -1 - V6

Hence, § 434, (x + 1 - V^)(x + 1 + \/"5) = a--^ + 2 x - 4 = 0. That is, the factors of x^ + 2 x - 4 are x + 1 - \/5 and x + 1 + V5 .

346 PROPERTIES OF QUADRATIC EQUATIONS

Factor :

11. iB2 + 4a;-6. 14. x^-^x + l.

12. y^-6y-\-3. 15. a^ + Sa-S.

13. z^-5z-l. 16. f + St-^7,

17. Factor 2 - 3 cc - 2 ic^^

SoGGESTiON. — Since 2 — 8 x - 2 x^ = - 2(x2 + f x - 1), factor x2 + f x — 1, in which the coefficient of x^ is +1, and multiply the result by — 2,

18. 2a;2 + 2a;-l. 21. 9a'-12a-\-5.

19. 9x'-4:X + l. 22. 16v(l-v)-9.

20. 24.x-16x'-S. 23. 16(3 + n)+3n2.

24. Factor 100 ic^ ^ ^q ^^ _ -1^;^9 ^3

Suggestion. — The coefficient of x^ being a perfect square, complete the square directly ; do not divide by 100.

25. 4 62 _ 48 6 4- 143. 28. 16p(p + l) -1517.

26. 9?'2_12r4-437. 29. 25e'-2h(o e-2h).

27. 4a24.i2a-135. 30. 3 ^(4 A;-3 ^) -7 Ar'.

31. Factor a;^ + 4a^ + 8a;2 + 8a;-5.

Solution '■

Let x* + 4x3 + 8x2 + 8x-5 = 0.

Completing the square,

(x* + 4 x3 + 4 x2) + 4(x2 + 2 x) + 4 = 9. Extracting the square root, x2 + 2 x + 2 = 3 or — 3.

.'. x* + 4 x8 + 8 x2 + 8 X - 6 = (x2 + 2 X + 2 - 3)(x2 + 2 X + 2 + 3) = (x2 + 2 X - 1) (x2 + 2 X + 5) .

Factor the following polynomials :

32. x*-i-6a^-^llx^-{-6x-S.

33. x^-\-2xf^-]-5x^-\-Sa^-\-Sa^ + Sx + S.

34. a^-4a:* + 6a;^ + 6a^-19ar^ + 10a; + 9.

35. 4a;« + 12a^ + 25a;^ + 40a3 + 40iB2 + 32aj + 15.

FKOPERTIES OF QUADRATIC EQUATIONS 347

36. Eesolve a;* + 1 into factors of the second degree. Solution

= («2 + 1)2 _ (a; ^)2

= (x^ + X V2 + l)(x^ - X V2 + 1).

Note. — Each of these quadratic factors may be resolved into two factors of the first degree by completing the square. The factors are: (x+i V2+^v/i:2), (a; + iV2_iV^2), (a; - i V 2 + i V^^r2), and(x- |\/2-i\AZr2).

Resolve into quadratic factors :

37. i»* + 16. 39. a;* + 2a2a52-f.4a*.

38. a*-\-b*. 40. v* - 4 n V - 2 w*.

439. Values of a quadratic expression.

An expression that has different values corresponding to different values of x is called a function of x.

a;2_2a; is a function of x, for when a; = 1, 2, 3, '■•,x'^—2x = — 1, 0,3, ••■^

In the following discussions only real values of x are con- sidered.

EXERCISES

440. 1. What values has the function a^ — 2x-~S corre- sponding to very large positive or negative values of ar?

Discussion. — When x is very large numerically and either positive or negative, the value of x^ — 2 a: — 3 is approximately equal to that of its largest term, x^. Thus when x = -± 100, a:2 - 2 a; - 3 = 10,000, approxi- mately ; when x= ± 1000, a;2 _ 2 a; — 3 = 1,000,000, approximately.

Since x^ is always positive, whether x is positive or negative, for very large numerical values of x, a:^ — 2 a; — 3 is very large numerically, and positive ; and by making x large enough we can make x^ — 2 x — 3 greater than any number that may be assigned, however great.

A number that may become greater than any assignable number is called an infinite number.

348 PROPERTIES OF QUADRATIC EQUATIONS

2. Interpret the conclusion of exercise 1 graphically.

1

I

I

/

\

\

?/

\

')l

\

^

\

\

\

[

il

\

/

\

f

\

J

'

—J

J

Discussion. — Draw the graph of x^ — 2 05 — 8, plotting values of x as abscis- sas and values ot x^ — 2x— S as ordi- nates (§ 418).

In the discussion of exercise 1, it is seen that when x = — oo , and also when x = + c»,a;2— 2x — 3 = + oo ; that is, when X increases without limit, either in the negative direction, or in the positive direc- tion along the x-axis, x^ — 2 x — 3, repre- sented by ordinates to the curve, increases without limit in the positive direction.

Referring to the graph of a^ — 2 x — 3 and observing the form of the function itself, a brief discussion for real values of X may be given as follows :

(a) As X increases continuously from — ooto -f-1, a? — 2x—3 decreases continuously from -f oo to its minimum value, — 4, crossing the a;-axis at ic = — 1, which is therefore a root of the equation aj^ — 2ic — 3 = 0.

(h) As X increases continuously from -flto -\- cc jX^ — 2x—3 increases continuously from its minimum value, — 4, to + oo , crossing the i»-axis at a; = 3, which is therefore the other root of the equation a;^ — 2 ic — 3 = 0.

(c) The function is positive for all values of x outside the limits a; = — 1 and a; = 3, and negative for all values of x within these limits.

Wben the coefficient of x^ is +1, the abscissa of the minimum point is half the coefficient of x with its sign changed (§ 418).

In a similar way discuss the following functions :

Z. x^ — bx + Q. 6. ar^ + 5a; + 4.

4. o^-2x^S. 7. aj2-9.

5. a^-f2a;-15. 8. ar^ + a;-hl.

PROPERTIES OF QUADRATIC EQUATIONS

19. Find the maximum value of 3 + 2 a; — a?^. First Solution Since 3+2a;-a;2=- (x^- 2 x -3), and aj'2 — 2 X — 3 has a minimum value at x = 1 (exercise 2), the given function has a maximum value at x — 1 .

When X = 1, 3 + 2 x — x^ = 4, the wiax- imum value.

Second Solution

349

Let 3 + 2 X

0 or a

^Bl

mi

Hi

I Solving f or X, X = 1 ± ^4 — y. Since x must be real^ 4 — y positive number. If4-y = 0, y = 4.

If 4 — y = a positive number, y is less than 4. Therefore, 4 is the maximum value of the function.

10. Complete the discussion of the values of o-{-2x—aP, Discuss the values of the following :

11. x-\-6-x^. 14. 2a^4-5.'?;-3.

12. 5-4.x-x^. 15. 2a:2^3^_^2.

13. 4ar^-16a; + 15. 16. 4a;-6-a:2

17. For what values of a; is a^ — 5a; + 6 positive?

Solution

a;2_5a.^6 = (x-2)(x-3).

x2 — 5 X 4- 6 is positive when both factors are positive or when both are negative ; that is, when x is less than 2 or greater than 3, these values being the roots of the equation x^ — 5x + 6 = 0.

18. For what values of x is ar^ — 3 a; — 28 positive ? nega- tive ?

19. Show that a^ — 6 a; + 12 is positive for all real values of X. What is the nature of the roots of a^ — 6a;H- 12 = 0?

20. Show fchat a; — a;^ — 1 is negative for all real values of x.

21. What is the condition that ax^-\-bx-\-c shall have the same sign for all real values of a;?

i

GENERAL REVIEW

441. 1. Define power; root; like terms; transposition; simultaneous equations; surd.

2. Distinguish between known and unknown numbers.

3. Why is the sign of multiplication usually omitted be- tween letters, and never omitted between figures ?

4. How is multiplication like addition ? division like sub- traction ? What two meanings has the minus sign in algebra ?

5. When x, -^, or both occur in connection with +, — , or both in an expression, what is the sequence of operations ?

6. State the law of exponents for multiplication; for divi- sion.

7. When is cc" — 2/" divisible by both x -^y and x — y?

8. When is a trinomial a perfect square ? When is a frac- tion in its lowest terms ? What are similar fractions ?

9. What operation is indicated by the radical sign? In what other way may this operation be indicated ?

10. When is an expression both integral and rational ? When

are expressions said to be prime to each other ?

I

11. By what principle may cancellation be used in reducing fractions to lowest terms ?

12. During 12 hours of a certain day, the following tempera- tures were recorded: -9°, -8°, -8°, -9°, -9°, -9°, -8°, +12°, +25°, +40°, +20°, +16°. Find the average temperature for the 12 hours.

860

GENERAL REVIEW 351

13. Define the terms conditional equation; identical equation.

14. Explain the meaning of a negative integral exponent j of a fractional exponent.

15. Define evolution ; radical; entire surd; binomial surd; similar surds.

16. Express the following without parentheses :

{a^x^y, - [- {ayY, {ay, {ay.

17. What is meant by the order of a surd ? Illustrate your answer by giving surds of different orders.

18. Tell how to rationalize a binomial quadratic surd.

19. What powers of V — 1 are real ? imaginary ?

20. What roots should be associated when the roots of a system of equations are given thus : ic=±2, 2/=q:3?

21. Illustrate how a root may be introduced in the solution of an equation ; how a root may be removed.

22. Why is it specially important to test the values of the unknown number found in the solution of radical equations ?

23. Upon what axiom is clearing equations of fractions based ? What precautions should be taken to prevent intro- ducing roots? If roots are introduced, how may they be detected ?

24. Define symmetrical equation ; quadratic surd ; coordinate axes ; imaginary number ; axiom ; coefficient ; homogeneous polynomial ; elimination.

25. Explain how, in the solution of problems, negative roots of quadratic equations, while mathematically correct, are often inadmissible.

26. Define negative number, subtraction, and multiplication, and show, from your definition, how the following rules may be deduced :

(1) "Change the sign of the subtrahend and proceed as in I addition ; "

(2) " Give the product the positive or the negative sign ac- cording as the two factors have like or unlike signs."

362 GENERAL REVIEW

27. What is a pure quadratic equation ? a complete quad- ratic equation ? Give the general form of each.

28. State two methods of completing the square in the solu- tion of affected quadratic equations.

29. What is the relation between the factors of an expression and the roots of the equation that may be formed by putting the expression equal to 0 ?

30. Outline the method of solving quadratic equations by factoring.

31. Write the roots of the quadratic equation ax^-i-hx-\-c = 0. Write the discriminant of the equation. What relation between the coefficients indicates that the roots are imaginary? reciprocals of each other ?

32. What is the advantage of letting ot^ = y in the graphic solution of quadratic equations of the form ax^ -i-bx-\-c = 0?

33. How does the graph of a quadratic equation show the fact, if the roots are real and equal ? real and unequal ? imaginary ?

34. Prove that a quadratic equation has two and only two roots.

35. Tell how to form a quadratic equation when its roots are given. Form the equation whose roots are | and ^.

36. What is the meaning of " function of a; " ? " infinite number " ?

37. Tell how the signs of the roots of a quadratic equation may be determined without solving the equation.

38. Derive the value of the sum of the roots of the equation or^ +^0? + g =0 ; the value of the product of the roots.

39. In clearing a fractional equation of its denominators, why should we multiply by their lowest common multiple?

Illustrate by showing what happens when the equation 2x 10 J 7

x — 1 x^—1 x + 1 is multiplied by the product of all the denominators.

GENERAL REVIEW 353

EXERCISES

442. 1. Add X Vy 4- y Vx -|- Vxy, x^y^ — -yfofy — '\/xy^, Va^y — -yjx]}- — Va;?/, and y -\/x — x V4 2/ — V9 xy.

2. What number must be subtracted from a — 6 to give

3. Simplify a- J&-a-[a-6-(2a + 2>) + (2a-6)-a]-6|.

4. Multiply ic V^ + a? V2/ + :y Vic 4- 2/ Vj/ by -yjx^^y.

a 0+5 a 0+6

5. Multiply 2 a; ^"-5 2/^ by 2x^'^-^^y~^,

6. Expand (a;'* — ?/''J(a;'*4-2/'')(a;2« + 2^").

7. Divide a;^ — 2/* by a; — 2/ by inspection. Test.

8. Divide a^ — 3 a;^ — 20 by a; — 2, by detached coefficients.

9. Show by the factor theorem that a;^ — 6^ is divisible by

10. Divide (a + 6) + a; by (a + h)^ + ^.

11. Factor 9 a^- 12a; + 4; 9a^ + 9a; + 2; y?-Zx-\-2', a^ + 1.

12. Show by the factor theorem that x — a is a factor of + 3 ax""-^ — 4 a".

13. Separate dP- — 1 into six rational factors.

14. Factor 4 {ad + hcf - {a^ -^ - & ^ d-f.

15. Find the H. C. F. of 3 a.-^ - a; - 2 and 6 a^ + a; - 2.

16. Find the H. C. F. of 2a)^- 7a^ + 4a:2 _^ 7^_ 5^ 2a;* + a^_4a^+7a;-15, and2x4 + a^-a;-12.

17. Find the L. C. M. of 4 a'hx, 6 ab^, and 2 axy.

18. Find the L. C. M. of x^ — y^,x-\- y, and xy — y\

q2 ^2 g2 2 6c

19. Reduce — to its lowest terms.

a2_62^c2 + 2ac

Milne's stand alg. — 23

354 Gti^NERAL REVIEW

20. Simplify-^ ^+ ^

X + 1 1 — X x^ — 1

21. Simplify ^ + -^ + ""-^ +1.^. ^ ^ 2x-2y^2x-\-2y^y'-x'

' 22. Simplify ^ 1.1

(a-6)(6-c) (c-6)(c-a) (c-a)(a-&)

23. Simplify 84. Simplify .	\ %+^ 1 + i 1	1 1 x-hl 1	•
a;	a; —	1 • X

1_1 ^-1

25. Raise a—b to the seventh power.

26. Expand (2 a 4-3 6/; (V^ + ^)«j (_1_V33)8.

27. Find the sixth root of 4826809.

28. Reduce V| to its simplest form.

29. Reduce V25 a* to its simplest form.

30. Find the value of — to 3 decimal places.

V2

31. Multiply 2 + V8 by 1-V2; 2-f-V^^byl-V^

32. Simplify ^^±^^.

V6-F2

33. Show that (axy = 1.

34. Show that 00?-'^=-.

x^

35. Show that a;^ = A/^; also that a* = (^«)2.

36. Find the value of 125*; of /"l^V^ when a; = .5.

GENERAL REVIEW

355

Solve the following equations for x :

37.

38.

-h

a-h6

1

X

a

+

X

1

X

&

a

+

a + 6

1

c

X

=0.

39. ma?^ — wx = mn.

42. Va;-9 = V»-l.

40. .^+l = 5i!!.

2 2

+

.J-g+x) = 20,

43. a;2 + Var^ + 16 = 14.

44. r^

45. (1 + a^)'* + (1 - aj)« = 242.

46. a; + a^ + (1 + a? + ar^^ = 55. 1 + « _^ 1 -l-a?

47.

= a —

1 + a;+Vl + ar^ Solve for «, 2/? and z :

48.

49.

50.

51.

52.

^^ + ^ = 10,

X y

X y

10.

2a; + 32/ + 2! = 9, a; + 22/ + 32; = 13, .3 a; + 2/ + 2^ = 11.

aa;+2/ + 2; = 2(a + l), a;4-a2/H-!^ = 3a + l, X -\- y -\- az — o? -\- ^.

a^ + a^ = 24, 2/^ 4- a^2/ = 12.

'0^ + 33^2/ = 7, a;2/ 4- 4 2/^'= 18.

53.

54.

55.

l-aj+Vl + x"

^ + x = 2Q-y'-y, xy = 8

(x^ + i [xy =

'V^=12,

X -{- y — Va; + y = 20. xy -xy^ = -6, . a; — xy^ = 9.

58

59.

{

(2x^ + 2 f = 9 xy, \x-\-y = 3.

xy=x-\- y, x' + f = 8. a^2/^ — 4 .^2/ = 5, + 4 2/2 = 29.

1 1

x^ -\-y^

x^ + y =

= 4, 16.

356 GENERAL REVIEW

Problems

[Additional problems are given on page 483.]

443. 1. The sum of two numbers is 72 and their quotient is 8. Find the numbers.

2. A man who had no room in his stable for 8 of his horses, built an addition, i the size of the stable. He then had room for 8 horses more than he had. How many horses had he ?

3. A woman on being asked how much she paid for eggs, replied, " Two dozen cost as many cents as I can buy eggs for 96 cents." What was the price per dozen ?

4. How far down a river whose current runs 3 miles an hour can a steamboat go and return in 8 hours, if its rate of sailing in still water is 12 miles an hour ?

5. In a mass of copper, lead, and tin, the copper weighed 5 pounds less than i of the whole, and the lead and tin each 5 pounds more than i of the remainder. Find the weight of each.

6. A person who can walk n miles an hour has a hours at his disposal. How far can he ride in a coach that travels m miles an hour and return on foot within the allotted time ?

7. There is a number whose three digits are the same. If 7 times the sum of the digits is subtracted from the number, the remainder is 180. What is the number ?

8. The greater of two numbers divided by the less gives a quotient of 7 and a remainder of 4 ; the less divided by the greater gives -^-^. Find the numbers.

9. The greater of two numbers divided by the less gives a quotient of r and a remainder of s; the less divided by the greater gives t Find the numbers.

10. A man received $ 2.50 per day for every day he worked, and forfeited $ 1.50 for every day he was idle. If he worked 3 times as many days as he was idle and received $24, how many days did he work ? •

1

GENERAL REVIEW 357

11. One machine makes 60 revolutions per minute more than another and in 5 minutes the former makes as many revolu- tions as the latter does in 8 minutes. Find the rate of each.

12. The value of a fraction is |. If 4 is subtracted from its numerator and added to its denominator, the value of the resulting fraction is f . Find the fraction.

13. A woman has 13 square feet to add to the area of the rug she is weaving. She therefore increases the length i and the width \, which makes the perimeter 4 feet greater. Find the dimensions of the finished rug.

14. A and B can do a piece of work in m days, B and C in n days, A and C in ^ days. In what time can all together do it ? How long will it take each alone to do it ?

15. It took a number of men as many days to pave a side- walk as there were men. Had there been 3 men more, the work would have been done in 4 days. How many men were there?

16. By lowering the selling price of apples 2 cents a dozen, a man finds that he can sell 12 more than he used to sell for 60 cents. At what price per dozen did he sell them at first ?

17. If the distance traveled by a train in 63 hours had been 44 miles less and its rate per hour had been 4| miles more, the trip would have taken 50 hours. Find the total run.

18. A merchant bought two lots of tea, paying for both $ 34. One lot was 20 pounds heavier than the other, and the number of cents paid per pound was in each case equal to the number of pounds bought. How many pounds of each did he buy ?

19. Two passengers together have 400 pounds of baggage and are charged, for the excess above the weight allowed free, 40 cents and 60 cents, respectively. If the baggage had be- longed to one of them, he woald have been charged $1.50. How much baggage is one passenger allowed without charge ?

358

GENERAL REVIEW

20. Find two numbers such that their sum, their product, and the difference of their squares are all equal.

21. The perimeter of a rectangle is 8 m and its area is 2m\, Find its dimensions. I

22. The volumes of two cubes differ by 296 cubic inches and their edges differ by 2 inches. Find the edge of each. J

23. A tank contains 400 cubic feet. Its height exceeds its width by 1 foot and its length is 5 times its width. Find its dimensions.

24. It takes A and B f of a day longer to tin and paint a roof than it does C and T>, and the latter can do 50 square feet more a day than the former. If the roof contains 900 square feet, how much can A and B do in a day ? C and D ?

25. The velocity acquired or lost by a body acted upon by gravity is given by the formula v = gt (take ^ = 32.16). A" bullet is fired vertically upward with an initial velocity of 2010 feet per second. Find in how many seconds it will return to the earth (neglecting the friction of the air).

Using the formula s=^gt% find how high the bullet will rise.

26. The load on a wall column for an office building is 360,000 pounds, including the weight of the column itself, and

Interioi Column

Girder acting as a lever

Wall_ Column* ■ 4'

I

	F	fhhl
""t			—	-\r.
f^^^:	tlis	^^^^	"^m^

is balanced, as shown in the figure, by a part of the load on an interior column. Neglecting the weight of the girder, find the load on the fulcrum.

GENERAL REVIEW

359

444. The following are from recent college entrance exami- ^ nation papers : [For other exercises see page 485.]

1. Determine graphically the roots of 4 aj + 5 ?/ = 24, iSx— 2y = — 5. Give the construction in full.

2. Solve

+

1

8

= 0.

1

2x-i-S ' x-5 2x^- 7x-W

3. Solve for x and y, (x — yy = c\ (y — d)(x — b)= 0.

4. Find x from the equations,

)x^ -{■ xy -\- z = 2, x-\-2y-\-z = 3, x — y-^z=:0.

Suggestion. — From the second equation subtract the third.

5 Solve I ^(^ + yy -{x + y){x-2y) = 70, \2{x^y)-3{x-2y) = 2.

6. Solve for a;, y, and z,

'x-\-y = xy, 2x-{- 2z = xz, _Sz -{- Sy = yz.

Suggestion. — Find y in terms of x, and z in terms of x ; substitute liese results in the remaining equation.

7. Solve x — y— ^/x — y = 2, x^ — f = 2044.

f 8. Solve for a; and i/ I ^'^ + ''^^ -(a-c)y = 2 ab, I' l (a + b)x —(a — b)y = 2 ac.

9. Factor 4 a;^ -f- 1 ; 27 a;2.-|_ 3 a; - 2 ; 4:X*-{-y^-5 xy.

10. Resolve into prime factors :

8(a -ly _ (1 + a); a'- a^¥ -b^-l-, S x-^ -\- 7 x'i - 6.

11. Solve the equation 4 a;^ -f ma; + 5 = 0. For what values of m are the roots imaginary ?

12. How much water must be added to 80 pounds of a 5 % salt solution to obtain a 4 % solution ?

360 GENERAL REVIEW

13. Construct the graph of the function x^ — 2x-{-l.

14. Under what conditions will the roots ot ax^ -\- bx -\- c = 0 be positive ? negative ? one positive and the other negative ?

15. Find to four terms the square root of ic^ — 3 a; + 1.

16. Find the square root of

4 612 63 + ^ 6« 3 9 *

17. Solve a^ + 7x-3 = V2a^ + 14a; + 2.

18. Simplify ^^(^ ~ ^) + "^^(^ ~ "^^ + "^^("^ ~ ^ .

a^ + 6c — ac — a6

19. Solve

f22c_5^_/3^_4^\ _2 3 12 V 2 3 y 3'

« + !/ 5

20. Solve

Vx — Vi/ = 2, (V^- V^)V^=30.

21. Show by the factor theorem that a" + 6** is exactly divisible by a + 6 for all positive odd integral values of n.

22. The area of the floor of a room is 120 square feet ; of one end wall, 80 square feet ; and of one side wall, 96 square feet. Find the dimensions of the room.

23. Show that V3 is greater than V6.

■x' + 2/' = 4,

24. Draw the graphs of the two equations

' 5 ic + 4 2/ = 20,

and tell the algebraic meaning of the fact that the two graphs

do not intersect.

25. A rectangular piece of tin is 4 inches longer than it is wide. An open box containing 840 cubic inches is made from it by cutting a ()-inch square from eacli corner and turning up the ends and sides. What are the dimensions of the box ?

i

INEQUALITIES

445. Any problem thus far presented has been such that bs conditions could be stated by means of one or more equa-

ms. In some problems and exercises, however, the condi-

itions are such as to lead to a statement that one number is

^eater or less than another. It is the purpose of this chapter

discuss such statements, for they often yield all the data

lecessary to the required solution.

446. One number is said to be greater than another when the jmainder obtained by subtracting the second from the first is

've, and to be less than another when the remainder )btained by subtracting the second from the first is negative.

Ita — b is a positive number, a is greater than b ; but if a — 6 is a egative number, a is less than b.

Any negative number is regarded as less than 0 ; and, of two legative numbers, that more remote from 0 is the less. Thus, — 1 is less than 0 ; — 2 is less than — 1 ; — 3 is less than — 2 ; etc.

An algebraic expression indicating that one number is greater >r less than another is called an inequality.

447. The sign of inequality is > or <. It is placed between two unequal numbers with the opening

)ward the greater.

a is greater than h ' is written a > & ; ' a is less than b ' is written <6.

The expressions on the left and right, respectively, of the sign of inequality are called the first and the second members of the inequality.

361

362

INEQUALITIES

448. The signs > and < are read, respectively, ^is not greater than ' and ' is not less than.'

449. When the first members of two inequalities are each greater or each less than the corresponding second members, the inequalities are said to subsist in the same sense.

When the first member is greater in one inequality and less in another, the inequalities are said to subsist in a contrary sense.

re > a and y > 6 subsist in the same sense, also oj < 3 and ?/ < 4 ; but ic > 6 and y < a subsist in a contrary sense.

450. The following illustrate operations with inequalities :

1.	Given 8 > 5	2.	Given		8>5
	Add, 2 2		Subtract,	8	2 2
	84-2>54-2	-2>5-2
	That is, 10 > 7.		That is.		6>3.
3.	Given 8 > 5	4.	Given		16>10
	Multiply, 2 2 8.2>5.2		Divide,		2 2
	16	--2>10-f-2
	That is, 16 > 10.		That is.		8>5.
5.	Given S> 5	6.	Given		16>10
	Multiply, - 2 - 2		Divide,		-2 -2
	8.-2<5.-2		16	-T--	_2<10--2
	That is, - 16 < - 10.		That is,^		-8<-5.

451. Principle 1. — If the same number or equal numbers be added to or subtracted from both members of an inequality, the resulting inequality will subsist in the same sense.

For, let a>b, and let c be any positive or negative number.

Then, § 446, a — b =p, a. positive number.

Adding c — c = 0, Ax. 1, a + c — (6 + c) = jt>.

Therefore, a -{- c^b + c.

Note. — Letters used in this chapter stand for real numbers.

I

INEQUALITIES 363

I

452. Principle 2. — If both members of an inequality are

multiplied or divided by the same riumber, the resulting inequality will subsist in the same sense if the multiplier or divisor is posi- tive, but in the contrary sense if the multiplier or divisor is , negative,

IFor, let a>b.

Then, § 446, a — b=p, a. positive number.

Multiplying by m, ma — mh = mp. If m is positive, mp is positive,

and therefore, § 446, jwa > mb.

If m is negative, mp is negative,

and therefore, § 446, ma < mh.

Putting — for m, the principle holds also for division. m

453. Principle 3. — A term Ttiay be transposed from one member of an inequality to the other, provided its sign is changed.

For, let a — b>c — d.

Adding b to each side, Prin. 1, a > 6 + c — rf.

Adding — c to each side, Prin. 1, a — c'>b — d.

454. Principle 4. — If the signs of all the terms of an in- equality are changed, the resultiJig inequality will subsist in the contrary sense.

For, let a — b>c — d.

Multiplying each side by — 1, Prin. 2,

— a + 6< —c-\-d,

455. Principle 5. — If the corresponding members of any number of inequalities subsisting in the same sense are added, the resulting inequality will subsist in the same sense.

For, let a>b, c>d, e >/, etc.

Then, § 446, a — b, c — d, e — f, etc., are positive.

Hence, their sum, a + c + e + ... — (6 + c? +/+ •••)> is positive; that is, a + c + e + ...>b + d +f+ '-.

364 INEQUALITIES

Note. — The studeut should bear in mind that the difference of two inequalities subsisting in the same sense, or tlie sum of two inequalities subsisting in a contrary sense, may have its Ji7'st member greater than, equal to, or less'than its second.

Thus, take the inequality 12 > 6.

Subtracting 7 > 3, or adding — 7 < — 3, the result is 5 > 3. Subtracting 8 > 2, or adding — 8 < — 2, the result is 4 = 4. Subtracting 8 > 1, or adding — 8 < — 1, the result is 4 < 6.

I

456. Pkinciple 6. — If each member of an inequality is sub- tracted from the corresponding member of an equation, the result- ing inequality will subsist in the contrail/ sense.

For, let a>6 and let c be any number. m

Then, § 446, a — 6 is a positive number.

Since a number is diminished by subtracting a positive number

f^o^it^ c-(a-b)<c.

Transposing, c — a<c — b.

That is, if each member of the inequality a > 6 is subtracted from the corresponding member of the equation c = c, the result is an inequality subsisting in a contrary sense. j

457. Principle 7. — If a>b and b > c, then a> c. For, § 446, a — 6 is positive and 6 — c is positive. Therefore, (a — 6) + (6 — c) is positive ;

that is, simplifying, a — c is positive.

Hence, § 446, a>c.

Note. — In a similar manner, it may be shown that if a < 6 and 6 < c, then a < c.

458. Principle 8. — If the corresponding members of two inequalities subsisting in the same sense are multiplied together, the result will be an inequality subsisting in the same sense, pro- vided all the members are positive.

For, let a > 6 and c>d, a, b, c, and d being positive. Multiplying the first inequality by c and the second by b, Prin. 2,

ac > be and be > bd.

Hence, Prin. 7, ac > bd.

INEQUALITIES 865

Notes. — 1 . When some of the members are negative, the result may be an inequality subsisting in the same or in a contrary sense, or it may be an equation.

Thus, take the inequality 12 >6.

Multiplying by - 2 > - 5, - 2 > - 3, and - 2 > - 4, the results are, respectively, - 24 > - 30, — 24 < — 18, and — 24 = — 24.

2. The quotient of two inequalities, member by member, may have its first member greater than, equal to, or less than, its second. Thus, take the inequality 12 > 6.

Dividing by 3 > 2, 4 > 2, and — 2 > — 3, the results are, respectively, 4>3, 3 = 3, and -6<-2.

EXERCISES

459. 1. Find the values of x in the inequality 3 a; — 10 > 11.

Solution

3a;-10>ll.

Prin. lor3, 3x>21.

Prin. 2, x>'J.

Therefore, for all values of x greater than 7, the inequality is true ; that is, the inferior limit of x is 7.

2. Find the values of x in the simultaneous inequalities 3 ic 4- 5 < 38 an d 4 a; < 7 a; - 1 8 .

Solution

3a; + 6<38. (1)

4x<7x-l8. (2)

Transposing in (1), Prin. 3, 3x<33.

.-.Prin. 2, x<ll.

Transposing in (2), Prin. 3, - 3 x < — 18.

.-.Prin. 2, x>6.

The result shows that the given inequalities are satisfied simultaneously any value of x between 6 and 11 ; that is, the inferior limit of x is 6, "and the superior limit 11.

366 INEQUALITIES

Find the limits of x in each, of the following :

3. 6x-5>13. (4,x-ll>^Xy

4. 5x-l<6x-\-4:. ' \20-2x>10.

5. 3a;-ia;<30. j3_4a;<7,

6. 4a; + l<6«-ll. * l5a; + 10<20.

9. a; + ^ + — >25and <30. 3 6

10. Find the limits of x and yinS x—y> —14 and x-\'2y=0 Solution (Sx-y>-U, (1)

\x + 2y = 0. (2)

Multiplying (1) by 2, Qx-2y>- 28. (3)

Adding (2) and (3), 7 x > - 28.

Dividing by 7, a; > - 4. (4)

Multiplying (2) by 3, Sx + 6y = 0. (5)

Subtracting (5) from (1), -7 i/> - 14.

Dividing by — 7, y < 2.

That is, X is greater than — 4, and y is less than 2.

Find the limits of x and y in the following, and, if possible, one positive integral value for each unknown number :

2x-Sy<2

rZx-6y<2, * l2a; + 52/ = 25.

12.

13.

3x-{-2y = A2,

3a;-^>16.

7

x-j-y = 10, 4:X<3y.

15.

16.

INEQUALITIES 367

17. Find the limits of a; in a;^ _|. 3 ^ ^ 28.

1 Solution

ic2 + 3a;>28. Transposing, Prin. 3, ic^ + 3 a; — 28 > 0. Factoring, (x - 4) (x + 7) > 0.

That is, (x — 4) (x + 7) is positive. I. Since (x — 4) (x + 7) is positive, either both factors are positive or both are negative. Both f actoi*s are positive when ic > 4 ; both factor* are negative when a; < — 7.

Hence, x can have all values except 4 and — 7 and intermediate values.

Eind the limits of x in each of the following :

18. a;2 _|_ 3 a; > 10. 22. a^ > 9 a; - 18.

19. a^+8aj>20. 23. aj^ + 40 a; > 3 (4 a; - 25).

20. a;2 + 5a;>24. 24. oc^ -\- bx > ax + ab.

21. (a;-2)(3-a;)>0. 25. (a; - 3)(5 - a;) > 0.

26. If a and b are positive and unequal, prove that

a'+b'>2ab.

Proof

Whether a — b is positive or negative, (a — 6)^ is positive: and since a and b are unequal, (a — 6)2 > 0 ;

that is, a2 - 2 a& + &2 > 0.

Transposing, Prin. 3, a^ + ft^ -^ 2 ab.

Note. —If a = 6, it is evident that a^ + b^ = 2 ab.

27. When a and b are positive, which is the greater,

a + b a -\-2b r,

a + 2b a + 36

368 INEQUALITIES

If a, 6, and c are positive and unequal :

28. Prove that a^ + Ir -\- c" > ab + ac + be

29. Prove that a^ -f- b^ > a^b + ab^.

30. Whichis the greater, ^^—t-^' or ^5l±_^'?

a + b a^ -\-b''

31 Prove that -^ -{- ?-5 >1, except when 2 a = 3 6. 3 6 4a

32. Prove that (a — 2 6) (4 6 — a) < b% except when a = 3 &.

33. Prove that a'' + 63+c3> 3 a6c.

34. Prove that the sum of any positive real number (except 1) and its reciprocal is always greater than 2.

35. Prove that a positive proper fraction is increased by adding the same positive number to each of its terms.

36. Find the smallest whole number such that J of it decreased by 1 is greater than i of it increased by 3.

37. If 5 times the number of pupils in a certain department, plus 25, is less than 6 times the number, minus 74 ; and if twice the number, plus 50, is greater than 3 times the number, minus 51, how many pupils are there in the department ?

38. At least how many dollars must A and B each have that 5 times A's money, plus B's money, shall be more than $ 51, and 3 times A's money, minus B's money, shall be $ 21 ?

39. Four times the number of passenger trains entering a certain city daily, minus 136, is less than three times the num- ber, plus 24 ; and 4 times the number, plus 63, is less than 5 times the number, minus 95. How many passenger trains enter the city each day?

40. Three times the number of soldiers in a full regiment, less 593, is less than 2 times the number, plus 608 ; and 8 times the number, minus 577, is less than 9 times the number, minus 1776. How many soldiers are there in a full regiment ?

I

RATIO AND PROPORTION

RATIO

460. The relation of two numbers that is expressed by the [uotient of the first divided by the second is called their ratio.

461. The sign of ratio is a colon (:).

A ratio is expressed also in the form of a fraction.

The ratio of a to 6 is written a :b or -

b The colon is sometimes regarded as derived from the sign of division y omitting the line.

462. To compare two quantities they must be expressed in irms of a common unit.

i'

^p Thus, to indicate the ratio of 20 ^^^ to $ 1, both quantities must be ex- pressed either in cents or in dollars, as 20^ : 100^ or $^ : .f 1. There can be no ratio betw^een 2 lb. and 3 ft.

The ratio of two quantities is the ratio of their numerical iasures.

Thus, the ratio of 4 rods to 5 rods is the ratio of 4 to 5.

463. The first term of a ratio is called the antecedent, and le second, the consequent. Both terms form a couplet.

The antecedent corresponds to a dividend or numerator ; the )nsequent, to a divisor or denominator.

In the ratio a : 6, or - , a is the antecedent, b the consequent, and b terms a and b form a couplet.

Milne's stand, alg. — 24 369

370 RATIO AND PROPORTION

464. A ratio is said to be a ratio of greater inequality, a ratio of equality, or a ratio of less inequality, according as the ante- cedent is greater than, equal to, or less than the consequent.

Thus, when a and 6 are positive numbers, a : & is a ratio of greater inequality, if a > & ; a ratio of equahty, if a = & ; and a ratio of less inequality, if a < 6.

465. The ratio of the reciprocals of two numbers is called the reciprocal, or inverse, ratio of the numbers.

It may be expressed by interchanging the terms of the couplet.

The inverse ratio of a to & is - : - . Since --=-- = -, the inverse ratio , a b aha

of a to 6 may be written -, or h -.a. a

466. The ratio of the squares of two numbers is called their duplicate ratio; the ratio of their cubes, their triplicate ratio.

The duplicate ratio of a to 6 is a^ ; &2 . \^q triplicate ratio, a^ : h^.

467. If the ratio of two numbers can be expressed by the ratio of two integers, the numbers are called commensurable numbers, and their ratio a commensurable ratio.

468. If the ratio of two numbers cannot be expressed by the ratio of two integers, the numbers are called incommen surable numbers, and their ratio an incommensurable ratio.

The ratio \/2 : 3 = — — = — cannot be expressed by any two

integers, because there is no number that, used as a common measure, will be contained in both \/2 and 3 an integral number of times. Hence, V2 and 3 are incommensurable, and \/2 : 3 is an incommensurable ratio. It is evident that by continuing the process of extracting the square root of 2, the ratio a/2 : 3 may be expressed by two integers to any desired degree of approximation, but never with absolute accuracy.

Properties of Ratios

469. It is evident from the definition of a ratio that ratios have the same properties as fractions ; that is, they may be reduced to higher or lower terms, added, subtracted, etc. Hence

RATIO AND PROPORTION 371

Principles. — 1. Multiplying or dividing both terms of a ratio by the same number does not change the value of the ratio.

2. Multiplying the ayitecedent or dividing the consequent of a ratio by any number multiplies the ratio by that number.

3. Dividing the antecedent or multiplying the consequent by any number divides the ratio by that number.

470. If the same positive number is added to both terms of a fraction, the value of the fraction will be nearer 1 than before, whether the fraction is improper or proper. The correspond- ing principle for ratios follows :

Principle 4. — A ratio of greater inequality is decreased and a ratio of less iyiequality is increased by adding the same positive number to each of its terms.

For, given the positive numbers a, h, and c, and the ratio - •

b

1. When a>b, it is to be proved that ^ < - •

6 + c 6

a + c a _ c(h — a) _ h + c l~ h(b ■\- c)

Since S — a is negative, because a > 6,

^^ ~ ^^ is negative :

'therefore, 2i^ — ^ is negative.

6 + c 0

Hence, §446, ^±l<^l.

h + c b

2. When a < &, it is to be proved that >— •

b -\- c b

Proceeding by the method used in 1, since a < ft it may be shown

that a + c _ « ^ positive.

b +c b ^

Hence, §446, ^L±£>E.

b + c b

1

372 RATIO AND PROPOKTIOJS

EXERCISES

471. 1. What is the ratio of 8 m to 4 m ? of 4 m to 8 m ?

2. Express the ratio 6 : 9 in its lowest terms ; the ratio 12 ic : 16 2/ ; am :bm; 20 ab : 10 be ; (m -\-n ) : (w? — n^).

3. Which is the greater ratio, 2:3 or 3:4? 4:9 or 2:5?

4. What is the ratio of i to i? ^ to i? | to f ? i

Suggestion. — When fractions have a common denominator, they have the ratio of their numerators.

5. What is the inverse ratio of 3 : 10 ? of 12 : 7 ?

6. Write the duplicate ratio of 2 : 3 ; of 4 : 5 ; the triplicate ratio of 1 : 2 ; of 3 : 4.

Reduce to lowest terms the ratios expressed by :

7. 10:2. 10. 3:27. 13. if. 16. 75^100.

8. 12:6. 11. 4:40. 14. ^. 17. 60-5-120.

9. 16:4. 12. 9:72. 15. f^. 18. 80 -- 240.

19. What is the ratio of 15 days to 30 days ? of 21 days to 1 week ? of 1 rod to 1 mile ?

Find the value of each of the following ratios :

20. 4a;:fa^. 23. 2^:7^. 26. a%^x^ : a'b^a?.

21. f a6 : 1 ac. 24. .7 m : .8 n. 27. {x^ - 2/^) : {x - yf.

22. ^xY'-ixy. 25. .4 a.-^ : 10 ic^^ 28. (a^-1) : (a' + a+i).

29. Two numbers are in the ratio of 4 : 5. If 9 is subtracted from each, find the ratio of the remainders.

30. Change each to a ratio whose antecedent shall be 1 :

5:20; 3a;:12a;; f:f; .4:1.2.

31. Reduce the ratios a : b and x:y to ratios having the same consequent.

32. When the antecedent is 6 a; and the ratio is 4, what is the consequent ?

RATIO AND PROPORTION 373

33. Given the ratio J and a positive number x. Prove that f > - by subtracting one ratio from the other.

Suggestion. — Proceed as in § 470.

34. The capital stock of a street railway company was 1 7,500,000, the gross earnings for a year $ 1,500,000, and the net earnings ^600,000. Find the ratio of gross earnings to capital stock ; of net earnings to gross earnings ; of net earn- ings to capital stock.

PROPORTION

472. An equality of ratios is called a proportion.

3 : 10 = 6 : 20 and a:x = b:y a,Te proportions.

The double colon (: :) is often used instead of the sign of [uality.

The double colon has been supposed to represent the extremities of the lines that form the sign of equality.

The proportion a:b = c:d, ov a:b::c:d, is read, ' the ratio of a to 6 is equal to the ratio of c to d/ or 'a is to 6 as c is to d'

473. In a proportion, the first and fourth terms are called the extremes, and the second and third terms, the means.

In a:b = c:d, a and d are the extremes, b and c are the means.

474. Since a proportion is an equality of ratios each of 'hich may be expressed as a fraction, a proportion may be

expressed as an equation each member of which is a fraction. Hence, it follows that :

General Principle. — TTie changes that may be made in a ^^(yportion without destroying the equality of its ratios correspond

the changes that may be made in the members of an equation nthout destroying their equality arid in the terms of a fraction nthout altering the value of the fraction.

374 RATIO AND PROPORTION

Properties of Proportions

475. Principle 1. — In any proportionj the product of the ex- tremes is equal to the product of the means.

For, given a:b = c:d,

a c or !f = r.

b d Clearing of fractions, ad = Ic.

Test the following by principle 1 to find whether they are true proportions :

1. 6:16 = 3:8. 2. || = if. 3. 7:8 = 10:12. ^

476. In the proportion a:m = m:b, m is called a mean proportional between a and b.

By Prin. 1, m^ = ab',

.*. m= -y/ab.

Hence, a mean proportional between two numbers is equal to the square root of their x>roduct.

1. Show that the mean proportional between 3 and 12 is either 6 or — 6. Write both proportions.

2. Find two mean proportionals between 4 and 2^.

477. Principle 2. — Either extreme of a proportion is equal to the product of the means divided by the other extreme.

Either mean is equal to the product of the extremes divided by the other mean.

For, given a:b = c:d.

By Prin. 1, ad = be.

Solving for a, d, 6, and c, in succession, Ax. 4,

be J be ■, ad ad

a = —,d=—,o=—f € = —— d a c b

1. Solve the proportion 3 : 4 = ic : 20, for a;.

2. Solve the proportion x:a = 2m:n, for x.

RATIO AND PROPORTION 375

3. If a: b = b:c, the term c is called a third proportional to a id b. Find a third proportional to 6 and 2.

4. In the proportion a:b — c:d, the terra d is called a fourth proportional to a, b, and c. Find a fourth proportional to ^, J, and :^.

478. Principle 3. — If the product of two numbers is equal to the product of two other numbers, one pair of them may be made the extremes and the other pair the means of a proportion.

For, given ad = be.

Dividing by &c?, Ax. 4, r = ^ »

0 a

that is, a:h = c:d.

By dividing both members of the given equation, or of be = ac?, by i the proper numbers, various proportions may be obtained ; but in all ] of them a and d will be the extremes and b and c the means, or viee versa, as illustrated in the proofs of principles 4 and 5.

1. If a men can do a piece of work in x days, and if b men can do the same work in y days, the number of days' work for one man may be expressed by either ax or by. Form a pro- portion between a, b, x, and y.

2. The formula pd = WD (See p. 173)

expresses the physical law that, when a lever just balances, the product of the numerical measures of the power and its distance from the fulcrum is equal to the product of the numerical measures of the weight and its distance from the ! fulcrum. Express this law by means of a proportion.

i

I 479. Principle 4. — If four numbers are in proportion, the I ratio of the antecedents is equal to the ratio of the consequents; I that is, the numbers are in proportion by alternation.

For, given a:b = e:d.

tThen, Prin. 1, ad = be.

Dividing by cd, Ax. 4, - = - ;

e d it is, a:c — bid.

376 KATIO AND PROPORTION

480. Principle 5. — If four numbers are in proportion, the ratio of the second to the first is equal to the ratio of the fourth to the third; that is, the ^lumbers are in proportion by inversion.

For, given	a : h = c : d.
Then, Prin. 1,	ad = be.
	:. be = ad.
Dividing by ac, Ax. 4,	b d
at is.	b:a = d :e.

481. Principle 6. — If four numbers are in proportion, the sum of the terms of the first ratio is to either term of the first ratio as the sum of the terms of the second ratio is to the corresponding term of the second ratio; that is, the numbers are in proportion by composition. ,|

For, given a:b = e:d, ^

a e or - = - .

b d

Then, 1+1 =£+1,

0 d

or a±b^e_±d

b d '

that is, a -\- b : b = c + d : d.

Similarly, taking the given proportion by inversion (Prin. 5), and adding 1 to both members, v^^e obtain ^

a -{- b : a = e + d : e. ' '^

482. Principle 7. — If four numbers are in proportion, the difference between the terms of the first ratio is to either term of the first ratio as the difference betiveen the terms of the second ratio is to the corresponding term of the second ratio; that is, the numbers are in proportion by division.

For, in the proof of Prin. 6, if 1 is subtracted instead of added, the following proportions are obtained :

a — b : b = e — d : d, and a — b:a = c — d'.c.

RATIO AND PROPORTION

377

483. Principle 8. — If four numbers are in proportion, the sum of the terms of the first ratio is to their difference as the sum of the terms of the second ratio is to their difference; that is, the numbers are in proportion by composition and division.

For, given			a:b =c:d.
By Prill. 6,			a -hb_c + d b d		(1)
By Prin 7,			a — b c— d b d		(2)
Dividing (1) by (2),	Ax. 4,		a + b _c -\- d , a — b c - d^
at is,	a	+ &	:a— b = c -i d:c -	-d.

484. Principle 9. — If four numbers are in proportion, their \like poivers, and also their like roots, are in proportion.

For, given

a	:b =	-.c:d,
	b~	_c ~d'
	a" bn~

Then, Ax. 6 and § 276, 3, that is, a" : 6" = c" : d\

Also, Ax. 7 and § 291, regarding only principal roots,

\/a _ \/c .

Vb~ ^d'

[that is, Va : y/b = ^/c : Vd.

485. Principle 10. — In a proportion, if both terms of a couplet, \or both aritecedents, or both consequents are multiplied or divided by the same number, the resulting four numbers form a proportion.

For, given a:b = c:d,

lor

Then, § 195, Also, Ax. 3,

mo a m b ' n

a _c b~ d

— = — , ov ma : mb = nc \ nd. '' nd

, or ma :nb = mc : nd.

378 RATIO AND PROPORTION

486. Principle 11. — The products of corresponding terms oj any number of proportions form a proportion.

For, given a:b = c:d,

k:l = m:n,

and x:y — z'.w.

Writing each proportion as a fractional equation, we have

a c k m ji X z - = -, - = -, and - = -. bain y ^

Multiplying these equations, member by member, Ax. 3, we have

akx _ cmz , bly dnw ' that is, akx : bly = cmz : dnw.

487. Principle 12. — If two proportions have a common couplet, the other tivo couplets will form a proportion.

For, given a-.b = x:y,

and c:d = x:y.

Then, Ax. 5, a:b = c:d.

488. A proportion that consists of three or more equal ratios is called a multiple proportion. i

2:4 = 3:6 = 5: 10 and a:b =c : d = e :f are multiple proportions. '

489. Principle 13. — In any multiple proportion the sum of \ all the antecedents is to the sum of all the consequents as any antecedent is to its consequent.

For, given a:b= c:d = e:f

or ^ = f = - = r, the value of each ratio.

b d f '

Then, Ax. 3, a = br, c = dr, e =fr\

whence, Ax. 1, a + c 4- e = (ft + c? +/) r,

g + c 4- e _ _«_£_«. ''' b-\-d-\-f~^ ~ b~ d~y

that is, a + c+tf:6 + rf +/ = a : 6 or c : </ or e :/,

1

RATIO AND PROPORTION 379

490. A multiple proportion in which each consequent is repeated as the antecedent of the following ratio is called a continued proportion.

2 : 4 = 4 : 8 = 8 : 16 and a:6 = 6:c = c:(?are continued proportions.

491. Principle 14. — If three numbers are in continued pro- portiouj the ratio of the extremes is equal to the square of either given ratio.

For, given a : ft = 6 : c,

Multiplying by ^, Ax. 3, |' = ^, (2)

By (1) and Prin. 9, ^' = 5?. (3)

By (2) and (3), Ax. 5, that is, a : c = a2 : 62 = 52 . ^2.

492. Principle 15. — If four numbers are in continued pro- portion, the ratio of the extremes is equal to the cube of any of the given ratios.

For, given		a-.h = h:c = c'.dy
or		a h _c b ~'c~ d	(1)
Then, Ax. 5,		a b c _a a a _a^, b ' c ' d b ' b ' b 63'	(2)
that is, canceling,		a:d = a^:b^	(3)
By (1) and Prin	.9,	a^:b^ = b^:c^ = c»: d\	(4)
By (4) and (3),	Ax. 5,	a:d = a^:b^ = b^:c^=c^:d^

exercises 493. 1. Find the value of x in the proportion S : 5 = x : 55. Solution. 3 : 5 = x : 55.

Prin. 2, a; = ^-^=33.

380 RATIO AND PROPORTION

Find the value of x in each of the following proportions :

2. 2:3 = 6:a;. 5. a; + 2 : a; = 10 : 6.

3. 5: a; = 4: 3. 6. a;:a; — 1 = 15:12.

4. l:x = X'.^. 7. x-\-2:x -2 = ^:1.

8. Show that a mean proportional between any two num- bers having like signs has the sign ± .

9. Find two mean proportionals between V2 and V8.

10. Find a third proportional to 4 and 6.

11. Find a fourth proportional to 3, 8, and 7^.

12. Find a mean proportional between

— and — — —

a;-f 4 x-\-2

Test to see whether the following are true proportions :

13. 5-i:3 = 4:li 15. 5 : 7a; = 10 : 14ic.

14. 4:13 = 2:6J. 16. 2.4a : .8a = 6a: 2a.

17. Given a:h=:^c: d,

to prove that 2a + 56:4a — 36 = 2c + 5d:4c— 3d.

Proof. — First form, from the given proportion, a proportion hav-^ ing as antecedents the antecedents of the required proportion.

a:b = c:d. (1)

Prin. 10, 2a:b = 2c:d. j

Prin. 10, 2a'.5b = 2c:5d. * 1

Prin. 6, 2 a + 5b :2a = 2 c + 6d :2 c. (2)

Next form from (1) a proportion having as antecedents the conse- quents of the required proportion.

Prin. 10, 4:a:b = 4:c:d.

Prin. 10, 4 a : 3 & = 4 c : 3 rf.

Prin.7, 4a - 3& : 4a = 4c - 3rf: 4c. (3)

Prin. 10, 4a-3&:2a = 4c-3</:2c. (4)

Next take (2) and (4) by alternation (Prin. 4) and apply Prin. 12 to the results.

Then, 2a + 5&:2c + 5f/ = 4a-36:4c-3<f,

or, Prin. 4, 2a + 56:4a-36 = 2c + 5c?:4c-3rf.

RATIO AND PROPORTION 381

When a:h = c:dj prove that the following are true propor- tions:

18. d:h = G:a, 21. a?:h''(?=l'.d:',

19. c:d = -:i. 22. ma:-==mc:-^ ha 2 2

20. ¥:d^ = a':<^. 23. achd^c^id?,

24. Va^:V6 = Vc:l.

25. a-|-6:c + (Z = a — 6:c — d.

26. a:a + & = a + c:a + 6H-c + d.

27. a + 6:c + Qj = VaM^: VcM^.

28. a^ + a?b-{-ah^+}^:a^ = c'-\-(^d-\-cd:^-\-d?:<^.

29. 2aH-36:3a-f 46 = 2c + 3d:3c-f 4c?.

30. 2a + 3c:2a-3c = 8&4-12d:85-12d

31. a4-&-4-c + d:a — 64-c — c? = a + 6 — c — c?:a — 6 — c + d

32. If a : & = c : d, and if a; be a third proportional to a and 6, "and 2/ a third proportional to h and c, show that the mean pro- portional between x and y is equal to that between c and d.

33. Solve for x, V^+7+_V^^ V^+7 - V^. 4 4- Va? 4 — V^

Solution By alternation, Prin. 4,

Va; + 7 — Vx 4 - Vx By composition and division, Prin. 8,

2Va; + 7^ 8 2y/x 2Vx Since the consequents are equal, the antecedents are equal.

Therefore, 2 Vx + 7 = 8.

Solving, a; = 9.

382 RATIO AND PROPORTION

34. Given V^n + 2 ^ ^2^+U + 2^^ ^^ ^^^^^ ^^^ ^^ Va; + ll-2 V2i» + 14-22 Solution By composition and division, Prin. 8,

2v/x + 11^2V2a; + 14 4 Jj^ '

Dividing both terms of each ratio by 2, Prin. 10,

■\'x + 11 _ \/2 a; + 14 2 I '

Dividing the consequents by |, Prin, 10,

V^ + n ^ V2 x + 14 3 4

By alternation, Prin. 4, .V^+IL = § . V2 a; + 14 ^

Squaring and applying Prin. 7,

a;4-3 _7 « + ll~9'

Solving, a; = 25.

Solve for x by the principles of proportion :

OK A^^+Vm m -- Vcc + & 4- Va; — h „

oo. — 3 = — • oo. — ::::^^ = Ci,

Va? — Vm ^ Va; + 6 — Va; — 6

gg Va; 4- V2 ft _ 2 ^^ Va + Va — a; _ 1 ^

Va; — V2 a 1 Va — Va — x ^

a; -|- Va; — 1 _ 13 Vo^ — 6 SVo^ — 2 6

X — Va; — 1 ' -y/ax + b 3 Vaa; + 56

41 Va + Va + X _ V64-Va; — 6 ^ Va — Va + a; V6 — Va; — 6

Va; + 1 + Va; — 2^Va; — 3 + Va; — 4 VaJ + 1 — Va; — 2 Va; — 3 — Va; — 4

RATIO AND PROPORTION 383

Problems

494. 1. Divide $ 35 between two men so that their shares shall be in the ratio of 3 to 4.

M 2. Two numbers are in the ratio of 3 to 2. If each is ■increased by 4, the sums will be in the ratio of 4 to 3. What Rare the numbers ?

J 3. Divide 16 into two parts such that their product is to the sum of their squares as 3 is to 10.

4. Divide 25 into two parts such that the greater increased by 1 is to the less decreased by 1 as 4 is to 1.

5. The sum of two numbers is 4, and the square of their sum is to the sum of their squares as 8 is to 5. What are the numbers ?

6. Find a number that added to each of the numbers 1, 2, 4, and 7 will give four numbers in proportion.

7. In the state of Minnesota the ratio of native-born in- habitants to foreign-born recently was 5:2. What was the number of each, if the total population was 1,760,000 ?

8. A business worth $ 19,000 is owned by three partners. The share of one partner, $6000, is a mean proportional be- tween the shares of the other two. Find the share of each.

9. What number must be added to each of the numbers 11, 17, 2, and 5 so that the sums shall be in proportion when taken in the order given ?

10. Four numbers are in proportion ; the difference between the first and the third is 2| ; the sum of the second and the third is 6|^; the third is to the fourth as 4:5. Find the num- bers.

11. Prove that no four consecutive integers, as w, n + Ij n -f- 2, and n -f 3, can form a proportion.

8 12. Prove that the ratio of an odd number to an even num- ber, as 2m-\-l:2n, cannot be equal to the ratio of another even number to another odd number, as 2 a; : 2 ?/ + 1.

i.

384

RATIO AND PROPORTION

13. The area of the right triangle shown in Fig. 1 may be expressed eitlier as ^ ah or as ^ ch. Form a proportion whose terms shall be a, b, c, and h. 1

Fig. 1.

Fig. 2.

Fig. 3.

14. In Fig. 2, the perpendicular p, which is 20 feet long, is a mean proportional between a and b, the parts of the diame- ter, which is 50 feet long. Find the length of each part. J

15. In Fig. 3, the tangent t is sl mean proportional between the whole secant c-\-e, and its external part e. Find the length of ^, if e = 9f and c = 50f .

16. The strings of a musical instrument produce sound by vibrating. The relation between the number of vibrations N and N' of two strings, different only in their lengths I and Z', is expressed by the proportion

]Sr:N' = V:l i

A c string and a g string, exactly alike except in length, ' vibrate 256 and 384 times per second, respectively. If the c string is 42 inches long, find the length of the g string.

17. If L and I are the lengths of two pendulums and T and t the times they take for an oscillation, then

T':f = L:l.

A pendulum that makes one oscillation per second is approxi-' mately 39.1 inches long. How often does a pendulum J 56.4 inches long oscillate ?

18. Using the proportion of exercise 17, find how many fee loDff a pendulum would have to be to oscillate once a minute

e^

II

VARIATION

495. Many problems and discussions in mathematics have do with numbers some of which have values that are con-

inually changing while others remain the same throughout the? liscussion. Numbers of the first kind are called variables; lumbers of the second kind are called constants.

Thus, the distance of a moving train from a certain station is a varia- ie, but the distance from one station to another is a constant.

Two variables may be so related that when one changes the bher changes correspondingly.

496. One quantity or number is said to vary directly as lother, or simply to vary as another, when the two depend

ipon each other in such a manner that if one is changed the bher is changed in the same ratio.

Thus, if a man earns a certain sum per day, the amount of wages he ens varies as the number of days he works.

497. The sign of variation is oc. It is read 'varies as.^ Thus, X oc ?/, read ' x varies as i/ ', is a brief way of writing the proportion

x-.x' =y:y', which x' is the value to which x is changed when y is changed to y'.

498. The expression a? oc 2/ means that if x is doubled, y is doubled, or if x is divided by a number, y is divided by the same number, etc. ; that is, that the ratio oi x to y is always the same, ov constant.

If the constant ratio is represented by Zc, then when a; x y,

- = A;, or a; = ky. Hence, If x varies as y, x is equal to y multiplied by a constant.

Milne's stand, alg. — 25 385

386 VARIATION

499. One quantity or number varies inversely as another when it varies as the reciprocal of the other.

Thus, the time required to do a certain piece of work varies inversely as the number of men employed. For, if it takes 10 men 4 days to do a piece of work, it will take 5 men 8 days, or 1 man 40 days, to do it.

1 . 1 ^

In a; oc - , if the constant ratio of a; to - is A;, r = ^» or xv = Tc. y ^ ' 1 ^

Hence, y

If X varies inversely as y, their product is a constant.

500. One quantity or number varies jointly as two others when it varies as their product.

Thus, the amount of money a man earns varies jointly as the number of days he works and the sum he receives per day. For, if he should work th7'ee times as many days, and receive twice as many dollars per day, he would receive six times as much money. !

In a; X yz, if ^the constant ratio of x to yz is 7c, *

— z=]c, or a; = kyz. Hence, yz

If X varies jointly as y and z, x is equal to their proditct multi- plied by a consta7it.

501. One quantity or number varies directly as a second and inversely as a third when it varies jointly as the second and the reciprocal of the third.

Thus, the time required to dig a ditch varies directly as the length of the ditch and inversely as the number of men employed. For, if the ditch were 10 times as long and 5 times as many men were employed, it would take twice as long to dig it.

1 y

In a; oc y • -, or a; oc -, if /c is the constant ratio, z z

x-^- = k, or x = k-. Hence, z ^ z

Iff ,

Ifx varies directly as y and inversely as z, x is equal to - multi-l

plied by a constant, '

VARIATION 387

502. If X varies as y when z is constant, and x varies as z '}hen y is constant, then x varies as yz when both y and z are iriable.

Thus, tke area of a triangle varies as the base when the altitude is con- mt ; as the altitude when the base is constant ; and as the product of le base and altitude when both vary.

Proof

Since the variation of x depends upon the variations of y and 2, sup- )se the latter variations to take place in succession, each in turn pro- ducing a corresponding variation in x.

While z remains constant, let y change to y^ thus causing x t<o change to x'.

Then, ^ = i. (1)

^ y\

Now while y keeps the value y^, let z change to z^, thus causing x' to change to x^.

Then, £; = ^. (2)

Multiplying (1) by (2), £- = J^ . (3)

x = ^'yz. (4)

Since, if both changes are made, Xj, «/,, and z^ are constants, — ^ is a constant, which may be represented by k. ^^ *

Then, (4) becomes x = kyz.

Hence, x oc yz.

Similarly, if x varies as each of three or more numbers, y, z, ), ••• when the others are constant, when all vary x varies as leir product.

That is, xacyzv--:

Thus, the volume of a rectangular solid varies as the length, if the width id thickness are constant ; as the width, if the length and thickness are mstant ; as the thickness, i( the length and width are constant ; as the toduct of. any two dimensions, if the other dimension is constant; and

the product of the three dimensions, if all vary.

388 VARIATION

EXERCISES

503. 1. If a; varies inversely as y, and .t = 6 when y = 8, what is the value of x when y = 12?

Solution Since xx -, let A; be the constant ratio of a; to -•

y y

Then, § 499, xy = k.

Hence, when oj = 6 and 2/ = 8, A; = 6 x 8, or 48. Since k is constant, k = A% when ?/ = 12,

and (1) becomes 12 a; = 48.

Therefore, when 2^ = 12, ic = 4.

2. If a? oc -, and if .t = 2 when y = 12 and z = 2, what is the

z

value of X when ?/ = 84 and z = 1 ?

3. If XCC-, and if x = 60 when ?/ = 24 and z = 2, what is the value of y when ic = 20 and z = 7?

4. If ic varies jointly as ?/ and z and inversely as the square of Wj and if a; = 30 when y = 3, z = 5, and 2<; = 4, what is the value of X expressed in terms of y, z, and w ?

5. li xccy and yocz, prove that xccz.

Proof

Since a; x ^ and y ccz, let m represent the constant ratio of x to y, and m the constant ratio of y to z.

Then, § 498, a; = my, (1)

and y = nz. (2)

Substituting W2 for y in (1), x = mnz. (3)

Hence, since ?/?n is constant, xccz.

6. If a;oc-, and 2/oc -, prove that a;oc2!.

y z

7. If a; X 2/ and :? (?c ?/, prove that (a; ± z)Qcy.

i

VARIATION 889

Problems

504. 1. The volume of a cone varies jointly as its altitude and the square of the diameter of its base. When the altitude is 15 and the diameter of the base is 10, the volume is 392.7. What is the volume when the altitude is 5 and the diameter of the base is 20 ?

Solution

Let V, H, and D denote the volume, altitude, and diameter of the base, respectively, of any cone, and V the volume of a cone whose altitude is 5 and the diameter of whose base is 20.

Since VccHD% or V=kHD\

and • F = 392.7 when H=\b and Z) = 10,

392.7 = A; X 15 X 100. (1)

Also, since V becomes V when H= 6 and D = 20,

F' = A; X 6 X 400. (2)

Dividing (2) by (1), Ax. 4, -^ = 5_xm^ ^ 4^ .g.

^ ^ ^ "^ ^ ^' ' 392.7 15 X 100 3 ^ ^

.-. F' = I of 392.7 = 523.6.

2. The circumference of a circle varies as its diameter. If the circumference of a circle whose diameter is 1 foot is 3.1416 feet, find the circumference of a circle 100 feet in diameter.

3. The area of a circle varies as the square of its diameter. If the area of a circle whose diameter is 10 feet is 78.54 square feet, what is the area of a circle whose diameter is 20 feet ?

4. The distance a body falls from rest varies as the square of the time of falling. If a stone falls 64.32 feet in 2 seconds, how far will it fall in 5 seconds ?

5. The volume of a sphere varies as the cube of its diameter. If the ratio of the sun's diameter to the earth's is 109.3, how many times the volume of the earth is the volume of the sun ?

6. If 10 men can do a piece of work in 20 days, how long will it take 25 men to do it ?

7. If a men can do a piece of work in 6 days, how many men will be required to do it in c days ?

390 VARIATION

8. The area of a triangle varies jointly as its base and altitude. The area of a triangle whose base is 12 inches and whose altitude is 6 inches is 36 square inches. What is the area of a triangle whose base is 8 inches and whose altitude is 10 inches ? What is the constant ratio ?

9. A wrought-iron bar 1 square inch in cross section and 1 yard long weighs 10 pounds. If the weight of a uniform bar of given material varies jointly as its length and the area of its cross section, what is the weight of a wrought-iron bar 36 feet long, 4 inches wide, and 4 inches thick ?

10. The weight of a beam varies jointly as the length, the area of the cross section, and the material of which it is com- posed. If wood is ^2^ as heavy as wrought iron (see exercise 9), what is the weight of a wooden beam 24 feet long, 12 inches wide, and 12 inches thick ?

11. What is the weight of a brick 2 in. X 4 in. x 8 in., if the material is \ as heavy as wrought iron ? (For the weight of wrought iron, see exercise 9.)

12. The distances, from the fulcrum of a lever, of two weights that balance each other vary inversely as the weights. If two boys weighing 80 pounds and 90 pounds, respectively, are balanced on the ends of a board 8|- feet long, how much of the board has each on his side of the fulcrum ?

13. A water carrier carries two buckets of water suspended from the ends of a 4-foot stick that rests on his shoulder. If one bucket weighs 60 pounds and the other 100 pounds, and they balance each other, what point of the stick rests on his shoulder ?

14. The horse power (H) that a solid shaft can transmit safely varies jointly as its speed in revolutions per minute (N) and the cube of its diameter. A 5-inch solid steel shaft making 150 revolutions per minute can transmit 585 horse power. How many horse power could it transmit at half this speed, if its diameter were increased 1 inch ?

i

VARIATION 391

15. The weight of a body near the earth varies inversely as ^the square of its distance from the center of the earth. If the [radius of the earth is 4000 miles, what would be the weight of

4-pound brick 4000 miles above the earth's surface ?

16. The weight of wire of given material varies jointly as fthe length and the square of the diameter. If 3 miles of wire

.08 of an inch in diameter weighs 288 pounds, find the weight of ^ mile of wire .16 of an inch in diameter.

17. The illumination from a source of light varies inversely the square of the distance. How far must a screen that is

[0 feet from a lantern be moved so as to receive one fourth as [much light ?

18. The number of times a pendulum oscillates in a given ;ime varies inversely as the square root of its length. If a )endulum 39.1 inches long oscillates once a second, what is the mgtli of a pendulum that oscillates twice a second? once in

iree seconds ?

19. Three spheres of lead whose radii are 6 inches, 8 inches, id 10 inches, respectively, are united into one. Find the idius of the resulting sphere, if the volume of a sphere varies

the cube of its radius.

20. The volume of a cone varies jointly as its altitude and le square of the diameter of its base. The altitudes of three mes, S, P, and R, are 30 ft., 10 ft., and 5 ft., respectively. )he diameter of the base of P is 5 ft. and that of R is 10 ft.

the volume of S is equivalent to that of P and R combined, rhat is the diameter of the base oi S?

21. A boy wishes to ascertain the height of a tower. He knows that it is 31 feet 6 inches from his window to the pave- ment below, and that the distance through which a body falls varies as the square of the time of falling. He drops a marble from his window and finds that it strikes the pavement in 1.4 seconds. Then throwing a stone upward he observes that it takes just 3 seconds for it to descend from the top of the tower to the ground. What is the height of the tower ?

392

VARIATION

505. Algebraic expression of physical laws.

If two wires exactly alike in all respects except in length (I) are stretched by equal weights, the greater number (ri) of vibra>

tions per second will be made by the shorter wire. If one wire

is half as long as the other, its rate of vibration will be twice

as great; if ^ as long, the rate will be 3 times as great; etc.

This result is expressed by the variation

n X

Next, experimenting with two wires alike in all respects except in diameter (d), it is found that

1

woe

Next, excluding all variable elements in the experiment except the stretching weight ( IF), it is found that

Finally, experimenting with wires of different niaterials, as steel and brass, which have different specific gravities (s),

1

noc

Vs

Since the number of vibrations per second varies inversely as I and cl, directly as the square root of W, and inversely as the square root of s, by § 502,

IdVs which is the expression of the law as a variation.

VARIATION 39a

It is found by measuring n, Z, d, TT, and s in any case that

the constant ratio of the first member to the second is \~' Hence, the law may be expressed by the equation ^

vw _

IdVs

EXERCISES

506. The approximate numerical value usually used for tt is 3.1416, and for g, 32.16 or 980 according as the distance unit is 1 foot or 1 centimeter.

Express by a variation, and when k is given by an equation, each of the following laws :

1. The distance (s) passed through in t seconds by a body falling freely from a state of rest varies as the square of the time. The constant ratio (k) is equal to i g.

2. The time required by a simple pendulum to make a complete oscillation varies as the square root of its length. A; = 2 TT -T- V^ is the constant ratio, at any given place.

3. The velocity (v) acquired by a body falling from a height (h) varies as the square root of the height. The constant ratio, for any given place, is V2 g.

4. The quantity (Q) of water flowing from a circular orifice, of diameter (d) and under a height, or head (h), of water varies las the square of d and as the square root of h. The constant ratio, under ordinary conditions, is Zc = .625 • | tt V2 g.

5. The intensity of a current (I) in an electric circuit varies directly as the electromotive force (E) and inversely as the re- sistance {R) in the circuit. The constant ratio is 1.

6. The heat loss (P) in an electric circuit varies directly as the intensity of the current (/) and the square of the resist- ance (E). The constant ratio is 1.

PROGRESSIONS

507. A succession of numbers, each of which after the first is derived from the preceding number or numbers according to some fixed law, is called a series.

The successive numbers are called the terms of the series. The first and last terms are called the extremes, and all the others, the means.

In the series 2, 4, 6, 8, 10, 12, 14, each term after the first is greater by 2 than the preceding term. This is the law of the series. Also since 1st term = 2 • 1, 2d term = 2 • 2, 3d term = 2-3, etc., the law of the series may be expressed thus :

nth term = 2 n. ^

In the series 2, 4, 8, 16, 32, 64, 128, each term after the first is twice the preceding term; or expressing the law of the series by an equation, or formula,

nth term = 2\

\

ARITHMETICAL PROGRESSIONS ^

508. A series, each term of which after the first is derived from the preceding by the addition of a constant number, is called an arithmetical series, or an arithmetical progression.

The number that is added to any term to produce the next is called the common difference.

2, 4, 6, 8, ••• and 15, 12, 9, 6, ... are arithmetical progressions. In the first, the common difference is 2 and the series is ascending ; in the sec- ond, the common difference is — 3 and the series is descending.

A. P. is an abbreviation of the words arithmetical progression. j

394 Jl

PROGRESSIONS 396

509. To find the nth, or last, term of an arithmetical series.

In the arithmetical series

1, 3, 5, 7, 9, 11, 13, 15, 17, 19,

the common difference is 2, or d=2. This difference enters once in the second term, for 3 = 1 -f- d ; twice in the third term, for 5 = 1 -|- 2 d ; three times in the fourth term, for 7 = 1 + 3 d ; and so on to the 10th, or last, term, which equals 1 + 9 d.

In a, a + d, a + 2d, a + 3 d, ••.,

which is the general form of an arithmetical progression, a representing the first term and d the common difference, observe that the coefficient of d in the expression for any term is one less than the number of the term. Then, if the nth, or last, term is represented by I,

l = a-{-{n-l)d. (I)

Note. — The common difference d may be either positive or negative. In the A.P. 25, 23, 21, 19, 17, 15, d= - 2.

EXERCISES

510. 1. What is the 10th term of the series 6, 9, 12, •••?

PROCESS Explanation. — Since the series 6, 9, 12, ••• is

,_ . .. an A.P. the common difference of whose terms is

~ ^ ^ 3, by substituting 6 for a, 10 for n, and 3 for d in

= t) -f (lU — 1)6 tjjg formula for the last term, the last term is found

= 33 to be 33.

2. Find the 20th term of the series 7, 11, 15, •••.

3. Find the 16th term of the series 2, 7, 12, •••.

4. Find the 24th term of the series 1, 16, 31, •••.

5. Find the 18th term of the series 1, 8, 15, •••.

6. Find the 13th term of the series —3, 1, 5, • •.

7. Find the 49th term of the series 1, IJ, If---.

4

396 PROGRESSIONS

8. Find the 15th term of the series 45, 43, 41, ..«. Suggestion. — The common difference is — 2.

9. Find the 10th term of the series 5, 1, — 3, •••.

10. Find the 16th term of the series a, 3 a, 5 a, •••.

11. Find the 7th term of the series x — 3y,x — 2y,-",

12. A body falls 16^2- feet the first second, 3 times as far the second second, 5 times as far the third second, etc. How far will it fall during the 10th second ?

511. To find the sum of n terms of an arithmetical series. ^ Let a represent the first term of an A.P., d the common dif- ference, I the last term, n the number of terms, and s the sum of the terms.

Write the sum of n terms in the usual order and then in the reverse order, and add the two equal series ; thus,

s = a+(a + c^)+(ot + 2d)H-(a + 3c?)-f ••• +?.

s= I + (I - d) + (I - 2 d) + (l - 3 d) + '" +a.

2s=(a + r) + (a4-0+(« + 0 + (« + 0+- + (a + 0- .\2s = n{a-{-l).

s = ^(a + l),ovn(^. (II)

EXERCISES

512. 1. Find the sum of 20 terms of the series 2, 5, 8, •

PROCESS

Z = a+(n-l)d = 2 + (20-l)x3 = 59

n(9^I\ = 20(^y610

Explanation. — Since the last term is not given, it is found by foi mula I and substituted for I in the formula for the sum.

PROGRESSIONS 397

Find the sum of :

2. 16 terms of the series 1, 5, 9, •••.

3. 10 terms of the series — 2, 0, 2, •••.

4. 6 terms of the series 1, 3^, 6, •••.

5. 8 terms of the series a, 3 a, 5 a, •••,

6. n terms of the series 1, 7, 13, •••.

7. a terms of the series Xy x-\-2 a, ••••

8. 7 terms of the series 4, 11, 18, •••.

9. 10 terms of the series 1, — 1, — 3, •••.

10. 10 terms of the series 1, J, 0, •••.

11. How many strokes does a common clock, striking hours, lake in 12 hours ?

12. A body falls 16^2^ feet the first second, 3 times as far le second second, 5 times as far the third second, etc. How

[far will it fall in 10 seconds ?

13. Thirty flower pots are arranged in a straight line 4 feet fcpart. How far must a lady walk who, after watering each )lant, returns to a well 4 feet from the first plant and in line

[with the plants, if we assume that she starts at the well ?

14. How long is a toboggan slide, if it takes 12 seconds for toboggan to reach the bottom by going 4 feet the first second id increasing its velocity 2 feet each second ?

15. Starting from rest, a train went .18 feet the first second, .54 feet the next second, .90 feet the third second, and so on, preaching its highest speed in 3 minutes 40 seconds. How far [did the train go before reaching top speed ?

16. In a potato race each contestant has to start from a mark and bring back, one at a time, 8 potatoes, the first of which is 6 feet from the mark and each of the others 6 feet farther than the preceding. How far must each contestant go in order to finish the race ?

398 PROGRESSIONS

513. The two fundamental formulae,

(I) Z = a + (n - l)d and (II) s = ^(a + Z),

contain j^ve elements, a, (?, Z, n, and s. Since these formulae are independent simultaneous equations, if they contain but two unknown elements they may be solved. Hence, if any three of the five elements are known, the other two may be found.

EXERCISES

514. 1. Given d = 3, Z = 58, s = 260, to find a and n.

Solution

Substituting the known values in (I) and (II), we have

58 = a+ (w-l).3, or a + 3w=61; (1)

and 260 = 1 7i(a + 58) , or a^i + 58 w = 520. (2)

Solving, n = ^^ or 5,

and, rejecting n = ^f^, a = 46.

Since the number of terms must be a positive integer, fractional or negative values of n are rejected whenever they occur.

2. Given a = 11, c? = — 2, s = 27, to find the series.

Solution Substituting the known values in (I) and (II), we have

Z = ll + (w- l)(-2), or Z=13-2n; (1)

and 27 = ^ n(ll + Z), or 54 = 11 n + In. (2)

Solving, n = 3 or 9 and Z = 7 or — 5. (3)

Hence, the series is 11, 9, 7, or 11, 9, 7, 5, 3, 1, _ l, _ 3, - 6.

3. How many terms are there in the series 2, 6, 10, •••, 66?.^

4. What is the sum of the series 1, 6, 11, •••, 61?

1

{

PROGRESSIONS 399

5. How many terms are there in the series —1, 2, 5, •••, if the sum is 221 ?

6. Find n and s in the series 2, 9, 16, •••, 86.

7. Find I and s in — 10, —81 — 7, ••• to 10 terms.

8. The sum of the series--, 22, 27, 32, ... is 714. If there are 17 terms, what are the first and last terms ?

9. If s = 113|, a=:^, andd = 2, findw.

10. What is the sum of the series —16, —11, —6, •••,34:?

11. What is the sum of the series •••, — 1, 3, 7, •••, 23, if the number of terms is 16 ?

12. What are the extremes of the series •••,8, 10,12, •••, if s = 300, andn = 20?

13. Find an A. P. of 14 terms having 10 for its 6th term, 0 for its 11th term, and 98 for the sum of the terms.

14. Find an A. P. of 15 terms such that the sum of the 5th, 6th, and 7th terms is 60, and that of the last three terms, 132.

From (I) and (II) derive the formula for :

15. I in terms of a, n, s. 18. d in terms of a, n, s.

16. s in terms of a, d, I. 19. d in terms of I, n, s.

17. a in terms of d, n, s. 26. n in terms of a, I, s.

515. To insert arithmetical means.

EXERCISES

1. Insert 5 arithmetical means between 1 and 31. Solution

Since there are 6 means, there must be 7 terms. Hence, in Z = a + (n — l)d, Z = 31, a = 1, » = 7, and d is unknown. Solving, d = 5,

Hence, 1, 6, 11, 16, 21, 26, 31 is the series.

400 PROGRESSIONS

2. Insert 9 arithmetical means between 1 and 6.

3. Insert 10 arithmetical means between 24 and 2.

4. Insert 7 arithmetical means between 10 and — 14.

5. Insert 6 arithmetical means between — 1 and 2.

6. Insert 14 arithmetical means between 15 and 20.

7. Insert 3 arithmetical means between a —h and a -f b.

516. If A is the arithmetical mean between a and h in the

series . , i

a, A,b, I

by §508, A-a = b-A.

,'.A = ^±^, That is, I

Principle. — 7%e arithmetical mean between two numbers is equal to half their sum.

EXERCISES

517. Find the arithmetical mean between :

1. I and J. ^ x + y ^^^ x-y

2. a + 6anda-6. ^~^ ^ + ^ ^

3. (a + &)2and(a-6/. ^- ^-^^^^^J^'*

Problems

518. Problems in arithmetical progression involving two unknown elements commonly suggest series of the form,

x,x-^y,x-}-2yjX + Sy, etc.

Frequently, however, the solution of problems is more readily accomplished by representing the series as follows :

1. When there are three terms, the series may be written,

x-y,x,x+y. ,

2. When there are Jive terms, the series may be written,

x-2y, x-y, X, x-\- y,x + 2y.

1

PROGRESSIONS 401

3. When there are four terms, the series may be written,

The sum of the terms of a series represented as above evidently con- tains but one unknown number.

1. The sum of three numbers in arithmetical progression is 30 and the sum of their squares is 462. What are the numbers ?

AJ-UlJUUt/JLO i	Solution
Let the series be	x-y,x,x + y.
Then,	(x-y) + a;+(x+y)=30.	0)
and	(a;-y)2 + x2 + (x + y)2 = 462.	(2)
From (1),	3a; = 30;	(3)
whence,	a; = 10,	(4)
From (2),	3a;2+2!/2=:462.	(5)
Substituting (4) in (5), 2 y^ = 162.
Solving,	2/ = ±9.

Forming the series from ic = 10 and y = i 9, the terms are 1, 10, 19 or 19, 10, 1.

2. The sum of three numbers in arithmetical progression is 18, and their product is 120. What are the numbers ?

3. The sum of three numbers in arithmetical progression is 21, and the sum of their squares is 155. Find the numbers.

4. There are three numbers in arithmetical progression the sum of whose squares is 93. If the third is 4 times as large as the first, what are the numbers ?

5. Find the sum of the odd numbers 1 to 99, inclusive.

6. The product of the extremes of an arithmetical progres- sion of 10 terms is 70, and the sum of the series is 95. What are the extremes ?

7. Fifty-five logs are to be piled so that the top layer shall consist of 1 log, the next layer of 2 logs, the next layer of 3 logs, etc. How many logs must be placed in the bottom layer ?

milne's stand, alg. — 26

402 PROGRESSIONS

8. It cost Mr. Smith $ 19.00 to have a well dug. If the cost of digging was ^1.50 for the first yard, ^1.75 for the second, $ 2.00 for the third, etc., how deep was the well ?

9. How many arithmetical means must be inserted between 4 and 25, so that the sum of the series may be 116 ?

10. Prove that equal multiples of the terms of an arith- metical progression are in arithmetical progression.

11. Prove that the difference of the squares of consecutive integers are in arithmetical progression, and that the common difference is 2.

12. Prove that the sum of n consecutive odd integers, beginning with 1, is n\

GEOMETRICAL PROGRESSIONS

519. A series of numbers each of which after the first is derived by multiplying the preceding number by some con- stant multiplier is called a geometrical series, or a geometrical progression.

2, 4, 8, 16, 32 and a*, a^, a^, a are geometrical progressions.

In the first series the constant multiplier is 2 ; in the second it is - •

G.P. is an abbreviation of the words geometrical progression.

520. The constant multiplier is called the ratio.

It is evident that the terms of a geometrical progression increase or decrease numerically according as the ratio is numerically greater or less than 1.

521. To find the nth, or last, term of a geometrical series.

Let a represent the first term of a G.P., r the ratio, n the number of terms, and I the last, or nth, term.

Then, the series is a, ar, ar^, ai^, ar*, •••.

Observe that the exponent of r is one less than the number of the term : that is,

I = ar--' (I)

(

PROGRESSIONS 403

EXERCISES

522. 1. Find the 9th term of the series 1, 3, 9, ....

Explanation. — In this exercise a = 1, r = 3, and

,; = a?-""^ n = 9.

1 V Q8 Substituting these vahies in the formula for Z, the

~~ ^ ^ ^ last term is found to be 6661.

= 6561

2. Find the 10th term of the series 1, 2, 4, •••.

3. Find the 8th term of the series J, |, 1, •••.

4. Find the 9th term of the series 6, 12, 24, ....

5. Find the 11th term of the series ^, 1, 2, •••.

6. Find the 7th term of the series 2, 6, 18, .••.

7. Find the 6th term of the series 4, 20, 100, ....

8. Find the 6th term of the series 6, 18, 54, ....

9. Find the 10th term of the series 1, |, -J, ....

10. Find the 10th term of the series 1, |, f, •••.

11. Find the 8th term of the series |, ^, |, ••••

12. Find the 11th term of the series d^^h, d^^b% ....

13. Find the nth term of the series 2, v^, 1, ....

14. If a man begins business with a capital of f 2000 and doubles it every year for 6 years, how much is his capital at the end of the sixth year ?

15. The population of the United States was 76.3 millions in 1900. If it doubles itself every 25 years, what will it be in the year 2000 ?

16. A man's salary was raised J every year for 5 years. If his salary was $ 512 the first year, what was it the sixth year?

17. The population of a city, which at a certain time was 20,736, increased in geometrical progression 25 % each decade. What was the population at the end of 40 years ?

404 PROGRESSIONS

18. A man who wanted 10 bushels of wheat thought $ 1 a bushel too high a price ; but he agreed to pay 2 cents for the first bushel, 4 cents for the second, 8 cents for the third, and so on. How much did the last bushel cost him ?

19. The machinery in a manufacturing establishment is valued at $20,000. If its value depreciates each year to the extent of 10 % of its value at the beginning of that year, how much will the machinery be worth at the end of 5 years ?

20. From a grain of corn there grew a stalk that produced an ear of 150 grains. These grains were planted, and each produced an ear of 150 grains. This process was repeated until there were 4 harvestings. If 75 ears of corn make 1 bushel, how many bushels were there the fourth year?

523. A series consisting of a limited number of terms is called a finite series.

524. A series consisting of an unlimited number of terms is called an infinite series.

525. To find the sum of a finite geometrical series.

Let a represent the first term, r the ratio, n the number of terms, I the nth, or last, term, and s the sum of the terms.

Then, s = a -^ ar + ar^ + ai^ -{- ••• +ar''-\ (1)

(1) X r, rs — ar -\- ar^ -\- ai^ + • " ^ ar""'^ + ar"", (2)

(2)-(l), 5(r-l) = ar"-a.

.-. s = ^^"-^. (II)

r — 1

But, since ar""^ = Z, ar'' = rl

Substituting rl for ar** in (II)

rtzS^, or ?^=I?. (Ill)

r — 1 1—r

PROGRESSIONS 405

EXERCISES

526. 1. Find the sum of 6 terms of the series 3, 9, 27, •••.

8 =

PROCESS ar^ — a Explanation. — Since the first term a, the

r — 1 ratio r, and the number of terms n, are known,

fornmla II, which gives the sum in terms of a, _ o Xo —o _ 1 092 r, and 7i, is used. 3-1

2. Find the sum of 8 terms of the series 1, 2, 4, •••.

3. Find the sum of 8 terms of the series 1, |, J, •••.

4. Find the sum of 10 terms of the series 1, 1^, 2J,

5. Find the sum of 7 terms of the series 2, — J, |, •

6. Find the sum of 12 terms of the series — |, J-, — i

7. Find the sum of 7 terms of the series 1, 2 ic, 4 x^.

8. Find the sum of 7 terms of the series 1, — 2 a;, 4 a;^,

9. Find the sum of n terms of the series 1, af, x\ •••.

10. Find the sum of n terms of the series 1, 2, 4, •••.

11. Find the sum of n terms of the series 1, ^, i, •••.

12. The extremes of a geometrical series are 1 and 729, and tne ratio is 3. What is the sum of the series ?

13. What is the sum of the series 3, 6, 12, •••, 192 ?

14. What is the sum of the series 7, •••, —56, 112,-224?

527. To find the sum of an infinite geometrical series.

If the ratio r is numerically less than 1, it is evident that the successive terms of a geometrical series become numeric- ally less and less. Hence, in an infinite decreasing geomet- rical series, the nth term I, or ar"~^ can be made less than any assignable number, though not absolutely equal to zero.

406 PROGRESSIONS

Formula (III), page 404, may be written,

a rl

s =

1-r 1

I

Since by taking enough terms I and, consequently, ?•? can be made less than any assignable number, the second fraction may be neglected.

Hence, the formula for the sum of an infinite decreasing

geometrical series is

(IV)

V		—1-/
		EXERCISES
528. 1.	Find the sum	of the series	^f TQ) TOTT?	....
		Solution
Substituting 1 for a and	,1^ for r in (IV),
		s- 1 - ^	_io
		1-tV ^	9'
Find the value of :
2.	i+i+i+-	••.	5.	-4"	1-	■i	—
3.	3+f+A+	•..	6.	-2 +	1-	^\	-h
4.	1-i+i-	• •.	7.	100-	10 + 1	—

8. l + ic + a^ + a^H , whenic = .9.

9. 1 — 07 + x*^ — ar^+ •••, when 07 = |. 10. Find the value of the repeating decimal .185185185 •■

Solution

.186 and

Since .186185185-. = .185 + .000185 + .000000185 + -., a r=.0{)l. Substituting in (IV), .185185185 ... =s= '^^^ =—- 1 — .001 27 Find the the value of :
11. .407407....		14. .020303....
12. .363636....		15. .007007....
13. 1.94444...		16. 5.032828....

PROGRESSIONS 407

529. To insert geometrical means between two terms.

EXERCISES

1. Insert 3 geometrical means between 2 and 162.

PROCESS Explanation. — Since there are three means, there are

^__^yn-i five terms, and w — 1 = 4. Solving for r and neglecting

1 fi2 — 2 4 imaginary values, r = ± 3.

~ Therefore, the series is either 2, 6, 18, 54, 162 or 2, - 6,

r = ± 3 18, - 54, 162.

2. Insert 3 geometrical means between 1 and 625.

3. Insert 5 geometrical means between 41 and ^f f ^. .

4. Insert 4 geometrical means between \*/- and ||.

5. Insert 4 geometrical means between 5120 and 5.

6. Insert 4 geometrical means between 4V2and 1.

7. Insert 5 geometrical means between a^ and If.

8. Insert 4 geometrical means between x and — y.

530. If G is the geometrical mean between a and h, in the

series ^ ,

a, G, by

by §519, ^ = 1.

a G

G= ± Vab. That is,

Principle. — The geometrical mean between two numbers is equal, to the square root of their product.

Observe that the geometrical mean between two numbers is also their mean proportional.

EXERCISES

531. Find the geometrical mean between :

1. 8 and 50. 4. (a + hf and (a - b)\

2. \ and 3|. 2,1- j, , ^2

3. Hi and |. ^- ,i^3^^"^a6-6^

408 PROGRESSIONS i

i

532. Since formula I with formula IT, or III, whicli is equivalent to II, forms a system of two independent simulta- neous equations containing live elements, if three elements are known, the other two may be found by elimination.

Note. — Solving for «, since it is an exponent, requires a knowledge of logarithms (§§ 558-598), except in cases where its value may be deter- mined by inspection. Only such cases are given in this chapter.

Problems

533. 1. Given r, I, and s, to find a.

2. The ratio of a geometrical progression is 5, the last term is 625, and the sum is 775. What is the first term ?

3. The ratio of a geometrical progression is J^-, the sum is ^, and the series is infinite. What is the first term ?

4. Find I in terms of a, r, and s.

5. Find the last term of the series 5, 10, 20, •••, the sum of whose terms is 155. j

6. If I+1V2 + 1+ ... =1|(1+V2), what is the last term, and the number of terms ?

7. Deduce the formula for r in terms of a, I, and s.

8. If the sum of the geometrical progression 32, •••, 243 is 665, what is the ratio ? Write the series.

9. The sum of a geometrical progression is 700 greater than the first term and 525 greater than the last term. What is the ratio ? If the first term is 81, what is the progression ? .

10. Deduce the formula for r in terms of a, n, and I.

11. The first term of a geometrical progression is 3, the last term is 729, and the number of terms is 6. What is the ratio ? Write the series. m

12. Find I in terms of r, n, and s. ^

13. A sled went 100 feet the first second after reaching the foot of a hill. How far did it go on the level, if its velocity decreased each second ^ of that of the previous second ?

PROGRESSIONS 409

14. Under normal conditions the members of a certain species of bacteria reproduce by division (each individual into two) every half hour. If no hindrance is offered, how many bacteria will a single individual produce in 8 hours ?

15. A ball thrown vertically into the air 100 feet falls and rebounds 40 feet the first time, 16 feet the second time, and so on. What is the whole distance through which the ball will have passed when it finally comes to rest ?

16. Show that the amount of $1 for 1, 2, 3, 4, 5 years at compound interest varies in geometrical progression.

17. Show that equal multiples of numbers in geometrical- progression are also in geometrical progression.

18. The sum of three numbers in geometrical progression is 19, and the sum of their squares is 133. Find the numbers.

Suggestion. — When there are but three terms in the series, they^may be represented by ic^, xy, y^, or by aj, Vxy, y.

19. The product of three numbers in geometrical progres- sion is 8, and the sum of their squares is 21. What are the three numbers ?

20. The sum of the first and second of four numbers in geo- metrical progression is 15, and the sum of the third and fourth is 60. What are the numbers ?

Suggestion. — Four unknown numbers in geometrical progression may

/V.2 y-2

be represented by — , x, y, ^. y X

21. From a cask of vinegar J was drawn off and the cask was filled by pouring in water. Show that if this is done 6 times, the contents of the cask will be more than ^^ water.

22. If the quantity, and correspondingly the pressure, of the air in the receiver of an air pump is diminished by -^ of itself at each stroke of the piston, and if the initial pressure is 14.7 pounds per square inch, find, to the nearest tenth of a pound, what the pressure will be after 6 strokes.

410 PROGRESSIONS 1

23. A man bouglit a farm for $5000, agreeing to pay prin- cipal and interest in five equal annual installments. Find the annual payment, interest at 6 %.

Solution

By the conditions of the problem the equal payments are to include the interest accrued at the end of eacli year plus a portion of the principal.

The principal for the second year will be less than the principal for the first year by the portion of the principal paid at the end of the first year ; therefore, the interest to be paid at the second payment will be less than the interest paid at the first payment by the interest for 1 year upon tlie first portion of the principal paid, or 6 % of the portion of the principal paid the first year.

Since the payments are to be equal, the portion of the principal to be paid at the second payment must be as much more than the portion paid at the first payment as the interest is less than the interest paid at the first payment ; that is, it must be 6% more than, or 1.06 of, the portion of the principal first paid.

By reasoning in the same way regarding subsequent payments, the third portion of the principal paid will be found to be 1.06 of the second, the fourth 1.06 of the third, and the fifth 1.06 of the fourth ; that is, the portions of the principal paid form a G.P., in which r = 1.06, w = 5 (the number of payments), and s = $ 5000. We desire to find a.

Substituting the known values in (I) and (III), we have

? = a 1.065-1, or Z = 1.064 a; (1)

and $5000 = 1^^^, or Z=?L±i|^2<^. (2)

Eliminating I from (1) and (2) and solving for a, we have

$5000x.06^^g3eQ3+^ 1.065-1

That is, the first portion of the principal paid = $886.98 ; but the first year's interest =Q% of $5000, or $300 ; hence, the entire first payment = $886.98 + $300 = $1186.98, which is also each annual payment.

24. A man borrowed $1500, agreeing to pay principal and interest at 6 % in four equal annual installments. Find the sum to be paid each year.

25. A father bequeathed to his son $ 10,000, the bequest and interest at 4 % to be paid in six equal annual installments. Find the annual payment.

INTERPRETATION OF RESULTS

534. A number that has the same value throughout a dis- cussion is called a constant.

Arithmetical numbers are constants. A literal number is constant in a discussion, if it keeps the same value throughout that discussion.

535. A number that under the conditions imposed upon it may have a series of different values is called a variable.

The numbers .3, .33, .333, .3333, ... are successive values of a variable approaching in value the constant |.

536. When a variable takes a series of values that approach nearer and nearer a given constant without becoming equal to it, so that by taking a sufficient number of steps the difference between the variable and the constant can be made numerically less than any conceivable number however small, the constant is called the limit of the variable, and the variable is said to approach its limit.

This figure represents graphically a variable x ap- o -y, x, x, X

proaching its limit OX =2. 1 \ \ \

The first value is OXi

= 1 ; the second is OX2 = 1^ ; the third is OX3 = If ; etc.

At each step the difference between the variable and its limit is diminished by half of itself. Consequently, by taking a sufficient number of steps this difference may become less than any number, however small, that may be assigned.

537. A variable that may become numerically greater than any assignable number is said to be infinite. The symbol of an infinite number is 00 .

411

412 INTERPRETATION OF RESULTS

538. A variable that may become numerically less than any assignable number is said to be infinitesimal.

An infinitesimal is a variable whose limit is zero.

The character 0 is used as a symbol for an infinitesimal num- ber as well as for absolute zero, which is the result obtained by subtracting a number from itself.

539. A number that cannot become either infinite or infini- tesimal is said to be finite.

THE FORMS a X 0, ?, ^, -^, ?, ?1 a 0 00 0 00

540. The results of algebraic processes may appear in the

forms, a X 0, -,-,—,-,— , etc., which are arithmetically a 0 00 0 00

meaningless ; consequently, it becomes important to interpret

the meaning of such forms.

541. Interpretation of a x 0.

1. Let 0 represent absolute zero, defined by the identity,

0 = n - w. (1)

Multiplying a = a by (1), member by member, Ax. 3, we have a X 0 = a(n — n) = an — an by def. of zero, = 0. That is.

Any finite number multiplied by zero is equal to zero.

2. Let 0 represent an infinitesimal, as the variable whose successive values are 1, .1, .01, .001, •••.

Then, the successive values of a x 0 are (§ 81)

a, .1 a, .01 a, .001 a, •••. Hence, a X 0 is a variable whose limit is absolute zero. That is,

Any finite number multiplied by an infinitesimal number is equal to an infinitesimal number.

INTERPRETATION OF RESULTS 413

542. Interpretation of -. a

1. Let 0 represent absolute zero, defined by the identity,

0 = 7i — w.

0 n n

Dividing by a,

a a a^

but by def . of zero, '^-'^ = 0, a a

Hence, Ax. 5, - = 0. That is,

a

Zero divided by any finite number is equal to zero,

2. Let 0 represent an infinitesimal^ as the variable whose successive values are 1, .1, .01, .001, •••.

mi. ^1, • 1 *0 1 .1 .01 .001

Then, the successive values or - are -, — , — , , •••;

a a a a a

whence, - is a variable whose limit is absolute zero. a

Hence,

Any infinitesimal number divided by a finite number is equal to

m infinitesimal number.

543. Interpretation of ^.

The successive values of the fractions, -, — , — , , etc.,

2 .2 .02 .002

[are .5, 5, 50, 500,- etc., and they continually increase as the .denominators decrease.

In general, if the numerator of the fraction - is constant while

X

the denominator decreases regularly until it becomes numer- ically less than any assignable number, the quotient will i-n- crease regularly and become numerically greater than any assignable number.

.-. - = 00 . That is, 0

If a finite number is divided by an infinitesimal number, the

quotient will be an infinite number.

414 INTERPKETATION OF RESULTS

544. Interpretation of — .

The successive values of the fractions, -, — , — -, -— — ,

' 2' 20' 200' 2000' etc., are .5, .05, .005, .0005, etc., and they continually decrease as the denominators increase.

In general, if the numerator of the fraction - is constant

X

while the denominator increases regularly until it becomes numerically greater than any assignable number, the quotient will decrease regularly and become numerically less than an} assignable number. i

.-. -=0. That is, i

oo

If a finite number is divided by an infinite number ^ the quotient will be an infinitesimal number.

545. Interpretation of -. '

Let 0 represent absolute zero.

Then, if a is any finite number, § 541,

ax 0 = 0; I

whence, - = a. That is,

' 0 '

When 0 represents absolute zero, - is the symbol of an indeter- minate number.

546.	Interpretation of — . 00
Let a	represent	any finite number and	X any	number what-
ever.		a
Then,		X a l~x X	•!=«•		(1)

I

If X decreases regularly until it becomes numerically less than any assignable number (§ 543), - and - each become oo .

INTERPRETATION OF RESULTS 415

Consequently, (1) becomes — = a, any finite number. Hence, — is the symbol of an indeterminate number.

547. Since (§ 543) ^ is infinite and (§ 545) - is indetermi- nate, it is seen that axiom 4 (§ 68) is not applicable when the divisor is 0; that is, it is not allowable to divide by absolute zero.

r The student may point out the inadmissible step or fal- lacy in: ^ o^ o -1^ •^ 7 a; — 35 = 3 a; — 15,

7(x - 5) = 3(x - 5).

.-. 7 = 3.

Suggestion. — Solve the equation to find what divisor has been used.

548. Fractions indeterminate in form.

Some fractions, for certain values of the variable involved,

give the result ^, which, however, is indeterminate only inform,

because a definite value for the fraction may often be found.

For example, when ic = 1, by substituting directly, ^ ~ = -.

a; — 1 0 Though ^_ZlI = (x + l)(x- 1) ^ a; + 1, it is not allowable to perform X- 1 x—l

this operation in finding the value of the fraction when x = 1, that is, when X — 1 = 0, for (§ 547) it is not allowable to divide by absolute zero.

^2 1

However, since the value of is always the same as the value of

x — l

♦JB + 1 so long as a; 9^ 1, let X approach 1 as a limit.

But (§ 536) X cannot become 1, and it is allowable to divide by x - 1.

y.2 1

Now as X approaches 1 as a limit, approaches x+ 1, or 2, as a

x — l

limit, and so 2 is called the value of the fraction. That is,

The value of such a fraction for any given value of the vari- able involved is the limit that the fraction approaches as the variable approaches the given value as its limit.

THE BINOMIAL THEOREM

549. The Binomial Theorem derives a formula by means of which any indicated power of a binomial may be expanded into a series.

POSITIVE INTEGRAL EXPONENTS

550. By actual multiplication, (a-\-xy=a^-i-2ax + ay^.

(a 4- x) ^ = a* + 4 A + 6 a^a^ 4- 4 aa^ + x\

These powers of (a + x) may be written, respectively i

(a + xy = a- + 2ax + ^ x^.

(a + xy = a'-\-Sa'x-^^aa^ + l^^a^.

f , x4 4,13 , 4 . 3 2^ , 4 • 3 . 2 3,4.3.2.14

{a _i- a;)^ = a^ + 4 a^x + - — - « V + - ,. ,. ax"- + a;^

If the law of development revealed in the above is assumed to apply to the expansion of any power of any binomial, as the nth power of (a -f x), the result is a

=a'+«a'-'x+"<"-^)a-^a^+"^"-^K"-2)„,-3^^.... (i) \ * Z \ ' Z • o

From formula (I) it is evident that in any term :

1. The exponent of a; is 1 less than the number of the term.

Hence, the exponent of x in the (r + l)th term is r.

416

THE BINOMIAL THEOREM 417

2. The exponent of a is w minus the exponent of x. Hence, the exponent of a in the (?• + l)th term is yi — r.

3. The number of factors in the numerator and in the denominator of any coefficient is 1 less than the number of the term.

Hence, the coefficient of the (?- + l)th term has r factors in the numerator and r factors in the denominator. Therefore, the (r + l)th, or general, term, is

n{n—l){n—2) "• to r factors „_;. ,. ^.j.

1 .2. 3-. -tor factors ^ ^ • ^ )

When there are two factors in the numerator, the last is 71 — 1 ; when there are three factors, n — 2; when there are four factors, ?^ — 3, etc. Therefore, when there are r factors, the last is 71 — (r — 1), or 71 — r + 1. Hence, (1) may be written

Therefore, the full form of (I) is

1-2 1.2.3

n(n — l)(7i — 2) '"(n-r-\-l) n_^ , , , „

1.2.3.-.r

This is called the binomial formula. It will now be proved be true for positive integral exponents.

551. Since it has already been proved, by actual multiplica- lion (§ 550), that the binomial formula is true for the second, hird, and fourth powers of a binomial, it remains to discover 'hether it is true for powers higher than the fourth. If the binomial theorem, when assumed to be true for the ith power, can be proved to be true for the (iz + l)th power, Tsince it is known to be true when the r^th power is the fourth power, it will then have been proved to be true for the Jifth power ; also for the sixth power, being true for the fifth power ; and in like manner for each succeeding power. milne's stand, alg. — 27

418

THE BINOMIAL THEOREM

Therefore, it remains to prove that if (I) is true for the 7<th power, it will hold true for the (??, + l)th power.

The (n -\- l)th power of (a + x) may be obtained from the nth power by multiplying both members of (I) by (a + x).

Then, we have

=a"+^4-n

+1

a-x 1 ^(''•"	-1)
a x-f-	• 2
a"a;+	n

a'-V4-

n(n-l){n-2) 1-2.3

yi(yi— 1)

1-2

a^

"1

Collecting the coefficients of like powers of a and x, we have Coefficient of a" a; = n +1.

Coefficient of a^-'x'^ n{n-l) 1-2

_n^ — 7i-{-2n_ (n + l)yi

1.2

1 .2

Coefficient of a«- V = <n-l)(n-2) n(n-l)

1.2-3 1-2

,-. (a + a;)"+i =a''+i + (^ + l)a''a; + (^ + l>^n-i^

n^ — n
1.2.3
(n 4- 1) n (ri -	■1)
1.2.3
1Wr« 1 (^ + 1>	r/"-

^(n4-l)ri(n-l)^„_,^3^ J.. ^ . o

(II)

Upon comparison it may be seen that (II) and (I) have the same form, w + 1 in one taking the place of n in the other. That is, (II) and (I) express the same law of formation.

Therefore, if the formula is true for the nth power, it holds true for the (n + l)th power.

By actual multiplication (§ 550) the formula is known to be true for the fourth power. Consequently, it is true fo;

THE BINOMIAL THEOREM 419

thQ fifth power; and then being true for the fifth power, it is true for the sixth power ; and so on for each succeeding power.

Hence, the binomial formula is true for any positive integral exponent.

This proof is known as a proof by mathematical induction.

552. If — a; is substituted for x in (I), the terms that con- tain the odd powers of — a; will be negative, and those that contain the even powers will be positive. Therefore,

= a» -"««-.! + '^i^^ a'-V - »(»--!)(«- 2) ^,_3^ . . ._ ^ jj j^ If a = 1, (I) becomes

553. From (I) it is seen that the last factor in the numerar tor of the coefficient is n for the 2d term, n — 1 for the 3d term, n — 2 for the 4th term, n — (n — 2), or 2, for the nth. terra, and n — (n — 1), or 1, for the (n + l)th term ; and that the coefficient of the (n + 2)th terra, and of each succeeding term, contains the factor n — n, or 0, and therefore reduces to 0.

' Hence,

When n is a positive integer, the series formed by expanding (a + xy is finite and has n -{-1 terms.

554. By formula (I) when n is a positive integer, (a + xy

n I n-l . ^(^ — 1) n-2 2 . i n(n — l) •••2-1 „

1-2 1 • 2.-- (7i — l)n

^^"^^^"n, n-l ,w(^-l)«-2 2, , 71 (^ - 1) • • • 2 • 1 „

= ic"-|- nx^^a -\ — ^^ ^a?" V -\ 1 — ^^ j a\

1 • 2 1-2 •"(n — l)n

A comparison of the two series shows that :

The coefficients of the latter half of the expansion of (a -{-xYy. when n is a positive integer, are the same as those of the first halff written in the reverse order.

420 THE BINOMIAL THEOREM

EXERCISES

555. 1. Expand (Sa-2 by.

Solution. — Substituting 3 a for a, 2 b for x, and 4 for n in (IH),

(3 a - 2 6)4 = (3 ay - 4 (3 ay (2 6) + il| (3 ay (2 by

1 • ^

-'^''^'^(Sa)(2by + ^'^'l'](2by = 81 a* - 216 a^b + 216 a^&a _ 96 ab^ + 16 6*.

2. Expand Z^^ + ftxY.

Suggestion. — Since ^^ + bxY= [^(1+2 x)T= |^(1 + 2 x)5,

55

(1+2 xy may be expanded by (IV), and the result multiplied by — <

Expand :
3. {b-7iy.	10.	(-I-)"	15.	(^-4)"
4. (1 + a-y.				n— 1 1
5. (2-3a;)6.	11.	(M)*	16.	(a; " _ x^.
6. (r^-a;)«.	12.		17.	(ax-'-bVx)\
7. (x + x-y.		18.	Na Vb Y Wb V^'l
8. (2a+V^)3. 9 (a4-«Va)^	13. 14.	(2V2-^/3)«.	19.	(Cy^yS^IJ'

556. To find any term of the expansion of (a + jr)**.

Any term of the expansion of a power of a binomial may be obtained by substitution in (1) or (2), § 550.

In the expansion of a power of the difference of two numbers (a — x)", since the exponent of x in the (r + l)th term is r, the sign of the general term is + if r is even, and — if r is odd.

THE BINOMIAL THEOREM 421

EXERCISES

557. 1. Find the 12th term of (a - &)".

Solution

12th term = 14.13.12.11.10.9.8.7.6.6.4

1.2. 3. 4. 6. 6.7 -8 -9.10. 11 ^ ^

= - ^^'^^'^^ a36ii = - 364 a^ftii. 1.2-3

Or, since there are 15 terms, the coefficient of the 12th term, or the 4tli term from the end, is equal to that of the 4th temi from the beginning.

.-. 12th terra = - ^^-^^'^^ a^ft" = - 364 a^b^\ 1.2.3

Without actually expanding, find the :

2. 4th term of (a + 2)io. 5. 20th term of (1 + x)^.

3. 8th term of (x - yf\ 6. 18th term of (1 - 2 xf^.

4. 5th term of (a; - 2 yy\ 7. 13th term of {p? - a'^^

8. Find the middle term of (a -f 3 hf,

9. Find the 6th term

of (-+9'

10. Find the middle term of ( - - ^

\y ^j

11. Find the two middle terms of {- - -Y.

\h a)

12. In the expansion of (a^ -f- a;)", find the term containing o^^. Solution. — Since (a;^ + a;)" = \x^ ( 1 + -^ T = x^^ /'l + 1\ , every

term of the series expanded from ( 1 + - ) will be multiplied by a;22.

Hence, the term sought is that which contains ( - j , or — ; that is, the (7 + l)th, or 8th term. ^^^ ^^

8th term = ^^^V,i:^l^^ f 1 V = ZZ^x^K 1.2.3.4 \x)

13. Find the coefficient of a^ in the expansion of (a^ -f of.

14. Find the term containing Q}^h^ in the expansion of (a — &)^ and obtain a simple expression for it when h = 15~^(143^ a*)~^.

I

422 THE BINOMIAL THEOREM

The binomial formula is true for the expansion of a binomial ^ when the exponent is negative or fra'itional, provided the tirst term of the binomial is numerically greater than the second. In such cases the expansion is an infinite series. For a more extended treatment of this subject, see the author's Advaiiced Algebra.

15. Expand (1 — y)~^ and find its (?' + l)th term. Solution. — Substituting 1 for a, y for x, and — 1 forw in (III),

(l-y)-i=l-i-(-l)l-2y+-Y-^)l-ay2--K-2)(^-3)l-4y3+...

= 1 + y + y2 + y3 + ....

The (r + l)th term is evidently y.

Since (1 — ?/)-i = , the above expansion of (1 — 2/)-i may be

verified by division. ~ ^

16. Expand (a + xy to five terms and find the 10th term.

Expand to four terms :

17. (l-a)-\ 21. (a + 6)i 25. (l + x)i

18. (1 + «)-'. 22. -\/(a-by. 26. {l-x)-K

19. (a-b)K 23. V(9-a;)^ 27. (J-x'^)^-

20. V4 + a;. 24. (a + &)~^. 28. (a^-x^)-\

29. Find the square root of 24 to three decimal places. Solution, v^ = (24)^ = (25 - 1)^ = (25)^(1 - ^^ = 6(1- ^)^

L 2V25y 1-2 V25; 1.2-3 1,25^ J

= 5 _ .1 _ .001 - .00002 = 4.89898 - = 4.899, nearly.

Find, to three decimal places, the value of :

30. V5. 32. v'M. 34. -v/9.

31. VTr. 33. ■\/25. 35. \/30.

LOGARITHMS

558. Early in the seventeenth century a scheme was devised to simplify long computations by representing all real positive numbers as powers of some particular number. The exponents of these powers, called logarithms, were arranged in tables for convenient reference; and in accordance with the principles of exponents, multiplication was replaced by addition, division by subtraction, involution by a single simple multiplication, and evolution by a single simple division.

Lord Napier, a Scotchman, was the inventor of logarithms and he published the first tables, but to Henry Briggs belongs the honor, next to Napier, for their development. He and Napier independently thought of the advantage of a system that would represent all numbers as powers of 10 to be used with our decimal system of notation, but after consultation with each other and because of Napier's declining health, it was left to Briggs to work out the system that is in common use.

559. The exponent of the power to which a fixed number, called the base, must be raised in order to produce a given num- ber is called the logarithm of the given number.

When 2 is the base, the logarithm of 8 is 3, for 8 = 2^.

When 10 is the base, the logarithm of 100 is 2, for 100 = 102 ; the loga- rithm of 1000 is 3, for 1000 = 10^ ; the logarithm of 10,000 is 4, for 10,000 = 10*.

560. When a is the base, x the exponent, and m the given number, that is, when a'' = m, x is the logarithm of the number m to the base a, written log„ 7n = x.

When the base is 10, it is not indicated. Thus, the logarithm of 100 to the base 10 is 2. It is written, log 100 = 2.

423

424 LOGARITHMS

561. Logarithms may be computed with any arithmetical number except 1 as a base, but the base of the common, or Briggs, system of logarithms is 10.

Since 10" = 1, the logarithm of 1 is 0. Since 10^ = 10, the logarithm of 10 is 1. Since 10^ = 100, the logarithm of 100 is 2. Since W = 1000, the logarithm of 1000 is 3. Since 10~^ = J^, the logarithm of .1 is —1. Since 10~^ = y^^, the logarithm of .01 is — 2.

562. It is evident, then, that the logarithm of any number between 1 and 10 is a number greater than 0 and less than 1. For example, the logarithm of 4 is approximately 0.6021.

Again, the logarithm of any number between 10 and 100 is a number greater than 1 and less than 2. For example, the logarithm of 50 is approximately 1.6990. " j

Most logarithms are endless decimals. All the laws estab- ' lished for other exponents apply also to logarithms, but the proofs have been omitted as being too difficult for the beginner.

563. The integral part of a logarithm is called the character- istic ; the fractional or decimal part, the mantissa.

In log 50 = 1.6990, the characteristic is 1 and the mantissa is .6990.

564. The following illustrate characteristics, mantissas, and their significance :

log 4580 =3.6609; that is, 4580 =103««».

log 458.0 = 2.6609 ; that is, 458.0 = lO^^^^.

log 45.80 = 1.6609 ; that is, 45.80 = 10'«^.

log 4.580 =0.6609; that is, 4.580 =10««»».

log .4580 =1.6609; that is, .4580 =10-^ + -^.

log .0458 = 2.6609 ; that is, .0458 =10-'+'^.

log .00458 = S.6609 ; that is, .00458 = 10 -»+•««».

From the above examples it is evident that :

LOGARITHMS 425

565. Principles. — 1. The characteristic of the logarithm of a nimiber greater than 1 is either positive or zero and 1 less than the number of digits in the iyitegral part of the number.

2. The characteristic of the logarithm of a decimal is negative and numerically 1 greater than the number of ciphers immediately following the decimal point.

566. To avoid writing a negative characteristic before a positive mantissa, it is customary to add 10 or some multiple of 10 to the negative characteristic, and to indicate that the number added is to be subtracted from the whole logarithm.

-Thus, 1.6609 is written 0.6609 - 10; 2.3010 is written 8.3010 - 10 or sometimes 18.3010-20, 28.3010 - 30, etc.

567. It is evident, also, from the examples in § 564, that in the logarithms of numbers expressed by the same figures in the same order, the decimal parts, or mantissas, are the same, and the logarithms differ only in their characteristics. Hence, tables of logarithms contain only the mantissas.

568. The table of logarithms on the two following pages gives the decimal parts, or mantissas, to the nearest fourth place, of the common logarithms of all numbers from 1 to lOQO

569. To find the logarithm of a number.

EXERCISES

1. Find the logarithm of 765.

Solution. — In the following table, the letter N" designates a vertical column of numbers from 10 to 99 inclusive, and also a horizontal row of figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first two figures of ?65 appear as the number 76 in the vertical column marked N on page 427, and the third figure 5 in the horizontal row marked N.

In the same horizontal row as 76 are found the mantissas of the loga- rithms of the numbers 760, 761, 762, 763, 764, 765, etc. The mantissa of the logarithm of 765 is found in this row under 5, the third figure of 765. It is 8837 and means .8837.

By Prin. 1, the characteristic of the logarithm of 765 is 2.

Hence, the logarithm of 765 is 2.8837.

426

LOGARITHMS

Table of Common Logarithms

N	0	1	2	3	4	5	6	7	8	9
lO	0000	0043	0086	0128	0170	0212	0253	0294	0334	0374
II	0414	0453	0492	0531	0569	0607	0645	0682	0719	0755
12	0792	0828	0864	0899	0934	0969	1004	1038	1072	1106
13	1 139	II73	1206	1239	1271	1303	1335	1367	1399	1430
14	1461	1492	1523	1553	1584	1614	1644	^(>n	^103	1732
15	1761	1790	1818	1847	1875	1903	1931	1959	1987	2014
16	2041	2068	2095	2122	2148	2175	2201	2227	2253	2279
17	2304	2330	2355	2380	2405	2430	2455	2480	2504	2529
18	2553	2577	2601	2625	2648	2672	2695	2718	2742	2765
19	2788	2810	2833	2856	2878	2900	2923	2945	2967	2989
20	3010	3032	3054	3075	3096	3118	3139	3160	3181	3201
21	3222	3243	3263	3284	3304	3324	3345	3365	3385	3404
22	3424	3444	3464	3483	3502	3522	3541	3560	3579	3598
23	3617	3636	3655	3674	3692	31^^	3729	3747	3766	3784
24	3802	3820	3838	3856	3874	3892	3909	3927	3945	3962
25	3979	3997	4014	4031	4048	4065	4082	4099	4116	4133
26	4150	4166	4183	4200	4216	4232	4249	4265	4281	4298
27	4314	4330	4346	4362	4378	4393	4409	4425	4440	4456
28	4472	4487	4502	4518	4533	4548	4564	4579	4594	4609
29	4624	4639	4654	4669	4683	4698	4713	4728	4742	4757
30	4771	4786	4800	4814	4829	4843	4857	4871	4886	4900
31	4914	4928	4942	4955	4969	4983	4997	501 1	5024	5038
32	5051	5065	5079	5092	5105	5"9	5132	5145	5^59	5172
33	5185	5198	5211	5224	5237	5250	5263	5276	5289	5302
34	5315	5328	5340	5353	5366	5378	5391	5403	5416	5428
35	5441	5453	5465	5478	5490	5502	5514	5527	5539	5551
36	55^3	5575	5587	5599	5611	5623	5635	5647	5658	5670
37	5682	5694	5705	5717	5729	5740	5752	5763	5775	5786
38	5798	5809	5821	5832	5843	5855	5866	5877	5888	5899
39	5911	5922	5933	5944	5955	5966	5977	5988	5999	6010
40	6021	6031	6042	6053	6064	6075	6085	6096	6107	6117
41	6128	6138	6149	6160	6170	6180	6191	6201	6212	6222
42	6232	6243	6253	6263	6274	6284	6294	6304	6314	6325
43	6335	6345	6355	6365	6375	6385	6395	6405	6415	6425
44	6435	6444	6454	6464	6474	6484	6493	6503	6513	6522
45	6532	6542	6551	6561	6571	6580	6590	6599	6609	6618
46	6628	6637	6646	6656	6665	6675	6684	6693	6702	6712
47	6721	6730	6739	6749	6758	6767	6776	6785	6794	6803
43	6812	6821	6830	6839	6848	6857	6866	6875	6884	6893
49	6902	6911	6920	6928	6937	6946	6955	6964	6972	6981
50	6990	6998	7007	7016	7024	7033	7042	7050	7059	7067
51	7076	7084	7093	7101	7110	7118	7126	7135 7218	7143	7152
52	7160	7168	7177	7185	7193	7202	7210	7226	7235
53	7243	7251	7259	7267	7275	7284	7292	7300	73°^	7316
54	7324	7332	7340	7348	7356	7364	7372	7380	7388	7396
N	0	1	2	3	4	5	6	7	8	9

LOGARITHMS

Table of Common Logarithms

427

N	0	1	2	3	4	5	6	7	8	9
55	7404	7412	7419	7427	7435	7443	7451	7459	7466	7474
56	7482	7490	7497	7505	7513	7520	7528	.7536	7543	7551
57	7559	7566	7574	7582	7589	7597	7604	7612	7619	7627
58	7634	7642	7649	7657	7664	7672	7679	7686	7694	7701
59	7709	7716	7723	7731	7738	7745	7752	7760	7767	7774
60	7782	7789	7796	7803	7810	7818	7825	7832	7839	7846
61	7853	7860	7868	7875	7882	7889	7896	7903.	7910	7917
62	7924	7931	7938	7945	7952	7959	7966	7973	79S0	7987
63	7993	8000	8007	8014	8021	8028	8035	8041	8048	8055
64	8062	8069	8075	8082	8089	8096	8102	8109	8116	8122
65	8129	8136	8142	8149	8156	8162	8169	8176	8182	8189
66	8195	82C2	8209	8215	8222	8228	8235	8241	8248	8254
67	8261	8267	8274	8280	8287	8293	8299	8306	8312	8319
68	8325	8331	8338	8344	8351	8357	8363	8370	8376	8382
69	8388	8395	8401	8407	8414	8420	8426	8432	8439	8445
70	8451	8457	8463	8470	8476	8482	8488	8494	8500	8506
71	8513	8519	8525	8531	8537	8543	8549	8555	8561	8567
72	8573	8579	8585	8591	8597	8603	8609	8615	8621	8627
73	8633	8639	8645	8651	8657	8663	8669	8675	8681	8686
74	8692	8698	8704	8710	8716	8722	8727	8733	8739	8745
75	8751	8756	8762	8768	8774	8779	8785	8791	8797	8802
76	8808	8814	8820	8825	8831	8837	8842	8848	8854	8859
77	8865	8871	8876	8882	8887	8893	8899	8904	8910	8915
78	8921	8927	8932	8938	8943	8949	8954	8960	8965	8971
79	8976	8982	8987	8993	8998	9004	9009	9015	9020	9025
80	9031	9036	9042	9047	9053	9058	9063	9069	9074	9079
81	9085	•9090	9096	9101	9106	9112	9117	9122	9128	9133
82	9138	9143	9149	9154	9159	9165	9170	9175	9180	9186
83	9191	9196	• 9201	9206	9212	9217	9222	9227	9232	9238
84	9243	9248	9253	9258	9263	9269	9274	9279	9284	9289
85	9294	9299	9304	9309	9315	9320	9325	9330	9335	9340
86	9345	9350	9355	9360	9365	9370	9375	9380	9385	9390
87	9395	9400	9405	9410	9415	9420	9425	9430	9435	9440
88	9445	9450	9455	9460	9465	9469	9474	9479	9484	9489
89	9494	9499	9504	9509	9513	9518	9523	9528	9533	9538
90	9542	9547	9552	9557	9562	9566	9571	9576	9581	9586
91	9590	9595	9600	9605	9609	9614	9619	9624	9628	9633
92	9638	9643	9647	9652	9657	9661	9666	9671	9675	9680
93	9685	9689	9694	9699	9703	9708	9713	9717	9722	9727
94	9731	9736	9741	9745	9750	9754	9759	9763	9768	9773
95	9777	9782	9786	9791	9795	9800	9805	9809	9814	9818
96	9823	9827	9832	9836	9841	9845	9850	9854	9859	9863
97	9868	9872	9877	9881	9886	9890	9894	9899	9903	9908
98	9912	9917	9921	9926	9930	9934	9939	9943	9948	9952
99	9956	9961	9965	9969	9974	9978	9983	9987	9991	9996
N	0	1	2	3	4	5	6	7	8	9

428 LOGARITHMS

2. Find the logarithm of 4.

Solution. — Although the numbers in the table appear to begin with 100, the table really includes all numbers from 1 to 1000, since numbers expressed by less than three figures may be expressed by three figures by adding decimal cipliers. Since 4 = 4.00, and since, § 567, the mantissa of the logarithm of 4.00 is the same as that of 400, which is .6021, the mantissa of the logarithm of 4 is .6021.

By Prin, 1, the characteristic of the logarithm of 4 is 0.

Therefore, the logarithm of 4 is 0.6021.

Verify the following from the table :

3. log 10 =1.0000. 9. log .2 =9.3010-10,

4. log 100 = 2.0000. 10. log 542 = 2.7340.

5. log 110 = 2.0414. 11. log 345 = 2.5378.

6. log 2 =0.3010. 12. log 5.07 = 0.7050.

7. log 20 =1.3010. 13. log 78.5 = 1.8949.

8. log 200 = 2.3010. 14. log .981 = 9.9917 - 10.

15. Find the logarithm of 6253.

Solution. — Since the table contains the mantissas not only of the logarithms of numbers expressed by three figures, but also of logarithms expressed by four figures when the last figure is 0, the mantissa of the logarithm of 625 is first found, since, § 567, it is the same as the mantissa of the logarithm of 6250. It is found to be .7959.

The next greater mantissa is .7966, the mantissa of the logarithm of 6260. Since the numbers 6250 and 6260 differ by 10, and the mantissas of their logarithms differ by 7 ten-thousandths, it may be assumed as sufficiently accurate that each increase of 1 unit, as 6250 increases to 6260, produces a corresponding increase of . 1 of 7 ten-thousandths in the mantissa of the logarithm. Consequently, 3 added to 6250 will add .3 of 7 ten-thousandths, or 2 ten-thousandths, to the mantissa of the loga- rithm of 6250 for the mantissa of the logarithm of 6253.

Hence, the mantissa of the logarithm of 6253 is .7059 + .0002, or .7961.

Since 6253 is an integer of 4 digits, the characteristic is 3 (Prin. 1).

Therefore, the logarithm of 6263 is 3.7961.

Note. — The difference between two successive mantissas in the table is called the tabular difference.

LOGARITHMS 429

Find the logarithm of :

16. 1054. 20. 21.09. 24. .09095.

17. 1272. 21. 3.060. 25. .10125.

18. .0165. 22. 441.1. 26. 54.675.

19. 1906. 23. .7854. 27. .09885.

570. To find a number whose logarithm is given.

The number that corresponds to a given logarithm is called its antilogarithm.

Thus, since the logarithm of 62 is 1.7924, the antilogarithm of 1.7924 is 62.

EXERCISES

571. 1. Find the number whose logarithm is 0.9472.

Solution. — The two mantissas adjacent to the given mantissa are .9469 and .9474, corresponding to the numbers 8.85 and 8.86, since the given characteristic is 0. The given mantissa is 3 ten-thousandths greater than the mantissa of the logarithm of 8.85, and the mantissa of tlie logarithm of 8.86 is 5 ten-thousandths greater than that of the logarithm of 8.85.

Since the numbers 8.85 and 8.86 differ by 1 one-hundredth, and the mantissas of their logarithms differ by 5 ten-thousandths, it may be assumed as sufficiently accurate that each increase of 1 ten-thousandth in the mantissa is produced by an increase of ^ of 1 one-hundredth in the number. Consequently, an increase of 3 ten-thousandths in the man- tissa is produced by an increase of | of 1 one-hundredth, or .006, in the number.

Hence, the number whose logarithm is 0.9472 is 8.856.

2. Find the antilogarithm of 9.4180 — 10.

Solution. — Given mantissa, .4180

Mantissa next less, .4166 ; figures corresponding, 261.

Difference, 14

Tabular difference, 17)14(.8

Hence, the figures corresponding to the given mantissa are 2618. Since the characteristic is 9— 10, or — 1, the number is a decimal with no ciphers immediately following the decimal point (Prin. 2). Hence, the antilogarithm of 9.4180 - 10 is .2618.

430 LOGARITHMS

Eind the antilogarithm of :

3. 0.3010. 8. 3.9545. 13. 9.3685-10.

4. 1.6021. 9. 0.8794. 14. 8.9932-10.

5. 2.9031. 10. 2.9371. 15. 8.9535-10.

6. 1.6669. 11. 0.8294. 16. 7.7168-10.

7. 2.7971. 12. 1.9039. 17. 6.7016-10.

J

572. Multiplication by logarithms. *

Since logarithms are the exponents of the powers to which a constant number is to be raised, it follows that : 1

573. Principle. — The logarithm of the product of two or more numbers is equal to the sum of their logarithms; that is^

To any base, log (mn) = log m + log n. For, let logo m=x and log^ n = y^ a being any base.

It is to be proved that logo (rnn) = x + y. § 559, a^ = m,

and a^ = n.

Multiplying, § 88, a'^+v = mn.

Hence, § 560, logo (mn) = x + y

= logo m + logo n.

EXERCISES

574. 1. Multiply .0381 by 77.

Solution Prin., § 573, log (.0381 x 77) = log .0381 + log 77. log .0381= 8.5809-10 log 77= 1.8865 Sum of logs = 10.4674 - 10 = 0.4674 0.4674 = log 2.934. .-. .0381 X 77 = 2.934.

LOGARITHMS 431

Note. — Three figures of a number corresponding to a logarithm may be found from this table with absolute accuracy, and in most cases the fourth will be correct. In finding logarithms or antilogarithms, allowance should be made for the figures after the fourth, whenever they express .5 or more than .5 of a unit in the fourth place.

Multiply :

2. 3.8 by 56. 6. 2.26 by 85. 10. 289 by .7854.

3. 72 by 39. 7. 7.25 by 240. ' 11. 42.37 by .236.

4. 8.5 by 6.2. 8. 3272 by 75. 12. 2912 by .7281.

5. 1.64 by 35. 9. .892 by .805. 13. 1.414 by 2.829.

575. Division by logarithms.

Since the logarithms of two numbers to a common base represent exponents of the same number, it follows that :

576. Principle. — The logarithm of the quotient of two num- bers is equal to the logarithm of the dividend minus the logarithm of the divisor; that is,

To any base, log (m -^ n) = log m — log n. For, let loga m = x and log^ n = y, a being any base.

It is to be proved that loga(m -f- n) = x — y. § 559, a^ = m,

and a^ = n.

Dividing, § 127, a'-" = m ^ n.

Hence, § 560, logo (m-^n) = x— y

= loga m - loga n.

EXERCISES

577. 1. Divide .00468 by 73.4.

Solution Prin., § 576, log (.00468 - 73.4) = log .00468 - log 73.4. log .00468 = 7.6702 - 10

log 73.4 =1.8657

Difference of logs = 5.8045 - 10

5.8045 - 10 =log .00000376. .-. .00468 -T- 73.4 = .00006376.

432 LOGARITHMS

2. Divide 12.4 by 16.

Solution

Prin., § 576, log (12.4 - 16) = log 12.4 - log 16. log 12.4 = 1.0934 =11.0934-10 log 16 = 1.2041

Difference of logs = 9.8893 - 10 9.8893 - 10 = log .775. /. 12.4 -- 16 = .775.

Suggestion. — The positive part of the logarithm of the dividend may be made to exceed that of the divisor, if necessary to avoid subtracting a larger number from a smaller one as in the above solution, by adding 10 - 10 or 20 - 20, etc.

Divide:

3. 3025 by 55. 8. 10 by 3.14. 13. 1 by 40.

4. 4090 by 32. 9. .6911 by .7854. 14. 1 by 75.

5. 3250 by 57. 10. 2.816 by 22.5. 15. 200 by .5236.

6. .2601 by .68. 11. 4 by .00521. 16. 300 by 17.32.

7. 3950 by .250. 12. 26 by .06771. 17. .220 by .3183.

578. Extended operations in multiplication and division.

Though negative numbers have no common logarithms, opera- tions involving negative numbers may be performed by con- sidering only their absolute values and then giving to the result the proper sign without regard to the logarithmic work.

Since dividing by a number is equivalent to multiplying by its reciprocal, for every operation of division an operation of multiplication may be substituted. In extended operations in multiplication and division with the aid of logarithms, the latter method of dividing is the more convenient.

579. The logarithm of the reciprocal of a number is called the cologarithm of the number.

The cologarithm of 100 is the logarithm of y^, which is — 2. It is written, colog 100 =—2.

LOGARITHMS 433

580. Since the logarithm of 1 is 0 and the logarithm of a quotient is obtained by subtracting the logarithm of the divisor from that of the dividend, it is evident that the cologarithm of a number is 0 minus the logarithm of the number, or the logarithm of the number with the sign of the logarithm changed ; that is, if log^ m = x, then, oology m = — x.

Since subtracting a number is equivalent to adding it with its sign changed, it follows that :

581. Principle. — Instead of subtracting the logarithm of the divisor from that of the dividend^ tJie cologarithm of the divisor may he added to the logarithm of the dividend; that is,

To any base, log (m -i- n) = log m + colog n.

EXERCISES

582. 1. Find the value of ^g^f|A|| by logarithn^s.

Solution .068 X 58.5 X 799 ^ ,,3 ^ ,3 , x 799 x JL x ^ X JL. 468 X 15.6 X .029 458 15.6 .029

log .063= 8.7993-10

log 58.5= 1.7672

log 799= 2.9025

colog 458= 7.3391-10

colog 15.6= 8.8069-10

coloff .029 = 1.5376

log of result = 31 .1526 - 30 = 1.1526. .-. result = 14.21. Find the value of : « no X 3.1 X. 653 3 15 X. 37x26.16

33 X 7.854 X 1.7 11 X 8 x .18 x 6.67

milne's stand, alg. — 28

434 LOGARITHMS

(- 3.04) X .2608 .4051 x (- 12.45)

*• 2.046 X .06219 *

600 X 5 X 29

5.

6.

.7854 X 25000 x 81.7 3.516 X 485 X 65

(-	8.988)	X .01442
78	x52x	1605
338	x767	X431'
.5	X.315	X428

3.33 X 17 X 18 X 73 .317 x .973 x 43.7

583. Involution by logarithms.

Since logarithms are simply exponents, it follows that :

584. Principle. — The logarithm of a power of a number is equal to the logarithm of the number multiplied by the index of the power ; that is,

To any base, log m** = n log m. j

For, let loga m = x^ and let n be any number, a being any base.

It is to be proved that loga w*" = nx.

§ 559, a* = m.

Raising each member to the nth power, Ax. 6 and § 276, 2,

Hence, § 560, loga ?/*" = nx = n loga m.

EXERCISES

585. 1. Find the value of .251

Solution Prin., § 584, log .262 ^ 2 log .25.

log .25= 9.3979-10. 2 log .25 = 18.7958 - 20 = 8.7958-10. 8.7958 - 10 = log .06249. .-. .252 = .06249.

Note. — By actual multipHcation it is found that .25^= .0625, whereas the result obtained by the use of the table is .06249.

Also, by multiplication, IS^ = 324, whereas by the use of the table it is found to be 324. 1. Such inaccuracies must be expected when a four-place table is used. ^ j

m

LOGARITHMS 435

Find by logarithms the value of :

2. 72. 7. .78^. 12. 4.073. 17. a^y,

3. 112. 8. 8.052. 13 5433, 18. (1)8.

4. (-47)2. 9. 8.332. 14, (_7)4. 19. (_l_2^S^)2.

5. 4.92. 10. 6.613. 15 1 025. 20. a^^y.

6. 5.22. 11, .7142^ 16. 1.7333, 21. (j^y

586. Evolution by logarithms.

Since logarithms are simply exponents, it follows that :

587. Principle. — TJie logarithm of a root ofanumber is equal

to the logarithm of the number divided by the index of the required

root; that is,

To any base, log "Vm —-^3-Hl.

n

For, let loga m = x and let n be any number, a being any base. It is to be proved that loga Vm = x ^ n. § 559, a* = m.

Taking the nth root of each member, Ax. 7 and § 290,

Hence, § 560, loga Vm = a: - n = !^^^.

EXERCISES

588. 1. Find the square root of .1296 by logarithms.

Solution

Prin., § 587, logV.129« = 1 log .1296.

log.1296 = 9.1126- 10.

2)19.1126-20 9.5563 - 10

9.5563»- 10 = log .360.

VU296 = .36.

436	; LOGARITHMS
Fin	d by logarithms the value of	:
2.	225l 8. (-1331)i	14.	V2.	20.	V-2.
3.	12.25i 9. 1024tV •	15.	V3.	21.	^.027.
4.	.2023^. 10. .6724^.	16.	V5.	22.	V30f.
5.	326l 11. 5.929i	17.	V6.	23.	V.90.
6.	.5121 12. .46241	18.	^.	24.	V.52.
7.	.1182*. 13. 1.464li	19.	^.	25.	■V/.032.
Sin	iplify the following : 176	31. 32.	14.5^- 11
9ft	-1.6
	15 X 3.1416 (-100)2 48 X 64 X 11
27.	/.4S4x96* ^(64 X 1500
28.	522 y. 300	33.	.32 X 5000 X 3.14 X .1222 )	18
12 X .31225 X 400000	<8
29.	/ 400 \55x 3.1416* 03.5	34. 35.	11 X 2.63 X 4.263 48 X 3.263
30.	/ 3500 V(-1.06)^
	36. 2ix(J)*	X^"	|xVT.

37. Applying the formula A = 7n^, find the area (A) of a^ circle whose radius (r) is 12.35 meters. (7r = 3.1416.)

38. Applying the formula F= f irr^, find the volume (F) of a sphere whose radius (r) is 40.11 centimeters.

39. The formula V= .7854 dH gives the volume of a right cylinder d units in diameter and I units long, V, d, and I being corresponding units. How many feet of No. 00 wire, which has a diameter of .3648 inches, can be made from a cubic foot of copper ?

LOGARITHMS 437

589. Solution of exponential equations.

Exponential equations, or equations that involve unknown exponents, are solved by the aid of the principle that, in any system, equal numbers have equal logarithms.

In simple cases the solution of such equations may be per- formed by inspection, but in general it is necessary to use a table of logarithms.

EXERCISES

590. 1. Find the value of x in the equation 2* = 32 V2.

Solution

2« = 32 \/2 =252^ = 2^; therefore, log (2*) = log (2 ^),

or, §584, a;log2 = J^log2.

Dividing by log 2, x = ^.

2. Find the value of x in the equation 2* = 48.

Solution Taking the logarithm of each member,

X log 2 = log 48. locr48

•.x =

log 2 1.6812 0.8010

= 5.59-

3. Solve the equation 3^* - 20 • 3^ + 99 = 0 for x.

Solution Factoring the given equation,

(3* -9) (3="^ -11) = 0.

• .-. 3=« = 9orll.

Solving the equation 3* = 9 by inspection, since 9 = 3^,

x=2.

Taking the logarithm ot each member of 3^ = 11,

a; log 3 = log 11.

^.^^10^^1,0414^2.18+. log 3 0.4771

Therefore, the value of x is either 2 or 2.18+.

438 LOGARITHMS

4. Given a^ = y^ and a^ = y", to find x and y.

Solution. — Raising the members of the first equation to the xth power, and those of the second equation to the 3d power,

and x^y = y^.

Hence, by inspection, 2x = Sy.

Squaring, since 4 x^ = 4 2/^, 4:X^ = 9y^ = iy^. .•.y = Oor|, and X = 0 or ^^.

5. Given 3^ =	2y and 2^ = y, to find x and	y-
Solution.		S- = 2y. 2=« =zy.			(1) (2)
Dividing (1) by (2),	(1.5)^ = 2.
		. •. X log 1 .5 = log 2.
Hence, by tables,		log 2 _ 0.3010 ^~ log 1.5 ~ 0.1761*		(3)
By logarithms,		log x = 0.2328;			(4)
whence, by tables.		a; =1.709.			(5)
From (2),		logy = x log 2.
Then,		log log y = log X + log log 2
by (4) and tables.		= 0.2328 + 1.4786
		= 1.7114.
Hence, by tables.		log y = 0.5145 ;
whence.		y = 3.270.
Solve the following :
6. 3^ = 81. 7. 4^ = 10.	12. 13.	2*' = 512. 17. (2^)2 = 256.	■2'+y ,2^+1	= 6, = 3^
8. 2^=80. 9. 3''=92^	14.	14^ = 20?/.	2^-u	= 32, = 4.
10. 23'=512. 11. 5^' =625.	15. 16.	32=" + 243 = 36-3^ 19. log logo; = log 2.	-2' =	y, l + logy.

LOGARITHMS 439

591. Logarithms applied to the solution of problems in com- pound interest and annuities.

Since the amount of any principal at 6 % interest, compounded annually, for 1 year is 1.06 times the principal ; for two years, 1.06 X 1.06, or 1.06^, times the principal; for 3 years, 1.06 X 1.06 X 1.06, or 1.06^, times the principal, etc., the amount (A) of any principal (P) for n years at any rate per cent (r) will be

A = P(l + ry.

Expressing this formula by logarithms,

log^ = logP-f-wlog(l + r). (1)

.-. logP=log^-nlog(l+r); (2)

also log(l + r)=l^^^l^MZ, (3)

n

and ^^log^-logP (

log(l + r)

EXERCISES

592. 1. What is the amount of |475 for 10 years at 6%

compound interest ?

Solution

^ = P(l+r)«.

log 475 = 2.6767

log 1.0610 = 0.2530

log^ =2.9297

.-.^=-$850.60.

Note. — In accordance with the note on page 431, antilogarithms are carried out only to the nearest fourth significant figure.

Find the amount, at compound interest, of : - 2. $ 225, 5 years, 8 %. 4. $ 400, 10 years, 3 %.

3. $ 700, 5 years, 6 % . 5. 1 1200, 20 years, 4 %.

440 LOGARITHMS

6. What principal will amount to $ 1000 in 10 years at 5 % compound interest ?

7. What sum of money invested at 4 % compound interest, payable semiannually, will amount to $ 743 in 10 years ?

8. What principal loaned at 4% compound interest will amount to $1500 in 10 years ?

9. What sum invested at 4 % compound interest at a child's birth will amount to $1000 when he is 21 years old?

10. In what time will $800 amount to $1834.50, if put at compound interest at 5 % ?

11. What is the rate per cent when $300 loaned at com- pound interest for 6 years amounts to $402 ?

12. A man agreed to loan $1000*at 6% compound interest for a time long enough for the principal to double itself. How long was the money at interest?

593. A sum of money to be paid periodically for a given number of years, during the life of a person, or forever, is called an annuity.

The payments may be made once a year, or twice, or four times a year, etc.

Interest is allowed upon deferred payments.

594. To find the amount of an annuity left unpaid for a given number of years, compound interest being allowed.

An annuity of a dollars per year, payable at the end of each year, will amount to a dollars at the end of the first year. If unpaid and drawing compound interest at a rate r, the accumu- lation at the end of the second year will be a-\-a{l -i-r) dol- lars ; at the end of the third year, a + a(l + »*) + a(l + ^y dollars; and so on.

Let a represent the annuity, n the number of years, r the rate, and A the whole amount due at the end of the wth year.

Then, A = a -{- a{l -\- r) + a(l -\- if + - - -\- a(l + ry-^ = a51-h(l+r) + (l-fr)2+...-f-(l-f-r)~-^}.

LOGARITHMS 441

Since the terms of A form a geometrical progression in which 1 + r is the ratio, § 525, the sum of the series is

^ = ^[(l4-r)--l]. r

EXERCISES

595. 1. What will be the amount of annuity of $100 re- maining unpaid for 10 years at 6 % compound interest ?

Solution

r

log 1.0610= .2530

.-. 1.0610 = 1.7904

and 1.0610 _ i _ .7904

log 100= 2.0000 log. 7904= 9.8978-10 colog.06= 1.2218

.•.log^= 13.1196 -10 = 3.1196. Hence, A = $1317, the amount of the annuity.

2. To what sum will an annuity of $ 25 amount in 20 years at 4 % compound interest ?

3. What will be the amount of an annuity of $17.76 re- maining unpaid for 25 years, at S^% compound interest ?

4. What annuity will amount to $1000 in 10 years at 5 % compound interest ?

5. AVhat annuity will amount to $5000 in 12 years at 3 % compound interest ?

596. A sum that will amount to the value of an annuity, if put at interest at the given rate for the given time, is called the present value of the annuity.

Sometimes annuities, drawing interest, are not payable until after a certain number of years.

442 LOGARITHMS

597. Let P denote the present value of an annuity due in n years, with compound interest at a rate r. Then, the amount of P at the end of the period will be found tlius :

By § 591, A = P{l + r)\

But, § 594, ^ A = -l(l + rY- 1].

Hence, Ax. 5, P(l + r)" = - [(1 + r)« - 1],

T EXERCISES

598. 1. What is the present value of an annuity of ^100 to continue 10 years at 6 % compound interest ?

Solution

p ^ a . 0 + r)" - 1 . r (1 + r)« log 1.0610= .2530 .-.1.0610= 1.7904 and 1.0610 - 1 = .7904

log 100= 2.0000

log .7904= 9.8978-10

colog .06 = 1.2218

colog 1.0610= 9.7470-10

.'. log P= 22.8666 -20 = 2.8666. Hence, P = $ 735.60, the present value.

2. What is the present value of an annuity of $300 for 5 years at 4 % compound interest?

3. What is the present value of an annuity of $ 1000 to con- tinue 20 years, if compound interest at 4J % is allowed ?

4. Find the present value of an annuity of £ 2000 payable in 10 years, interest being reckoned at 3 %.

COMPLEX NUMBERS

599. The student has learned that the indicated even root of a negative number is called an imaginary number, and that operations involving such numbers are subject to the condition that

(V— ly, or i^, equals — 1, not + 1.

600. Including all intermediate fractional and surd values, the scale of real numbers may be written

3 2 ... - 1 ... 0 ... + 1 - + 2 ... + 3 ..., (1)

and the scale of imaginary numbers, composed of real multiples of + i and — i, may be written

SI 2 i i ... 0 ... + i ... -f- 2 1 ... + 3 i .... (2)

Since the square of every real number except 0 is positive and the square of every imaginary number except 0 i, or 0, is negative, the scales (1) and (2) have no number in common except 0. Hence,

An imaginary number cannot he equal to a real number nor cancel any part of a real manber.

601. The algebraic sum of a real number and an imaginary number is called a complex number.

2 -I- SV— 1, or 2 4- 3 1, and a + bV— 1, or a + bi, are complex numbers. a^ + 2 abV — 1 — b'^ is a complex number, since a^ -{- 2 abV — 1 — b'^ =

602. Two complex numbers that differ only in the signs of their imaginary terms are called conjugate complex numbers.

a + bV— 1 and a — 6V— 1, or a 4- bi and a — bi, are conjugate com- plex numbers.

443

444 COMPLEX NUMBERS

603. Operations involving complex numbers.

EXERCISES

1. Add3-2V^^and2-f 5V^3.

Solution

Since, § 600, the imaginarj'^ terms cannot unite with the real terms, the simplest form of the sum is obtained by uniting the real and the imaginary terms separately and indicating the algebraic sum of the results .

Simplify the following :

2. (S+V^ + CV^O-S).

3. (2-V-16) + (3+V-4).

4. (3-V^^) + (4 + V^=^l8).

5. (V^20-Vi6)4-(V^=^4-V4).

6. (4 + V^=^)-(2 4-V^.

7. (3_2V^^)-(2-3V^=^).

8. (2-2V^ri+3)_(Vl6"-V^=l6).

10. Expand (a + 6V^^)(a + 6V^T).

Solution §105, (a-hbV^^)ia + by/^n.) = a^ + 2abV^^ + {bV'^)

§ 599, = a2 + 2 aby/^n. - b^.

11. Expand ( V5 — V^^y.

Solution ( V5 _ VZr3)2 = 5 -2\/^n5 + (- 3) = (5_3)_2V- 15 -2- 2V^^=T5.

COMPLEX NUMBERS ' 445

Expand :

12. (2 + 3V^(l+V^n). 15. (2 + 31)2.

13. (5_V^ri)(l_2V^. 16. (2-3 1)2.

14. (V2 4-V'^=^)(V8-V^. 17. {a-bi)\

Show that :

18. (H-V^(l+V^(l+V^3)=-8.

19. (-1+V^(-1+V^(-1 + V"^)=8.

20. (_|+|V^(-i + lV^3)(-i + iV^=l.

21. Divide 8 + V^I by 3 + 2 V^^.

First Solution

3 + 2V^^

2-\/-l

8 + V^nr = 6 + V^^\ + 2

_ 3V^l 4- 2 -3\/^n^+ 2

The real term of the dividend may always be separated, as above, into two parts, one of which will exactly contain the real term of the divisor.

Second Solution g+yHT ^ (8+V31)(3-2V31) ^26-13AAri^^ ^^ — - 3+2\/^ (3 + 2V^l)(3-2V^l) 9+4

Divide :

22. 3b3^1-V-2. 25. a^ + h^hy a-hV^^.

23. 2 by 1 + V^^. 26. a — hi by ai + 6.

24. 4+V4by 2-V^=^. 27. (1 + 0' by 1 - /.

28. Find by inspection the square root of 3 + 2V— 10. Solution

3 + 2>ArTon:(5-2)+2\/5 . -2=5 + 2V5. -2+(-2). .-. Vs + 2 V^=l0 = Vs + 2 V'S'T"^ + (- 2) = V5 + V^=^

446 COMPLEX NUMBERS

Find by inspection the square root of :

33. 4V'^3-1.

29.	4 + 2V"	-21
30.	1 + 2V~-	-6.
31.	6-2V~	^.

34. 12\/-l-5.

35. 24V-1-7. 32. 9 + 2V^^22. 36. b^ + 2abV^^-a\

37. Verify that — 1-hV — 1 and — 1— V — 1 are roots of the equation x'^-\-2x + 2 = 0.

38. Expand (i- + 1 V - 'Sf.

*

604. Tlie sum and the product of two conjugate complex num- bers are both real.

Eor, let a + &V — i and a — 6 V — 1 be conjugate complex numbers.

Their sum is 2 a.

Since ( V — 1)"^ =— 1, their product is,

§ 114, cfi - (&V^n)2 = a2 - ( - 62)

= a2 -}. &2.

605. If two complex numbers are equal, their real parts are equal and also their imaginary parts.

For, let a + 6\/ — 1 =x -{-yV —1.

Then, a - ic = (i/ - 6) V^^,

which, § 600, is impossible unless a = x and y =b.

606. Ifa-\- 6V — 1 = 0, a and h being real, then a = 0 and 6 = 0.

For, if	a + 6V_l = 0,
then,	6V-l=-a,
and, squaring, ,	- 62 = a2 ;
whence.	a2 + 62 = 0.

Now, a2 and 62 are both positive ; hence, their sum cannot be 0 un- less each is separately 0 ; that is, a = 0 and 6 = 0.

SUPPLEMENTARY EXERCISES

POSITIVE AND NEGATIVE NUMBERS

607. Add by columns ; then by rows : [Supplementing § 55.]

1. 2. 3. 4. 5. 9. 10. 11. 12. 13.

6. 3 -8 +7 +5 -4 14. 7 +8 -4 +7 -6

7. 5 +6 -2 -8 +6 15. -9 -5 +8 +9 -8

8. 7-4-8+3+5 16. 4 -6 -9 +8 +7

Subtract the lower number from the upper one : [Supple- menting § 56.]

17. 8 19. 9 21. -10 23. 25 25. 19 27. -32 4 -8 _12 40 -19 -16

18. -9 20. -43 22. 35 24. -18 26. -24 28. 16

IQ _22 -40 _J^ -24 -34

Find the indicated products : [Supplementing § 85.]

29. (5)(-3). 34. (-16)(6). 39. (2a2)(30).

30. (-8)(9). 35. (-7)(12). 40. (24)(-2).

31. (0)(-7). 36. (25)(-2). 41. (-16)(-8).

32. (-5)(m). 37. (-3)(-20). 42. (-8c)(-5).

33. (8)(15). 38. (-9)(-3a). 43. (-4a;)(-15).

Divide as indicated: [Supplementing § 124.]

44. §§. 48. ^- 52. -?1. 56. ^^.

8-2 7-9

45. -2)m 49. 5)-60: 53. -8) -96. 57. -6)84.

46. 7) -63. 50. -l)-50. 54. 3) -87'. 58. -9) -99.

47. -32-4. 51. -9--(-3). 55. 8--(-2). 59. -75-^5.

447

448 • SUPPLEMENTARY EXERCISES

ADDITION

608. Add : [Supplementing § 59.]

1. 5x 2. Sy 3. -b 4. 6ax 5. -Sb^y

2x -6y Sb -Tax -2b^y

I

6.	4a	7.	-8 c	8.	10 .r	9.	-3 6c	10.	4:y^
	4a 32/	12.	9c Iz	13.	-^x	14.	-8 6c -9xy	15.	-6yz^
11.	15 a	-2aY
	9y		-62!		-20 a		9xy		-7 ay
16.	b	17.	4a;	18.	14 a2	19.	4a6	20.	-7(}d^
	lb		— X		-8a2		-2a6		5c'd'
	66		^x		2 a}		-6a6		-dc'd'

Simplify : [Supplementing § 60.]

21. a2-c2 + 62 + a2 + c2. 24. a6 + 6c-6c-a6.

22. 5r — 6s-\-t — r + 5s. 25. x'^z - 5 y^ + 3 x-z + y\

23. a^-2f-:^-\-2f-4:^. 26. cW -\- a'b^ - 2 a'b^ - cW.

27. x-\-3y — z — 4y—5x-\-3z + 4:X-{-y — 5z.

28. 62 + a^c" _ 5 ^2 - 3 aV + 4 62 4-3c?2 4.2 a^c^ - b\

29. a62 + a262 _ 0^6 - 3 a262 -f 2 06^ + 4 a26 - a62 + a262.

30. yz + a} -xz + y^ + 2xz-\-4:yz — Sx'^ — 3y^ — 2yz.

Add:

31. b-3d,5b + d. 33. 3aV - 5 6^, 36^ - 2a2c2.

32. 5x — y,y — 3x. 34. m^w^ _ ab, 2ab — 3 mhi^.

35. 2 a 4- 6, 3 6 — c, d — 2 a, c 4- d, and 6 — c?.

36. \x-\-\y — z,\x—y-\-\z,?ii\diX-\-\y-\-\z.

37. (c + d) — (a + 6), (a + 6) - (c + d), and (a + 6) + (c + d).

38. 5Va— 3V6+6Vci V6 — Va+Vc, and V6 — Vc— Va.

39. (a-l-l)V^- (6_H-2)V2/, (a + 1) V^- (6 + 2)V^, and (6 + 2) V2/-(a + l)Va?.

SUPPLEMENTARY EXERCISES 449

SUBTRACTION

609. Subtract the second expression from the first : [Supple- mentmg § 61.]

1. 6 a, a. 4. 6 ?/, — 2 y. 7. 2 ab, 4 ab.

2. 3 &, 7 6. 5. — 8 c, 4 c. 8. xy, — 7 xy.

3. ic, - 8 «. 6. - 8 a;, - 2 a;. 9. - 4 6=^, - 8 ¥.

10. 1 cd, — 3 cd. 13. — 4 xyz, — 11 xyz.

11. 5 V&, - 10 V6. 14. 10 a'bc, - 8 a^^c.

12. -bc'd, -^ bcH. 15. 2(a4-6), -3(a + &). 16-45. In exercises 1-15, page 448, subtract the lower ex- pression from the upper one ; the upper from the lower one.

What expression added to the first will give the second ?

46. a 4- 4, a 4- 9. 65. 6 — c — c?, 0.

4n. c — d,c + d. 66. a + 6 - 5, a + 10.

48. aj + 5, 5 aj + 8. Q1. x-\-y,2x+y-\-z.

4:9. y -2,7 y -6. 68. b + c -hd, 2 b - S d.

50. 2 a -\- b, a -\- S b. 69. x— y -j-z, x-\-y + z.

51. x— 5y, Sx-{-y. 70. a—b — c,a-\-b-]-c.

52. 6 — 3 c, 4 6 — 5c. 71. r -^ s — v, r — s -{-v.

53. 2 y — z, y -\-'iz. 72. x -\-y — z, 3 x — y + z.

54. 5x + y,9x — Sy. 73. 2 a + & + c, a + 2 6 + c.

55. 3c-4d, 7c + 6d. 74. x—2y-{-z,x + 2y+z.

56. 4p-4g, -5p + g'- 75. b-c — 8d,b-\-c-\-2d.

57. —3x-{-y, 2x — 9y. 76. p — g + 5 r, 3 p + g — r.

58. 5 m + 71, — m + 6 n. 77. 2 a — 6 — c, 4 a + & — c.

59. 10x-\-3y,lSx-7y. 78. x^ - xy -^y^, x"^ -i-xy -\-y\

60. -12a-5 6, 7a + 2 6. 79. a2+2 a&H-6^ a^- 2 aft + ft".

61. 16c+9d, -10c-5d. 80. c^ - cc? + d^, 5 c^ - cc? - d^.

62. 15p-8g, 18p — 4g. 81. 4 ac + c^, a^ -|-6 ac H-c^.

63. 14 a; - 10 2/, 25 2/ -f 6 a;. 82. a^ - a^b - ab\ a'b + ab^.

64. 10 m — 5 n, 5 n — 10 m. 83. a;"* — 2 a;"*?/" + 2/"> ^"'2/" + 2/"-

milne's stand, alg. — 29

450 SUPPLEMENTARY EXERCISES

Parentheses

610. Simplify : [Supplementing § 62.]

1. x-\-{y + z). 3. a-h(a + 6). 5. 5a-{2a-Sb).

2. b—(c-\-d). 4. Sx — (2x + y). 6. x — (x — y--z).

7. a-\-(b — c — d). 10. 4:X — {x—y) — (y — z).

8. x-\x-{-(a-\-b)\. 11. 3a6-|-a2-(a2H-2a6+&2)^

9. 2 a — (a — b) + (a-\-b). 12. m+n— [(m + 7z) — (??i— ti)].

[Supplementing § 65.]

13. (a-\-b)x-{-(a—b)x. 18. (a + 6)a; + (2 a + &)a;.

14. (c-{-d)y-(c-\-d)y. 19. (c + d)2/H-(c-2d)2/.

15. {a — d)x — (a-\-d)x. 20. (a - c)a; — (2 a-f 2 c)a;.

16. (6-c)aj + (& + c)a;. 21. {b^ -\-c^)y -(b"" -Sc^)y.

17. (a2 + 62)2_f.(a2_52);2. 22. {m^ - 7i^)z-(2 m^-2n^)z.

Problems I

[Supplementing § 78.] J

611. 1. What number diminished by 14 is equal to 29 ? i

2. Find two numbers whose sum is 38 and whose difference is 10. J

3. Find five consecutive numbers whose sum is 80. '

4. Find four consecutive numbers the sum of the first and last of which is 61. i

5. Separate 52 into two parts the larger of which is 3 times the smaller.

6. One number exceeds 3 times another by 10 and their sum is 38. Find the numbers.

7. A basket-ball team played 26 games and won 2 more than it lost. How many games did the team win ?

8. A basket-ball team scored 322 points in a season's play. The captain scored 62 points less than all the other players. How many points were scored by him ?

SUPPLEMENTARY EXERCISES 451

9. If 3 times a certain number is decreased by 3, the result is 24. What is the number?

10. If from 6 times a certain number 8 is subtracted, the remainder is 34. What is the number ?

11. Twelve times a number is decreased by 6. The result is 8 more than 5 times the number. Find the number.

12. A rectangle whose perimeter is 26 feet is 5 feet longer than it is wide. Find its dimensions.

13. A lawn is 17 rods longer than it is wide. If the dis- tance around it is 194 rods, find its dimensions.

14. One number is 5 times as large as another number and their sum is 28 less than twice the larger number. Find the numbers.

15. The number of boys who tried for a football team was 47, or 17 more than twice as many as received letters. How- many received letters ?

16. If I double a certain number and subtract 8 from the product, the result thus obtained is 6 more than the number. Find the number.

17. The height of a railroad bridge is 300 feet, or 80 feet less than twice the width of the chasm it spans. What is the width of the chasm ?

18. The average net cost of a 2-year-old dairy heifer is $61.41, which is $5.09 more than 8 times her initial cost. Find her initial cost.

19. The average cost of feed for a yearling heifer is $ 24.67, which is $2.42 more than 5 times the cost of labor for her care. Find the cost of such labor.

20. The average cost of feed for a 2-year-old heifer is $ 40.83, which is $ 1.78 more than 5 times the cost of labor for her care. Find the cost of the labor.

452 SUPPLEMENTARY EXERCISES

21. A real estate dealer sold a house for $6500 and gained y% of what the house cost. How much did he gain ?

22. The combined weight of two horses is 1900 pounds. The younger weighs -^ as much as the older. What is the weight of each ?

23. Two boys caught 48 trout, which were 4 less than twice as many as they caught the day before. How many did they catch the day before ?

24. A 110-dollar cruise to Panama costs $ 40 less than twice as much as a cruise to St. Johns. Find the cost of the latter.

25. Divide $ 168 among 3 boys so that the first shall receive ^ as much as the second, and the second ^ as much as the third.

26. A, B, and C together had $430. f of A's money was $15 less than B's and $ 35 more than C's. How much had each ?

27. One year the profit from an acre of raspberries was $ 756. If this was $ 108 less than twice the average annual profit per acre, what was the average profit per acre ?

28. A farmer raised 42 crates of raspberries one year, which was 12 more than -^ of what he raised the next year. How many crates of raspberries did he raise the second year ?

29. The price paid for picking a crate of raspberries was 25 f^, which was 5^ more than ^ of the money for which they were sold. Find the selling price per crate.

30. The hoist shaft of a mine is 698 feet deep, which is 26 feet more than 16 times the height of the boiler stack of this mine. How high is the stack ?

31. A certain cow's thirty-day record for butter fat is 102.5 pounds, which is 34.5 pounds less than 5 times her seven-day record. Find her seven-day record.

32. The thirty-day butter-fat record for a certain cow was 111.8 pounds, which was 1.4 pounds more than 4 times her seven-day record. Find her seven-day record.

SUPPLEMENTARY EXERCISES 453

33. Two pillars together weigh 24 tons. One has lost ^ of its original weight by being cored. Both originally had the same weight. Find the present weight of each.

34. A fruit orchard of 1800 trees contained twice as many peach trees as plum trees and 200 more plum trees than pear trees. Find the number of each.

35. A man bored for water to a depth of 75 feet. The dis- tance through rock was 5 feet less than 3 times the distance through clay. Find the number of feet of clay bored through.

36. The air pressure gauge on an automobile tire indicates 92 pounds per square inch, or 12 pounds less than 4 times the pressure on a bicycle tire. Find the pressure on the latter.

37. A motor boat cost $ 5000 more than i as much as a steam yacht. The yacht was sold at a loss of $ 10,000, or $ 15,000 more than the cost of the motor boat. Find the cost of each.

38. A man sold 2 acres more than | of his farm and had 4 acres less than ^ of the farm left. Find the number of acres in the farm.

39. The inside diameter at the top of a chimney is lOi feet less than that at the base, and the latter is 7^ feet less than 3 times the former. Find the diameter at the base.

40. The sum of the ages of a father and his son is 64 years, and the age of the father is 4 years more than 3 times the age of his son. Find the age of each.

41. The length of a ship is 7 feet less than 6 times its width. If the former were 345 feet less and the latter 108 feet more, they would be equal. Find the length and the width.

42. One number is twice another number. If the smaller is subtracted from 32, the remainder is 11 less than the remainder when the larger is subtracted from 50. Find the numbers.

454 SUPPLEMENTARY EXERCISES

MULTIPLICATION

612. Multiply:	[Supplementing §	89.]
1.	3a^	7.	4a6c	13.	- 4 a'bf	19.	-7m«-2
	4a^		- a'bc'		- 3 aby		-9 m'
2.	ab	8.	— 5afyz	14.	Ga^Y	20.	8 5m+i
	a%		- 3 xy^^		-x-y^		-4 6--1
3.	-&	9.	4 m^n^	15.	9 aWy^	21.	^+c
	ac -2f	10.	— mn*	16.	2 a-'b-'y 3x-y	22.	^f-c
4.	1 a^by	4^n+3
	-hy		Zb'^cif		2a?y-^		1 X--'
5.	^aJ^b''	11.	- 8 x^y'	17.	2f-x	23.	— a^b'^C'
		12.	4 x'y' 10 m'n'	18.	-xr	24.	— a^b^'c'^
6.	xl'f^H
	-5a^f		-Sn'm^		X		vf'y-'^z

Find the products indicated : [Supplementing § 91.]

25. (-4)(-5aa;)(4aar^). 27. {-2xy){-'ix^y){4::x?){^f).

26. (- 6 6)(- a262)(_ aft^). 28. (- 32)(- 23)(22 . 3)(2 • 32).

Multiply : [Supplementing § 94.]

29. 5 6c2 + 63by 2 62. 34 a^ - 2 xy -^ y^ by xy.

30. 4a2 — 3a6 by 4a6. 35. 5 b^<^ - 7 abc hy - be.

31. a?f-^hy—xyz. 36. 2 a2_ 3 a6-|-&2 ^y 4 ^5

32. 2 rs — ^ by 4 r^s^. 37. 2 «?/ — 3 jcs; 4- 2/2; by xyz.

33. 6 62^3 _ cc2 by 3 cd. 38. 4 c* + 3 02^2 - d^ by - cd2.

[Supplementing § 96.]

39. a; + 7 by a? — 5. 41. 4 ?/ — 9 2 by 8 ?/ — 5 2.

40. 5 a? -I- 8 by 3 a; + 6. 42. a^ -\- xy + y^ by x — y.

43. 3c2-2cd4-5(i2by c2-cd-c«2.

44. xl'y* + a?^?y222 -|- z^ ^y ^2/2 - z\

SUPPLEMENTARY EXERCISES 455

Special Cases in Multiplication 613. Write out the square indicated : [Supplementing § 106.]

1. ic-\-d)\ 6. 2P. 11. (3 a + 5)2. 16. {\a-\-h)\

2. (6 + c)2. 7. 322. 12. (4 a; + 7)2. 17. (r + 2")'.

3. (a + 5)2. 8. 412. 13, (5?/ + 3 2)2. 18. {:^x-\-\yf.

4. (a;2 + 8)2. 9. 2012. 14, (^o^^^ff. 19. (3a6 + 4cd)2.

5. (^ + 9)2. 10. 1022. 15. (3a3 + 3 63)2. 20. (2 c» + 2 d«)2.

[Supplementing § 109.]

21. (a-d)2. 26. 192. 31. {x^-y^. 36. {y?-f)\

22. iy-zf, 27. 282. 32. (4c? -6)2. 37. {^x-2y)\

23. (6-5)2. 28. 692. 33, (2 a; -4^)2. 38. (5-3a6)2.

24. (ar^-9)2. 29. 982. 34, (4c-5d)2. 39. (xy'^-z'^f.

25. (c2-7)2. 30. 1992. 35. {^y'^-zy. 40. {2h^-c^d)\

Write out the product : [Supplementing § 115.]

41. (a2 - 5)(a2 + 5). 47. 22x18. 53. (^ y - ^){^ y Jf- ^).

42. (52_|.7)(52_7), 48. 37x23. 54. (^ « - 2 5)(i. ^^ + 2 5).

43. (a^_8)(a^ + 8). 49. 28x32. 55. (2 a^b - y)(2 x'b -\- y).

44. (9 + ic2)(9_a;2). 50. 41x59. 56. {4^ x^ - y'^)(4: x^ -\- y^).

45. (a^_4)(a^ + 4). 51. 68x52. 57. (2 c-- (?")(2 C" + cZ").

46. ((? + 62)(d _ 52) 52 74x66. 58. (5aa;-662)(5aa;+662).

[Supplementing § 118.]

59. (a; + 6)(a; + 8). 65. (a-3 6)(a + 6 6).

60. (aj-8)(x-9). 66. {2x-8y){2x-5y).

61. (a;-6)(a; + 9). 67. {m^n^ - 9) {m^n^ + 10).

62. (a2 + 5)(a2-8). 68. (a^ + c)(a^ - d).

63. (f-^)(f-5y 69. (xy-aXxy-b).

64. (c2 + 8)(c2 + 12). 70. Qr^-S)(a^l)-\-4:),

456 SUPPLEMENTARY EXERCISES

Problems

[Supplementing § 120.]

614. 1. The sum of two numbers is 44. The larger is 2 more than twice the smaller. Eind the numbers.

2. Separate 28 into two parts, such that one part shall be 4 less than 3 times the other.

3. A man's age is 3 years more than 3 times his son's age, and the sum of their ages is 47 years. Find the age of each.

4. The difference between the squares of two consecutive numbers is 121. Find the numbers.

5. If each dimension of a square signal flag were 4 inches more, its area would be increased 1 square foot. Find the dimensions of the flag.

6. If 122 marbles were divided among three boys so that the first had twice as many as the second and the second had 6 more than the third, how many marbles had each ?

7. The weight of 27 new one-dollar bills equals that of a twenty-dollar gold piece. If 20 of these bills weigh 140 grains less than the gold piece, what is the weight of each ?

8. One race course is 2 kilometers longer than another. If 20 times the length of the longer course equals 21 times that of the shorter one, what is the length of each ?

9. The diameter of a tower is 15 feet more than that of its belfry, and twice the latter is 23 feet more than the former. Find the diameter of the tower ; of the belfry.

10. If two tracts of land contain together 1473 trees and the number in one is 15 less than 15 times the number in the other, what is the number of trees in each tract ?

11. A grocer bought 3 cases of eggs. If he had bought 1 case less and each case had contained 2 dozen eggs more, the total number of eggs would have been 768. How many eggs did he buy ?

SUPPLEMENTARY EIXERCISES 457

12. A's age exceeds B's by 12 years. Four years ago A was twice as old as B. Find the present age of each.

13. Leon is 26 years younger than his father. If he were

2 years younger than he is, his age would be ^ of his father's age. How old is Leon ?

14. The difference between the squares of two consecutive odd numbers is 64. Find t||^ numbers.

15. Separate 100 into two parts such that the greater exceeds 60 by twice as much as 34 exceeds the less.

16. If a boat were 10 feet narrower, its length would be 9 times its width ; if it were 4 feet wider, its length would be

3 times its width. Find its length and its width.

17. A woman receives 4 ^ more per day for picking olives than a child receives. If in 6 days the two together receive $ 2.16, what are the daily wages of each ?

18. The present price of an article is 40% of its former price and it now costs $ 3.75 less than it did formerly. Find the present price of the article.

19. A purse contains 18 coins, some of which are quarters and the remainder dimes. If the coins are worth $2.40 alto- gether, how many coins are there of each Jiind ?

20. Three liners, A, B, and C, together made 30 trips. A and C together made twice as many as B, and B made 1 more trip than A. How many trips did each make ?

21. Find three consecutive numbers such that if the square of the least is subtracted from the product of the other two the remainder will be 47.

2.2. The length of a box is 4 inches more than its width and 12 inches more than its height. If the area of the bottom is 264 square inches more than the area of one end, what are the dimensions of the box ?

458 SUPPLEMENTARY EXERCISES

23. One number exceeds another by 4, and the square of the first is 48 more than that of the second. Find the numbers.

24. A farmer bought 12 calves. If he had bought 3 more for the same money, each calf would have cost him $ 1 less. How much did he pay for each ?

25. The central arch of a bridge is 35 meters longer than each of the 6 other arches, and the total length of the bridge is 175 meters. Find the length of each arch.

26. A North Carolina poplar made 15 cuts. The 12-foot cuts were 2 more than twice as many as the 14-foot cuts, and there was 1 10-foot cut. How many 12-foot cuts were there ?

27. The diameter of one circle is 10|^ feet less than that of another, and the diameter of the latter is 7^ feet less than 3 times that of the former. Find the diameter of each circle.

28. A steamer carries 90,000 bushels of grain. The amount of corn is 5000 bushels more than i that of the wheat and the same as that of the barley. Find the amount of each grain.

29. The combined weight of two of the largest gold nuggets ever found is 4715 ounces. If 439 times the larger equals 504 times the smaller, find the weight of the larger.

30. A and B each shoot 30 arrows at a target. B makes twice as many hits as A, and A makes 3 times as many misses as B. Find the number of hits and misses of each.

31. How many pounds each of two grades of tea costing 50 cents and 80 cents a pound, respectively, must a dealer use to make 12 pounds of a blend worth 60 cents a pound?

32. How many pounds each of two grades of coffee costing 20 cents and 28 cents a pound, respectively, must a dealer use to make 40 pounds of a blend worth 25 cents a pound ?

33. The total height of a monument is 160 feet. The height of the pedestal is 5 feet more than that of the statue, and the height of the shaft is 5 feet more than twice that of the pedes- tal. Find the height of each part.

SUPPLEMENTARY EXERCISES 459

DIVISION 615. Divide as indicated : [Supplementing § 128.]

1.	3«-32. 3.	27-2». 5. e^H-e^	7. 8-(-2
2.	4^-4. 4.	58_=_53, 6. -21-1-7.	8. x^^x^.
9.	— &-i-C.	12. -S¥^ib\ 15.	12 c'd' ^2 cd\
10.	fH-y'y	13. — 5-a2a;H-(— 1). 16.	^a?z^^\ xz.
11.	-a'^(-a).	14. .6a2"_j_.2a". 17.	— it'"'?/"* -T- iC^y^

18. \x'^fz^-^4.y^z. 19. 9(a -|-6)^H-3(a -f- 6).

Divide : [Supplementing § 131.]

20. 8 xz^ + 4 0^2; by 4 xz. 22. .6 aj^ — ,^xy-\- .4 by .2.

21. —x + y — z^uhj-1. 23. aft^ -f a^ft — a^ft^ by ^ a6.

24. a^" — 5 a" -I- 6 a^" -h 3 a^" -f- 4 a^" by tt^

25. 3(a + 6)-9(aH-6)2-6(a + 6)3by (a-h6). [Supplementing § 132.]

26. a^ — 8 a + 7 by a — 1. 28. 35 — 31 x -f- 6 aj^ by 2 a; - 7.

27. ic + a;^ — 30by aj — 5. 29. a}"" -{-^ a^'h^A^h'^hj a^'-^-h.

30. aj«+3 -f- xf+^ -h a!«-i by a;^ — a; -h 1.

31. a^ + Whj a + h to five terms of the quotient.

Special Cases in Division 616. Mvide: [Supplementing § 137.]

1. x^-f by a;2 — y. 10. .16 — 4 2/^ by .4 — 2 ?/.

2. c« -h rf^ by c^ -h (^l 11. 216 - a^ft^ by 6 - a6.

3. 0^52 _ 9 by a^h + 3. 12. a^ + 6^ by a'* -|- &".

4. cs-h32byc-h2. 13. ^0^2-39^2/2 by ^aj_|2,.

5. a^—¥c^hjo?-h^&. 14. i- a^ - 2^ 6^ ^^ ^ ^ _ 2 ft,

6. d«-64bydH-2. 15. (c-d)2-4 by c-d -2.

7. a^ - 144 by a^ - 12. 16. (a 4- hf+ 1 by a + 6 -|- 1.

8. x' + y'^hy x + y. 17. S-{c -\-d)^ hy 2- c -d.

9. 2/' -243 by 2/ -3. 18. (a;+2/)2-100 by aj-f-2/+10.

460 SUPPLEMENTARY EXERCISES

Problems

[Supplementing § 139.]

617. 1. Fourteen years ago a man was ^ as old as lie is now. Find his present age.

2. A baseball team one year won 99 games, or | of the number it played. How many games did it play ?

3. The sum of two numbers is 42 and ^ of the difference between them is 4. Find the numbers.

4. Mary's age is ^ that of her sister Ruth. In 6 years Mary will be | as old as Ruth. Find the age of each.

5. A father is m times as old as his son. In^ years he will be n times as old. Find the age of each.

6. Separate 25 into two parts such that |- the larger part equals twice the smaller part.

7. Separate the number c into two parts such that a times the larger part equals b times the smaller part.

8. One boy weighs 100 pounds, which is 85 pounds more than -J- of another boy's weight. Find the weight of the latter.

9. A jointed fish rod in three parts is 7 feet long. The butt is ^ as long as each of the other parts. Find the length of each part.

10. After playing 17 games of checkers, the champion had won 4 times as many games as he had lost, and 6 times as many as were drawn games. How many games did ie win ?

11. The age of a man exceeds 3 times the age of his son by 3 years, and the age of the son exceeds ^ of the age of the father by 5 years. Find the age of each.

12. A house and a flagstaff on top of it together are 180 feet high. The height of the house is 5 times the length of the flagstaff. Find the height of the house; the length of the flagstaff.

13. Each year a certain state uses 30,000 more balls than bats, and twice the number of balls equals 5 times the number of bats. How many bats are used ?

SUPPLEMENTARY EXERCISES 461

14. A textbook contains 496 pages. If the body of the book were decreased 34 pages, it would contain 65 times as many pages as the index. How many pages are there in the index ?

15. A storage room contains twice as many coats as it does muffs, and ^ of the number of coats is 5000 less than the num- ber of muffs. Find the number of each.

16. A factory each day makes 1000 more doors than pairs of blinds and ^ as many pairs of blinds as windows. If it makes 500 more windows than doors, what is the daily output of each?

17. The length of a card is 4 inches greater than its width. If its length were 2 inches more and its width 1 inch less, its length would be twice its width. Find its dimensions.

18. The length of a ship is 384 feet more than its width. If its width were 1^ times as great, it would be 10 feet less than ^ of its length. Find the length of the ship.

19. The length of a stadium is 196 feet more than its width. If its length were 70 feet less and its width 26 feet more, its width would be f of its length. Find its dimensions.

20. A picture is 4 inches longer than it is wide. Another picture which is 12 inches longer and 6 inches narrower has the same area. Find the dimensions of each picture.

21. The cost of a balloon with gas and outfit was $775. The balloon cost 4 times as much as the gas to fill it, and the gas cost $25 less than the rest of the outfit. How much did the balloon cost ?

22. The length of a projectile discharged by a large gun is 4 inches less than 4 times its diameter. If the diameter were 4 inches less than it is, it would be ^ of the length. Find the length and the diameter of the projectile.

23. One year 4116 vessels passed through the Suez Canal. This number was 2 more than 34 times the increase of that year over the preceding year. How many vessels passed through the canal the preceding year?

462 SUPPLEMENTARY EXERCISES

FACTORING 618. Factor : [Supplementing § 146.]

1. c2 + 2cd. 3. Sy-2y\ 5. da'-Sa^b.

2 4:0^- Sx. 4. 6x'-Sx. 6. 4: x'^y -\- 12 xy^

7. 3mV + 7mri^ 13. b' + Sb'- 2b^- 5 ah\

8. 5 6^ + 10 62 -f 15 6. 14. a(x + 22/)H-3(a;+22/).

9. 3x* + 3x^y — 9xY. 15. mV + m??-^ + 5 m V.

10. 9xY + Sxy-6xy\ 16. a6(2 c-3d)-4(2c-3d).

11. 4r3s2_8^,2g2_^2rs*. 17. a6 - 2 a^^ - 3 a6c + a^^/c^.

12. 7 a6 + 14 a^c + 7 a'^6l 18. Saa^f -IS ba^y*-h 15 cxy.

[Supplementing § 147,]

19. 3a-\-ab-{-3b-\-b\ 25. 2 m + am + 2a -f-a^

20. iic^ + xy-\-xz-\-yz. 26. 2 a + 3 6 + 2 am + 3 bm.

21. a^-{-3x-\-xy + 3y. 27. 3 op^ — a^^ _^ 3 fep^ _ ^,^2

22. ac — ad 4- 6c - bd. . 28. 2 a^i/^ — 2yz-{- o^yz^ — 2;^.

23. ex — dx — cy-\- dy. * 29. 3 a6 — ac + 6 6d — 2 cd.

24. 2y^-Jr4:y-yz — 2z. 30. aV + 2 a?'2s - a26rs — 2 68^

[Supplementing § 151.]

31. 16a^-Sxy + y\ 36. 4: xr"" -{- 16 x'^y'' -{- 16 f\

32. 4 + 12 m^ + 9 m*. 37. 1 + 10 mw + 25 mhiK

33. 9 - 12 a6 + 4 a262. 38. ax^ -\- 6 axy^ + 9 ay\

34. s2-12rs + 36r2. 39. a262 - 2 a6cfi + c^d^.

35. 4 a^ 4- 4 axy + ay. 40. 9 xY + 12 xyrs + 4 r^sl

[Supplementing § 153.]

41. 25 -tt^. 45. 9a262-l. 49. 4a2"-25.

42. a;* - 4 f. 46. 4-49 c'd^ 50. 75 ar^ - 27 /.

43. 9c2-4. 47. 4b^-a^d^ 51. 36y*-Slz\

44. a62-4a3. 48. t^^-16y^ 52. 9 a262 - 16 c^d^.

71.	?-2 + 2ar-3a2.
72.	24: - 2 m^-m\
73.	x^-hSaxy-Aay.
74.	2/2-7ac?/-8aV.
75.	z'-S abz - 18 a'b''.
76.	62_l_6a5c-16aV.
77.	4 m^ — 4 m?i — 48 7i^.
78.	5 0^0 + 20 abc + 15 b^c.

SUPPLEMENTARY EXERCISES 463

Factor : [Supplementing § 15.5.]

63. «2-3aj4-2. 66. x'' -{- bx - 20 b\

54. a2 + 9a + 14. 67. 2x'-2xy-12y\

55. ?/2-5 2/ + 6. 68. -12r2-dr + d2.

56. 62^26-15. 69. 4i»y-a;2-|-2l2/^

57. m^ 4- w — 30. 70. cx"^ -\-'dcx — 40 c.

58. a;2 4-7i» — 60.

59. p2_9j3_9o.

60. 13aj + 42 + icl

61. _5a; + a;2-36

62. 16 + 62/-2/^

63. n^-9n2-52.

64. — a;2 + a;4.l2.

65. y'^ — ay — 2aK

[Supplementing § 166.]

79. 4i»2-lla;-3. 89. 5ap-p^-Qa\

80. 6 + 5 ?• — 6 r^. 90. 3 a;2« -j- 19 a;" — 14.

81. 122/2 + 11^ + 2. 91. 6a^ + 2a;i/-482/^

82. 4aj2+16«+15. 92. aV - 3 a6a; - 4 ft^.

83. 8 aj2 + 26 oj + 15. 93. 4 «y + 6 icz/ - 40.

84. 6 a;6- 23 05^-35. 94. 1Q> q^ -^r ^ pq - ^ p\

85. SxY-^xy-Q. 95. 9ic2 + 21 (^a;+ 10(f.

86. -16 2/2 + 20 2/ + 24. 96. 18 6^ + 3 aft^ _ 10 a^.

87. 2tt2-3a6-262. 97. 10a2_25a62_ 60 6^

88. 6 a2 - 7 ac- 20 c2. 98. 5 a^ft^ + 9 a6cri + 4 c^dl

[Supplementing § 158.]

99. 1-f. 104. a^+125. 109. f^'+^ — y.

100. z^^-f. 105. 64-82/^ HO. 3^3+8161

101. 16 a^- 2. 106. ^a^-Wc\ 111. ^Qi? + 21y^.

102. a;i»+ 0^2/3^ 107. 64c3 + f^6 112. 64a^ + 1252/3.

103. ^3-216. 108. 21a?-a?f. 113. 27 (a + 6)3 + 8.

464 SUPPLEMENTARY P^XERCISES

Review of Factoring

619. Factor orally : [Supplementing § 171.]

1. a;2-4. 7. a^ + dl 13. dV-1.

2. 16-2/1 8. x'-^x-^h. 14. a;Y-64a2.

3. 2a5 + a. 9. 7^ + 6 r + 9. 15. m^ + Sm + T.

4. a2-16d2. 10. 4a2«-16 62. le. (/ + 4)2-4 2/2.

5. aj2 + 5aji/. 11. aa^ + aV. 17. {a -lif - {c -df.

6. 9r2-25s2. 12. a^ + 2a;2+l. 18. a;^ + 20 a;?/ + 100 y.

Factor :

19. ^® — 8. 24. a^ft^— 1. 29. a;^+/.

20. 68-81. 25. 3a^-12a;. 30. a^h-h&.

21. c^d^— 1. 26. 0^2/ - 2/^- 31. a;^ + a? — 6.

22. d^y''' - z\ 27. 27 6^ + 6cl 32. 64 j^^ - 53.

23. c* + 64 d\ 28. a^^'^+i - a;?/^". 33. 6a;^ - 16 d^h.

34. ^2 - 8 2 - 9. 48. a2 H- 2 ac - 62 + cl

35. 9 + 12 d + 4 d2. 49. Q ax - hy + ^ ay -2 bx.

36. ^2 + 6;2 _ 12 62. 50. 8 ar^ - 8 c2 - 32 a; + 32.

37. aj^ + 2/4 _ 7 ^2/2. 51. 81 p^ - 54: pq + 9 q\

38. c2 — 17 c + 72. 52. 3 ca; + 9 c?/ — 4 da; — 12 dy.

39. 4 a2 + 32 a + 39. 53. 4 a^ - ?/' + a^^V _ 4 2^2/2.

40. d'-d2-d4-l. 54. 36 c - 21 ca;- 135 ca?^.

41. a2 - 6 ad - 72 ^2. 55. 9 a262c 4- 6 ab^c + b*c.

42. 16 a2 + 8 a6 - 15 b\ 56. 12 d2a;2 - 4 d2a;2/ - 16 ^22/2.

43. 9 a;4 + 12 a^^2 -f_ 4 2/^. 57. 48 ar^ - 72 axy + 27 a23/2.

44. 3 a*-30 a262+75 6^ 58. 6 aV -^a'x-G aV - a'a^.

45. 28 a; -15-12 ar^. 59. a;* - 2 a;2 (a - 2/) + (a - 2/)'-

46. (m — 2 a)2 — 4(a + 6)2. 60. apq + bcpq — az- bcz.

47. 12a2-llad-15d2. 61. (a-a;)2 4-8(a -»)« + 16a^.

SUPPLEMENTARY EXERCISES 465

FRACTIONS

620. Reduce to similar fractions having their L. C. D. : [Supplementing § 203.]

7. ^^ ^

1.	Scd 5bd 7 ' 3 '
2.	2m Q„
3.	a 6 3 rs' 6 crs
4.	m w
5.	3b 1 C2_c?=^' C + rf
6.	32 + 2 32/~2
8 a;/ ' 12«2;2
Sm 13.	iplify : [Suppleme 8^_3a^ ab be
14.	2x X X 3 "^6 4*

(x + yy Sa{x-\-y)

2 2

a? — 1 35 — 2

:> «:

10. a, ,

13 4a

11. 12.

5 a 3 a a6

a;2 4_a;-2' «2_4' ^j-l

-^ a — 26 a + 7 6

2a 8a

a; + 7 a 2a; + a

18.

26 56

15. ^ + A+ 27. ^^ 2^+^_ 13 _i. a; 2ic 8ar^ 3a; Sx^ 2x

16. y + 2 ^ y + 3 2^^ ^Lzj? + ^L±^_f_^[!zi^.

3 5 * a; a 2 aa;

Reduce the following mixed expressions to fractions :

21. 3y-^. 24. 3r-^l+i. 27. 4.,- 1^.

4 2 ic — 2/

22. a6 + ?:. 25. -^ + 3. 28. a-64--^.

a; a; + l a + 6

23. ^^-4.. 26. ^-1. 29. x^y-t±t.

milnb's stand, alg. — 30

466 SUPPLEMENTARY EXERCISES

Simplify : [Supplementing § 208.]

30. ^Xl6. 37. -li^.^^ZlJ?.

4 5 a - 10 4 a^

31. ^^X — - 3Q a;'^ + 2a; + l ^y\ ' he ah y x^ — 1

32. ^x^. 39. ^-^ 1-.

12 a" 10 6 2a^6^ 5c^ 18 6c

* 5 62 ^*9c ' * 3c3 • 453 ■ 25 a^*

34 «^y^^. 4, c+c? 10 a;'' ac — 2 gc?

* or^^/ abc' ' 6x ' &- cd-2 d"^' 2x _^ 3a'c2 \2xz a-h a'-h'- 4

4:xyz 27 aV (« + 6)2 3 o?-2ah + h''

36 ^f!zL^x-i^. ^3 a;2 + a?-2 3 x-Z

' a^ + ah a-h * 4 a; -12 ' a^-x-Q, x-1

[Supplementing § 212.]

44. 4a6 . 12 6^^ gg a^ + 2/^ . a?-xy^y^

xy ' xy^ ' ahx-\-ahy ' a^

45 5^^25 a6-6^^^-2a6+62

' Sc^ ' Oc'* * 3 0^2/3 • 27a^f '

46. L^^21c3d. 54. iziM._j_(4a^ + 2a; + l). 3 a — 2 aa;

^^ xhi-\-yz . ay + 6v , ^^^ ^^ ^^2^2) . 3a; + 6yg

" 18 9 ot

48 8 . 2a;+2y 4«-86 . a2-a6-262

aW-a%' ah ' ' '(a-hf'' a^-1^

^ — \p- . x-\-y {r — 2 sf . r^ — rs — 2^

4 * 12 a * ' ~T- s ' ?-2 _ ^^2

50. ^-^ ^(2+a;). 58. «^-2^.V-3.V-^(a^-3yy 2-3a;+a;2 ^ ^ a?' + 2 a;?/ + 2/^^ a; + 2/

-, xy-x^y'^ a-axy ^^ 2a^-\-ah-Sh^ ,a^-W

01. — ^ -T~ ' . 5»7. -J" '

axy a^x+a^y 2a^-{-5ah + Sh^ a + h

SUPPLEMENTARY EXERCISES 467

SIMPLE EQUATIONS Problems

[Supplementing § 235.]

621. 1. The sum of two numbers is 142 and the first ex- ceeds the second by 14. Find the numbers.

2. The sum of two numbers is 40. The second is 4 more than twice the first. Find the numbers.

3. Six times a certain number exceeds 4 times the number by 24. Find the number.

4. The sum of a number and .03 of itself is 24.72. What is the number ?

5. The sum of .5 and .7 of a certain number is 36. Find the number.

6. If 7 times a certain number is diminished by 2.4, the result is 53.6. Find the number.

7. A and B together pay $ 126 in taxes. A pays twice as much as B. Find the tax that each pays.

8. In an election A received a majority over B of 72 votes, 584 votes being cast. Find the number of votes cast for each.

9. A certain typewriter has been increased $40 in price. The present price is $20 less than twice the original price. Find the original price.

10. An oak waste basket costs $ 8, which is 20 % less than the same style made in mahogany. Find the cost of the latter.

11. A firm made a certain kind of shoe for 30 years, which was 4 years more than twice as long as it had manufactured another kind. How long had it made the latter kind ?

12. A tribe of Indians live 50 miles from the railroad. To reach them, the distance traveled by wagon is 5 miles more than twice the distance traveled by trail. What distance is traveled by trail ? by wagon ?

468 SUPPLEMENTARY EXERCISES

13. A certain clock sells in Canada for $3, which is 20 % more than the price received for it in the United States. Find the selling price in the United States.

*

14. The perimeter of a rectangle is 112 feet and the rec- tangle is 8 feet longer than it is wide. Find its dimensions.

15. A field is 10 rods longer than it is wide. The perimeter is 132 rods. Find its dimensions.

16. The length of a rectangle is 120 feet more and its width is 40 feet less than the side of a square of equal area. Find the dimensions of each.

17. A field is 5 rods longer than it is wide. If its length is increased 3 rods and its width 4. rods, the area is increased 116 square rods. Find its dimensions.

18. A rectangle is 4 feet longer than it is wide. If the width is increased 5 feet and the length decreased 3 feet, its area is increased 29 square feet. Find its dimensions.

19. A certain rug is 3 feet longer than it is wide. If 3 feet were added to each side, it would contain 72 square feet more. Find the original dimensions.

20. The length of a field is m times its width. Another field is a yards longer and b yards wider and contains c^ square yards more than the first. Find the dimensions of the first field.

21. A building with its tower is 909 feet high. If the height of the main structure is 69 feet more than that of the tower, what is the height of the tower ?

22. On a trial run a gasoline torpedo boat used 220 gallons of gasoline. If it had used 3 quarts more per mile, 352 gallons would have been used in all. Find the length of the run.

23. The length of a bridge is 10 times its width. If its length were 50 feet less and its width 5 feet more, the former would be 5 times the latter. Find the length of the bridge.

SUPPLEMENTARY EXERCISES 469

24. A and B together have m cents. If A has n cents more than half as many as B, how many cents has each?

25. A maple grove contained originally 2868 trees. It was thinned out so that the number of trees left was 200 less than the number removed. Find the number of trees removed.

26. A city had 506 miles of streets. The number of miles paved was 20 more than 80 times the number of miles unpaved. How many miles were paved ?

27. A train traveled a certain distance in 5 minutes. If its rate per minute had been ^ of a mile more, it would have covered the distance in 41 minutes. Find its rate per minute.

28. A certain well was deepened to 4 feet more than twice its original depth. If it was finally 22 feet deeper than it was at first, what was its depth after it was deepened ?

29. The per cent of copper contained in United States standard bronze is 30 times the per cent of zinc, which is f that of tin. Find the per cent of each metal in the bronze.

30. The amount of copper contained in a five-cent piece is 3 times the amount of nickel. Find the per cent of each metal in a five-cent piece.

31. A loin of beef was sold for 11 ^ more than twice its cost. If the gain was lOlf %, what was the cost ?

32. A man invests | of his capital at 4^ % and the rest of it at 4 %. His annual income is $176. Find his capital.

33. A house cost a dollars and rents for n dollars a month. What per cent per annum is the income from the investment ?

34. At what time between 7 and 8 o'clock are the hands of a clock together ?

35. At what two different times between 5 and 6 o'clock are the hands of a clock at right angles to each other ?

36. Find two different times between 7 and 8 o'clock when the hands of a clock are 14 minute spaces apart.

470 SUPPLEMENTARY EXERCISES

37. A can do a piece of work in 4^ days, B in 5 days, and C in 7i days. In how many days can all together do it ?

38. A can do a piece of work in c days, B in e days, and C in r days. In how many days can all together do it ?

39. A cistern is being filled by two pipes. One pipe fills it in m hours and the other in n hours. How long does it take both flowing together to fill it ?

40. In a number of two digits, the units' digit is 2 more than the tens' digit. If the number less 6 is divided by the sum of the digits, the quotient is 4. Find the number.

41. In a number of two digits, the tens' digit exceeds the units' digit by 4. If the digits are reversed, the number formed is ^ of the original number. Find the number.

42. A man has \ of his property invested at 4 %, ^ at 5 %, and the remainder at 3%. For how much is his property valued, if his annual income is $ 610 ?

43. Two men starting from Albany travel east and west, respectively, one traveling 3 times as fast as the other. In 6 hours they are 360 miles apart. Find the rate of each.

44. A man went to a village and back in 1 hour and 45 minutes. He walked out at the rate of 4 miles per hour and rode back at the rate of 7 miles per hour. How far away was the village ?

45. In an alloy of 40 ounces of silver and copper, there are 8 ounces of silver. How much copper must be added so that 60 ounces of the new alloy may contain 4 ounces of silver ?

46. In a mixture of 5 ounces of water and alcohol, there is 1 ounce of alcohol. How much alcohol must be added to make the new solution 40 % pure alcohol ?

47. A regiment drawn up in the form of a solid square lost 60 men in battle, and when the men were rearranged with 1 less on a side, there was 1 man left over. How many men were there in this regiment ?

SUPPLEMENTARY EXERCISES

471

SIMULTANEOUS SIMPLE EQUATIONS

622. Solve by addition or subtraction : [Supplementing § 246.]

2.

4.

5.

6.

13.

14.

15.

16.

17.

Sx-{-4.y =	29.
3x- y =	4.
2x+ y =	-2,
5x-[-3y =	-3.
6x-\-5y =	13,
4.x + 2y =	10.
'4a;+ y =	7,
6x+2y=	13.
x-h y =	-5,
3x-\-2y =	-13
2x-4.y:=	5,
5x+Sy=	17.
ve by substi	tutior
x + Sy =	:9,
2x-\-4:y =	:14.
\3x + 4.y =	:34,
1 ^- y =	:-5.
x-5 =	--2y,
2x-^6y =	:15.
y-{-2x = -	-8,
x+2y=-	-7.
3 a- 46=	0,
2a=& + 10.

7.

8.

9.

10.

11.

12.

18.

19.

20.

21.

22.

3 a - 6 = 2 6,

4 a - 3 = 3 6.

.5a; + .2 2/ = 2,

3 m 4- 6 ?i = 5,

6 m + 4 = 9 ri.

.4r-.3s = -l, .6r-.3s = 0.

3p4-4g = 100, 8^-3^ = 48.

( 8 a + 81 = - 5 6,

4 a + 23 = 6.

4 c - 3 d = 1, 2c = 3d

y- aj = 11, x + 3?/ = l.

^ + 2/ = 0,

3 ?/ + 2 a; = 4.

48 = 3i) + 2g, 4 = 5 p — 3 g'.

4 a; - 13 = - 2/, 2/_ 8 = 16 a;.

472 SUPPLEMENTARY EXERCISES

Problems [Supplementing § 254.]

623. 1. Find two numbers the sum of which is 24 and the difference between which is 12.

2. The sum of two numbers is 18 and J of the difference between them is 2. What are the numbers ?

3. Find two numbers the sum of which is s and the differ- ence between which is d. ■

4. The length of a rectangle is 14 inches more than its width and its perimeter is 124 inches. Find its dimensions.

5. A man deposited in the bank 35 bills valued at $ 100. Some were 2-dollar bills and the rest were 5-dollar bills. How many bills of each kind did he deposit ?

6. Two women shelled 165 pounds of hazelnuts. If one had shelled 10 pounds more and the other 5 pounds less, they would have shelled equal amounts. How many pounds did each shell ?

7. Two large apples together weighed 64 ounces. If the weight of the smaller apple had been 13 ounces less, it would have been ^ of the weight of the larger one. Find the weight of each.

8. One man earned $ 6 more per week than another. If the former earned in 5 weeks as much as the latter did in 8 weeks, how much was the weekly wage of each ?

9. The sum of 2 times one number and 5 times another is 20 and the sum of 3 times the former and 4 times the latter is 23. Find the numbers.

10. On its first trip, an ocean liner made 762 knots in two days. If the second day's run was 152 knots more than the first day's run, how many knots did it make each day ?

11. A large canal lock is 7} times as long as it is wide. If the width had been 5 feet less, it would have been ^ as much as the length. Find the length and width of the lock.

SUPPLEMENTARY EXERCISES 473

12. If 3 yards of ribbon and 2 yards of lace cost $ 1.64 and, at the same price, 2 yards of ribbon cost 1 ^ more than 3 yards of lace, how much does each cost per yard ?

13. If 3 cotton grain bags and 2 burlap bags hold 15|- bushels, and 4 cotton grain bags and 3 burlap bags hold 22 bushels, what is the capacity in bushels of a cotton grain bag ? of a burlap bag?

14. I find that I can buy 10 boxes of the best quality of matches and 4 boxes of the poorest quality, or 8 boxes of each, for 24 ^. Find the cost per box of matches of each quality.

15. A bird flying with the wind goes 55 miles per hour, and flying against a wind twice as strong it goes 30 miles per hour. What is the rate of the wind in each case ?

16. A man can row m miles downstream in c hours, and m miles upstream in d hours. Find his rate of rowing in still water ; the rate of the current.

17. If a pipe line had been 100 feet longer, its length would have been 200 times its diameter. If its diameter had been 11 feet more, its length would have been 100 times its diameter. Find the length of the pipe line ; of the diameter.

18. The average weight of cast iron per cubic foot is 30 pounds less than that of wrought iron. If 16 cubic feet of the former weigh the same as 15 cubic feet of the latter, what is the average weight of each per cubic foot ?

19. Watermelons contain 10% more water than currants do, and 17 times the per cent of water in the former equals 19 times the per cent of water in the latter. Find the per cent of water in each.

20. Bricks in common use are of two sizes, a brick of one size weighing 21 pounds more than a brick of the other size. If 6 bricks of the large size and 5 bricks of the small size weigh 92 pounds, what is the weight of a brick of each size ?

474 SUPPLEMENTARY EXERCISES

21. The sum of the reciprocals of two numbers is 4|-, and one number is 32 times the other. Find the numbers.

22. The sum of the reciprocals of two numbers is m, and one number is I times the other. What are the numbers ?

23. The sum of the two digits of a number is 12, and the units' digit is twice the tens' digit. Find the number.

24. A two-digit number is 8 times the sum of its digits. If 45 is subtracted from the number, its digits are reversed. Find the number.

25. The sum of 3 times the reciprocal of one number plus 6 times that of another number equals 4, and the first number is 1^ times the other. What are the numbers ?

26. If 1 is subtracted from the numerator of a certain frac- tion, its value becomes f ; if 10 is added to its denominator, its value becomes ^. What is the fraction ?

27. The smallest size of pecan nuts marketed has 75 more nuts to the pound than the largest size. If 2 pounds of the former and 3 pounds of the latter contain together 275 nuts, how many nuts of each size are there in a pound ?

28. A car of sacked grain is unloaded in 50 minutes less time than a car of grain in bulk. If 9 cars of the former are unloaded in the same time as 4 cars of the latter, how long does it take to unload a car of each ?

29. A bridge contains 5600 tons of steel and iron together. If 17 times the weight of the steel equals 11 times the weight of the iron, what is the number of tons of each metal in the bridge ?

30. If the bleachers at the ball grounds seated 2000 less, they would provide ^ of the total seating capacity. If the total seating capacity were 1000 more, it would be 4 times, that of the bleachers. How many do th^ l^l^chers seat ?

SUPPLEMENTARY EXERCISES 475

31. A man invested % 3250, a part at 5 % and the rest at 3 %. If the annual income from both investments was $ 117.50, what was the amount of each investment ?

32. A man invested $ 3500, a part at 4 % and the rest at 3%. If the annual income from each investment was the same, what was the amount of each investment ?

33. The number of bridges on a certain railroad is 23 more than 15 times the number of tunnels. If the number of tun- nels were 5 more, it would be -^^ of the number of bridges. Find the number of bridges ; of tunnels.

34. When weighed in water, tin loses .137 of its weight and lead loses .089 of its weight. If a solder of tin and lead weighing 24 pounds loses 2.904 pounds when weighed in water, what is the amount of each in the piece ?

35. A pier jutted into the water a distance that was 70 feet less than the length of the part upon land, and | of the total length was 190 feet more than that of the part over water. Find the total length of the pier.

36. A mixture of Canadian and Kentucky bluegrass seed is worth 7 ^ per pound and contains 3 pounds more of the former than of the latter. If the former is worth 5 i^ and the latter 15^ per pound, how many pounds of each are there in the mixture ?

37. One pipe is discharging water into a tank and another is emptying it. If the first runs r minutes and the second s minutes, there will be t gallons in the tank. If the first runs s minutes and tjie second u minutes there will be v gallons in the tank. Find the capacity of each pipe per minute.

38. The total height in feet of a monument is a number of two digits, the sum of which is 15, and which, when reversed, represents the height of the shaft in feet. If the height of the monument exclusive of the shaft is 27 feet, what is the total height of the monument ?

476 SUPPLEMENTARY EXERCISES

Problems

[Supplementing §257.]

624. 1. Separate 432 into three parts, such that the sum of the first, I of the second, and i of the third is 257 ; and the sum of the third, i of the second, and |- of the first is 267.

2. Henry, James, and Ralph together have 50 cents. Henry's and Ralph's money amounts to 35 cents ; James's and Ralph's amounts to 40 cents. How many cents has each ?

3. Three boys together have 57 marbles. If the first boy gives 1 marble to the second and 5 marbles to the third, each will have the same number. How many marbles has each?

4. A, B, and C together earn $ 7.50 a day. A and B to- gether earn twice as much as C, and A and C together earn $ 1.50 more than B. Find the daily wages of each.

5. A house, garden, and stable cost $ 12,000. The house cost twice as much as the stable. If the garden had cost $ 250 more, it would have cost ^ as much as the stable. Find the cost of each.

6. Separate the number n into three parts such that the quotient of the first divided by the second is a and the quotient of the second divided by the third is b.

7. The sum of the digits of a three-digit number is 8 and the units' digit exceeds the tens' digit by 3. If 396 is added to the number, the digits will appear in reverse order. What is the number?

8. A man deposited 20 bills in the bank, their value being $ 52. The number of 1-dollar bills was 4 times the number of 10-dollar bills. If the rest were 2-dollar bills, how many bills of each kind did he deposit ?

9. Raisins are packed in boxes of three sizes: 3 of the smallest and 2 of the largest boxes hold 55 pounds ; 2 of the middle-sized and 4 of the smallest boxes hold 50 pounds ; and 1 of the largest and 3 of the middle-sized boxes hold 65 pounds. How many pounds does a box of each size hold ?

SUPPLEMENTARY EXERCISES 477

RADICALS

625. Simplify : [Supplementing § 339.]

1. V75 + Vi2+V243. 3. vj+Vi + Vf+Ve.

2. v'128 + v^250 + \/i. 4. V44b-Vi75 + 3V63,

5. (|)^_2Vi+(5)-^-V5.

6. 6.3-^ + 9^-3.2-1.3^+^.

7. V^-A/acy22 + -v/8^W-

8. V2 a^ - 12 a^b + 18 ab^ - V2 a^ + 16 a'b + 32 aft^.

Find products : [Supplementing § 34L]

9. V5xV8. 17. Va6x\/a36.

10. 3V7x2Vl2. 18. ^24x2V32.

n. 3V8x5Vl6. 19- 4^/2x^/32.

12. 2V7X3V15. 20. 3^6x4Vi2.

13. 4-^x3^21. 21. Vlx-^Ix-v/f

14. V^x7v/^. 22. ^|xV|xV|.

15. 2^x2Vl8. 23. (12)^x(6)^x(8)i

16. Vl5x3V24. 24. (V7 + V2)(V7-V2).

25. (2V6 + 5V2)(3V6-4V2).

Find quotients : [Supplementing § 343.]

26. V27-r-V6. 30. ^12--V8. 34. Vf-f-Vf.

27. V30-2V3. 31. 3^3-^12. 35. v^l-A/f.

28. 5V5-f-V20. 32. Vax--r-\/ab. 36. V|-^^|.

29. V125--V5. 33. -s/^^Vbx, 37. Vi-^^f

38. (V6-6V3 + 9)h-V3.

39. (V42 + 6V5)-^(V7 + V30).

478 SUPPLEMENTARY EXERCISES

QUADRATIC EQUATIONS

Problems

[Supplementing § 380.]

626. 1. If a number is multiplied by | of itself, the prod- uct is 216. Find the number.

2. The square of a certain number plus the square of i of the number equals 80. What is the number ?

3. The area of a blotter is 32 square inches and its length is twice its width. Find its dimensions.

4. The sum of two numbers is 10 and their product is 24. Find the numbers.

5. Find two numbers the difference between which is 6 and the sum of the squares of which is 68.

6. The perimeter of a floor is 42 feet and its area is 110 square feet. Find its length and its width.

7. A bushel of seed cost 90 cents. If the number of pounds in a bushel was 10 times the number of cents the seed cost per pound, what was the number of pounds in a bushel?

8. The weight of some boxes was 192 pounds. If the number of boxes was 3 times the number of pounds that each weighed, what was the number of boxes ?

9. A factory made 180 tons of cardboard. If the number of tons made daily was 5 times the number of days necessary to make the whole amount, what was the daily output ?

10. A cow's annual yield of butter cost $30. If the number of pounds was 30 times the number of cents it cost to produce each pound, what was the annual yield of butter ?

11. There are 250 needles in a bundle. The number of packages in each bundle is f of the number of needles in each package. Find the number of needles in each package.

supplemp:ntary exercises 479

Affected Quadratics 627. Solve by factoring, and verify: [Supplementing § 385.]

1. a;2-a;-6 = 0. S. 3x^ + x-4: = 0.

2. a;2-3a;-10 = 0. 9. 2 s^- 7 s-j-5 = 0.

3. a;24-9a;-22 = 0. 10. 8 r^-f- 10 r- 3 = 0.

4. x^-\-2x-24: = 0. 11. 7(a;2-l) = 48aj.

5. a;2-6a; = 40. 12. 6(2 z^ -1) -\-z = 0.

6. i»2 = 5(4a;-15). 13. 8 a;^ -f 29 a; - 12 = 0.

7. x^-x = 6{x-{-10). 14. 10 a;2 -I- 21 a; -10 = 0.

Solve by completing the square, and verify: [Supplementing 387.]

15. x^-6x = 7. 22. x^-x = ^2.

16. x^-\-2x = 35. 23. 'v2_^7v = 120.

17. a;2_4a. = 77. 24. x(x — 5) = 104:.

18. a;2-12« = 13. 25. r2-3r + 2 = 0.

19. a;2 + 24 a; = 112. 26. 2 a;^- 5 a; + 3 = 0.

20. y^-\-12y = 2SS. 27. 3a;2 + 8 a;- 28 = 0.

21. z^-10z = 200. 28. 5a;2 + 2a;-51 = 0.

Solve by formula, and verify: [Supplementing § 391.]

29. 2aj2 + a;-6 = 0. 36. 3a;2 = 7a;-2.

30. 2ar^-5a; + 2 = 0. 37. 5a;2 + 8a; = -3.

31. 15a;2-a;-2 = 0. 38. a;^ z= 3 (1 - 2 a;).

32. 12aj2-5a;-2 = 0. 39. a;2 4-9a; + 4 = 0.

33. 11 a;2- 10 a;- 1 = 0. 40. 4a;2 + 7a; + 3 = 0.

34. 16a;(a; + l) = -3. 41. 2 (a;^ + 2 a;) = 5.

35. 14»2 = 25(l-a;). 42. 3(a;2 - 1) + 5 aj = 0.

480 SUPPLEMENTARY EXERCISES

Problems

[Supplementing § 398.]

628. 1. Separate 26 into two parts whose product is 165.

2. Find two consecutive integers whose product is 272.

3. Find two consecutive integers the sum of whose squares is 113.

4. Find two consecutive odd integers the sum of whose squares is 2 (a" -f. 1).

5. The area of a floor is 180 square feet. If its length is 12 feet more than 3 times its width, what are its dimensions ?

6. If the length of a square ventilator were 4 feet more and its width 6 feet less, its area would be 200 square feet. Find its dimensions.

7. A shipment of glass consisted of 80 sheets. If the number of sheets in each box was 2 more than the number of boxes, how many sheets of glass were there in each box ?

8. The number of cords of wood that a man bought was 30 less than the number of cents he paid per cord. If the total cost was $ 130, how many cords of wood did he buy ?

9. In a paper of pins, the number of pins in each row is 6 more than twice the number of rows. If the paper con- tains 360 pins, how many pins are there in each row ?

10. The height of a box is twice its length. If the area of the bottom is 6^ square inches and its perimeter is 10 inches, what are the dimensions of the box?

11. The area of a sheet of paper is 208 square centimeters. If it were twice as wide, its width would exceed its length by 10 centimeters. Find its length ; its width.

12. A train carried 12,800 barrels, the number in each car being 40 more than 7 times the number of cars. How many cars were there? how many barrels in each car?

SUPPLEMENTARY EXERCISES 481

13. The sum of a certain number and its reciprocal is ^-. Find the number.

14. Find two consecutive integers the sum of the reciprocals of which is |.

15. If c times the reciprocal of a number is subtracted from the number, the result is c — 1. Find the number.

16. A poultry car accommodates 6720 fowls. The number of crates lacks 8 of being twice the number of fowls in each crate. How many crates are there ? how many fowls in each crate ?

17. If a dredge could lift 50 tons more of sand per minute, it would take 10 minutes less time to lift 10,000 tons. How many tons can the dredge lift per minute?

18. Two Alaskan dogs hauled freight 20 miles. If it had taken them 2 hours longer, the distance traveled per hour would have been f of a mile less. How long did it take them?

19. A man earned $1.60 by picking cherries. If he had filled 5 baskets more and received 2 ^ more per basket, he would have earned $ 2.50. How many baskets did he fill ?

20. A machine makes 6250 square feet of paper per minute. The number of feet in its length is 50 more than 3 times the number of inches in its width. Find the length of the paper.

21. A party of people agreed to pay $12 for the use of a launch. As 2 of them failed to pay, the share of each of the others was 50 cents more. How many persons were there in the party ?

22. A tank can be filled by two pipes in 35 minutes. If the larger pipe alone can fill it in 24 minutes less time than the smaller pipe, in what time can each fill the tank ?

23. A man shipped 13,440 pineapples packed in crates. If there had been 4 pineapples less in each crate, 112 extra crates would have been required. How many pineapples were there in each crate ?

milne's stand, alg. — 31

482 SUPPLEMENTARY EXERCISES

Problems

[Supplementing § 417.]

629. 1. The sum of two numbers is 8 and the sum of their squares is 40. Find the numbers.

2. The difference between two numbers is 2 and the differ- ence between their cubes is 26. Find the numbers.

3. The product of two numbers is s^ and the difference be- tween them is 8 times the smaller number. What are the numbers ?

4. The perimeter of a floor is 44 feet and its area is 120 square feet. Find its length and its width.

5. An electric sign is 10 feet longer than it is wide and its area is 6375 square feet. Find its dimensions.

6. The perimeter of a right triangle is 12 feet and its hypotenuse is 1 foot longer than its base. Find its base.

7. If a two-digit number is multiplied by the sum of its digits, the result is 198. If it is divided by the sum of its digits, the result is 5^. Find the number.

8. The denominator of a certain fraction exceeds its nu- merator by 1, and if the fraction is multiplied by the sum of its terms, the result is 3^. Find the fraction.

9. The amount of a sum of money for one year is $ 3990. If the rate were 1 % less and the principal were $ 200 more, the amount would be $4160. Find the principal and the rate.

10. A man packed 2000 pounds of cherries in boxes. If each box had contained 6 pounds more, he would have used 75 boxes less. How many boxes did he use and how many pounds of cherries did each contain ?

11. A farmer received 20^ less per bushel for oats than for rye, and sold 3 bushels more of oats than of rye. The receipts from the oats were $4.50 and from the rye $4.20. Find the number of bushels of each sold and the price per bushel.

SUPPLEMENTARY EXERCISES 483

GENERAL REVIEW

Problems

[Supplementing § 443.]

630. 1. Separate 60 into two parts such that J of the larger exceeds J of the smaller by 6.

2. The sum of two numbers is 85 and the difference between them is 19. Find the numbers.

3. Nine times a certain number divided by 6 less than the number equals 1^ times the number. What is the number?

4. The difference between two numbers is a and the dif- ference between their square roots is V6. Find the numbers.

5. A rectangle is 6 feet longer and 3 feet narrower than a square of equal area. Find the dimensions of the rectangle.

6. A rectangle is u feet longer and v feet narrower than a . square of equal area. How long is the square ?

7. Some cotton seed yielded 600 pounds of oil, this being 15 % of the weight of the seed. How much did the seed weigh ?

8. In a bronze, the per cent of copper is 5 more than that of iron, ^\ times that of aluminium, and 3 more than twice that of nickel. Find the per cent of each metal.

9. If the length of a square were 6 inches more and its width 6 inches less, its area would be 2f square yards. Find the dimensions of the square in feet.

10. If it had taken an automobile 7 seconds less time to travel 5 miles, its average speed would have been 10 rods per second. Find the time required to travel the whole distance.

11. The greater of two numbers divided by the less gives a quotient of 4 and a remainder of 2. If the greater exceeds the less by 50, what are the numbers ?

12. A wooden block contains 112 cubic inches and its length is twice its width. If its width were increased 1 inch, it would contain 140 cubic inches. Find its dimensions.

484 SUPPLEMENTARY EXERCISES

13. If 20 bushels of rye brought $ 5 more than 15 bushels of corn, and 15 bushels of rye brought $7.50 more than 5 bushels of corn, what was the price per bushel of each ?

14. The height of a viaduct is 6 feet less than .15 of its length and 10 times the former is 1030 feet more than the latter. Find the height of the viaduct ; the length.

15. A and B can do a piece of work in I days, B and C in m days, and A and 0 in n days. How long will it take them to do the work, if they work together ?

16. A box contains 2560 firecrackers. The number of pack- ages is 24 less than the number of firecrackers in each package. How many packages are there in the box ?

17. The length of a box is equal to the sum of its width and its height, and its volume is 120 cubic inches. If 5 times its length equals 8 times its height, what are its dimensions ?

18. A man earned $ 9 by gathering peanuts. The number of cents he earned per day was 4 less than 3 times the number of days he worked. How many days did he work ?

19. The area of a large electric sign was 2250 square feet. If the width had been 5 feet more, it would have been ^ of the length. What were the dimensions of the sign ?

20. A and B together can do a piece of work in 4|- days, but after they have worked 3 days, C joins them and they finish it in I of a day. In what time can C do it alone ?

21. If 125 gallons less of water had flowed into a well per minute, it would have taken 1 minute longer for 3750 gallons to flow in. How many gallons flowed in per minute ?

22. A man raised celery enough to fill 8500 crates. To pro- duce the same amount on 2 acres less, each acre would have to yield 212i cratefuls more. Find the yield per acre and the number of acres devoted to celery.

3. Solve

SUPPLEMENTARY EXERCISES 485

[Supplementing § 444.]

631. The following questions were asked in a recent Ele- mentary Algebra Examination of the Regents of the University of the State of New York.

References show where the text provides instruction necessary to answer these questions.

1. Solve and check ^ - ^^^ = 12 - ^±i - x. (§ 232.)

2. Extract the square root of 4c^— 4c^H-5c^— 2c + l. (§ 293.)

a;H-22/ = a,

2x-y = h. (§253.)

4. Simplify V| 4- V|. (§§ 331, 339.)

( V5 - V2)(2 V5 + 3 V2). (§ 341.)

5. In 5 years A will be twice as old as B ; 5 years ago A was 3 times as old as B. Find the age of each at the present time. (§ 235 or § 254.)

6. Find the quotient to three terms and the remainder, v^rhen 11 a« - 5 a + 12 - 82 a^ + 30 a^ is divided by 2 a - 4 + 3 a^. (§ 132.)

7. (a) What is the dividend which, divided by x, gives a quotient of y and a remainder of z? (§§ 138, 234.)

(6) If a apples are sold for a dime, how many can be bought for c cents? (§§ 119, 138.)

8. Two men, A and B, can dig a trench in 20 days. It would take A 9 days longer to dig it alone than it would take B. How long would it take B alone ? (§ 398 or § 417.)

9. Find three successive even numbers whose sum is | of the product of the first two. (§ 398.)

10. Simplify j-^. (§ 214.)

X + - + 1

X

486 SUPPLEMENTARY EXERCISES

A recent Regents Examination in Intermediate Algebra.

1. Factor a*H-a262_^^4^ (§167.) 4.a^-[-4.h\ (§§146,157.) 2 a^c - 2 ahc + 2 hh. (§ 146.) Find their H. C. F. and L. C. M. (§§ 179, 183.)

2. Simplify each of the following expressions :

(a) V3x^2x-\/5. (§ 341.) (6) ^±^^. (§§355,371.) (0)-^. (1371.)

4. (a) Solve by the graphic method ar* = 2 a; -4- 3. (§ 418.) (6) Solve by the formula a^ + a; + 1 = 0. (§ 390.)

5. If the series f, fV ••• is arithmetical, find the sum of the first 15 terms. (§ 511.) If geometrical, find the 5th term. (§ 521.)

6. Determine, without solving, the nature of the roots of the equation ^ x" -^ x-[-3 = 0. (§ 428.)

7. (a) If a;"^ : 2 = 1 : a;^, what is the value of a; ? (§§ 313, 493.)

(6) Simplify and express with positive exponents : a-ftV3 JgFy ^3^3^

8. (a) Form the quadratic equation whose roots are 5 and -I- (§434.)

(b) State the relation between the roots and the coefficients in a quadratic equation. (§ 433.)

9. A merchant bought a number of barrels of apples for $120. He kept 2 barrels and sold the remainder at an ad- vance of $2 per barrel, receiving $154 for them. How many barrels did he buy ? (§ 398 or § 417.)

10. A train traveled 273 miles at a uniform rate. If the rate had been 3 miles an hour less, the journey would have taken ^ hour longer. Find the rate of the train. (§ 398 or § 417.)

SUPPLEMENTARY EXERCISES 487

College Entrance Board Examination in Algebra to Quadratics.

1. (a) Find the H. C. F. (§ 179) and the L. C. M. (§ 183) of 12 a;2 -f- a; - 6 (§ 156), 6 a;^ - 19 x + 10 (§ 156), 3i^-2x''- 12aj + 8 (§§ 147, 152).

(6) Simplify ^_^-^^^_^-^_,_^-^_^. (§206.)

Check by substituting c = 0, /i = I in the original fractions and in the answer. (§ 41.)

2. (a) Solve 3^z:l _ 2^ + 5 ^ 5|L^2^

(6) Solve a^ + a-6 a^-«-^>^2a(2c-a.)^ 233.) {2x-'dy-{- z = -2,

3. Solve 1 4:X—4:y — 3z

[6x+ 2/ -4 2 = 6. (§256.)

4. (a) Solve (3 ic- 11)^-2 = 1(27 X- 243) i (§365.)

(&) Find the value of a^ - 2 a; + 1 when x=~^'^^. (§345.)

5. (a) Simplify l^ + Va^^ll^. (§313.)

(6) Find the value of 8"^ x 16^ x 2». (§ 313.)

6. If the digits of a certain number of two figures are inter- changed, the result is 6 less than twice the original number. The sum of the digits is \ of the original number. Find the number. (§ 254.)

7. A passenger train traveling m miles an hour starts t hours later than a freight train whose rate is r miles an hour. In how many hours will the former overtake the latter ? (§ 235.)

8. At an election there are two candidates, A and B. A's supporters are taken to the polls in carriages holding 8 each, and B's are taken in carriages holding 12 each. If the voters, 740 in all, just fill 75 carriages, find which man wins the elec- tion and by what majority. (§ 254.)

488 SUPPLEMENTARY EXERCISES

College Entrance Board Examination in Quadratics and Beyond.

1. (a) Solve -^ - -A- 4- ;^ = 0. (§§ 232, 390.) x — 2 x-\-3 2 — x

x-y = l,

oo__y^5^ Check the results. (§ 403.)

y X 6

(b) Solve for x and y,

2. (a) Solve V7 a; -f- 1 - V3 a; + 10 = 1. (§ 397.) Test the values found and explain why one of them will not satisfy the equation. (§ 366.)

(5) Solve 3(a^ + 3a; + l)2-7(x2 4-3a; + l)+4 = 0. (§401.)

3. Two men, A and B, start at the same time from a certain point and walk east and south, respectively. At the end of 5 hours A has walked 5 miles farther than B, and they are 25 miles apart. Find the rate of each. (§ 417.)

4. A man worked a number of days and earned $ 75. If he had received 50 cents more per day, he would have earned the same amount in 5 days less. How many days did he work ? (§ 398 or § 417.)

5. Construct with respect to the same axes of reference the graphs of y-{-2x'^ — Sx—9 = 0f

and a; + 2/ — 3 = 0.

Estimate the values of x and y which satisfy both equations.

(§ 422.)

6. An elastic ball bounces to f of the height from which it falls. If it is thrown up from the ground to a height of 15 feet, find the total distance traveled before it comes to rest. (§ 527.)

7. (a) Find the coefficient of x^ in f2 ^^-j^^' (§ ^^^O

(b) Solve x^-\-5x-i-d = 0. What is the least integer, which, when substituted for d, makes the roots imaginary ? (§ 428.)

8. The sum of 3 numbers in geometrical progression is 70. If the first is multiplied by 4, the second by 5, and the third by 4, the resulting numbers are in arithmetical progression. Find the three numbers. (§ 533.)

SUPPLEMENTARY EXERCISES 489

Questions from Entrance Examinations at various colleges.

1. Solve a(x - a) = b{x - h). (§ 233.)

University of California.

2. Factor 2x- + ^xy + 2 y^. (§ 156.) {x - 3/)^ + 4 {x" - y^) - 21 {x 4- yy. (§ 168.) (a - hf + (c - d)^ (§ 157.) Bryn Mawr.

3. Find the L. C. M. (§ 183) of 12 o?y - 12 xf, (§§ 146, 152) 2 x2(a; + 2/)', 3 2/V - 6 a^2/3 _^ 3 ^ (§§ 146^ 14g^)^

Washington University.

4. Simplify (^^ + V'6^)(v^a» -^^2 _^-v^2)^ (§313.)

5. Solve V2 a: + 5 - V^^^ = 2. (§397.)

University of Alabama.

^ + 7f = n,

6. Solve ^^^_^^_^^^_^^^2^ (§416.) Yale.

7. Write the middle term of (w^ — 2 m"^^)^^^ (§ 556.)

Princeton.

8. If ^ = ^, prove that ^ = ^ =^^' + ^'\ (§ 493.) M. I. T.

b d 6 c? v^2_^^2 ^

9. Find the square root of 29 + 12 V5. (§ 356.)

U S. Naval Academy.

10. Express ^~ • correct to two decimal places. (§ 355.)

3V2 + 4 ^ ^

Cornell.

11. The product of two numbers is 30 and the difference between their squares is 221. What are the numbers ? (§ 417.)

Uiiiversity of Missouri.

12. For what values of k will the equation x^ + 2 (A; + 2) a? + 9 A; = 0 have equal roots ? (§ 428.) Vassar.

13. A man paid a bill of $16 in 25-cent pieces and 5-cent pieces. If the number of coins was 80, how many of each kind were there ? (§ 254.) University of Illinois.

14. Insert an arithmetic mean between 9 tt and 4 tt. (§ 515.) A geometric mean. (§ 529.) How do they compare ?

Dartmouth.

INDEX

(The numbers refer to pages.)

Abscissa, 201. Absolute number, 13. Absolute term, 68, 285. Absolute value, 25. Addend, 25.

Addition, 25-27, 30-33, 61-62, 135-140, 249-250, 269-270, 444, 447, 448, 465, 477.

defined, 25.

elimination by, 178-179, 471.

of complex numbers, 444.

of fractions, 135-140, 465.

of imaginary numbers, 269-270, 444.

of monomials, 30-31, 448.

of polynomials, 32-33, 448.

of radicals, 249-250, 477. Affected quadratic equations, 285-302, 326-329, 337-349, 479, 480-481.

defined, 285.

general directions for solving, 291.

graphic solution of, 326-329, 337-338.

solved by completing square, 286-290, 479.

solved by factoring, 285, 479.

solved by formula, 290, 479. Aggregation, signs of, 15. Algebra, 7. Algebraic expression of physical laws,

392-393. Algebraic expressions, 19-20. Algebraic fraction, 126. Algebraic numbers, 24, 25. Algebraic representation, 16-17, 44-45, 69,

84, 161-162. Algebraic signs, 14-17. Agebraic solutions, 8-12. Algebraic sum, 25. Alternation, proportion by, 375. Annuities, 439-442. Antecedent, 369. Antilogarithm, defined, 429.

finding, 429-430. Arithmetical means, 399-400. Arithmetical numbers, cube root of, 227- 230.

defined, 13.

square root of, 221-224. Arithmetical progressions, 394-402. Arithmetical series, defined, 394.

last term of, 395-396.

sum of, 396-397. Arrangement of polynomial, 58. Associative law, for addition, 30.

for multiplication, 52. Axioms, 41, 45, 208, 215.

Base of system of logarithms, 423, 424. Binomial formula, 211-213, 239, 417-

422. Binomial quadratic surd, 256.

square root of, 259-261. Binomial surd, 256.

Binomial theorem, 211-213, 239, 416-422. Binomials, defined, 19.

product of two, 67-68, 455. Biquadratic surd, 243. Braces, 15.

Brackets, 15.

Briggs system of logarithms, 424.

Characteristic, 424.

Circle, 330.

Clearing equations of fractions, 155-161.

Coefl&cients, defined, 17.

detached, 59.

law of, for division, 73.

law of, for multiplication, 53.

literal, 17.

mixed, 17.

numerical, 17. Cologarithm, 432. Commensurable numbers, 370. Commensurable ratio, 370. Common difference, 394. Common factor, 92, 119. Common multiple, 123. Common system of logarithms, 424. Commutative law, for addition, 30.

for multiplication, 52. Comparison, elimination by, 179-180. Complete quadratic, 285. Completing the square, 286-290, 479.

factoring by, 345-347.

first method of, 286-288, 479.

Hindoo method of, 289-290.

other methods of, 288-290. Complex fractions, 145-148. Complex numbers, 443-446. Composition, proportion by, 376. Composition and division, proportion by,

377. Compound expression, 19. Compound interest, 439-442. Condition, equation of, 153. Conjugate complex numbers, 443. Conjugate surds, 256. Consequent, 369. Consistent equations, 177. Constant, 385, 411. Continuation, sign of, 16. Continued fractions, 147-148. Continued proportion, 379. Coordinates, 201. Couplet, 369. Cross-products, 68. Cube, 17. Cube root, 224-230.

defined, 18.

of arithmetical numbers, 227-230.

of polynomials, 224-227, 240. Cubic surd, 243.

Deduction, sign of, 16. Definitions and notation, 13-22. Degree, of expression, 119.

of term, 119. Denominator, defined, 126.

lowest common, 134. Dependent equations, 176. Detached coefficients, 59. Difference, common, 394.

defined, 25.

of cubes, factoring, 104, 463.

of even powers, factoring, 105-106.

490

INDEX

491

Difference, of odd powers, factoring, 104- 105.

of squares, factoring, 97-99, 109, 462.

of two numbers, 28-29, 447.

square of, 63-64, 455.

tabular, 428. Direction signs, 24. Discontinuous curve, 332. Discriminant, 339. Dissimilar fractions, defined, 134.

reduction to similar fractions, 134- 135, 465. Dissimilar terms, 19. Distributive law, for division, 74.

for evolution, 216.

for involution, 208.

for multiplication, 55. Dividend, 72.

Division, 72-87, 143-148, 238-239, 241, 253-254, 271, 431^34, 445, 447, 459, 466, 477.

by logarithms, 431^34.

defined, 72.

of complex numbers, 445.

of fractions, 143-148, 466.

of imaginary numbers, 271, 445.

of monomials, 73-74, 238, 241, 447, 459.

of polynomial by monomial, 74-75, 238, 459.

of polynomial by polynomial, 76-80, 239, 459.

of radicals, 253-254, 477.

proportion by, 376.

special cases in, 81-83, 459. Divisor, 72. Duplicate ratio, 370.

Elimination, by addition, 178-179, 471.

by comparison, 179-180.

by substitution, 181, 471.

by subtraction, 178-179, 471.

defined, 177. Ellipse, 332. Entire surd, defined, 244.

reduction of mixed surd to, 247. Equations, 41-44, 70-71, 85-87, 115-118, 152-206, 240, 262-267, 279-349, 437- 438, 467-470, 471-476, 478-482.

affected quadratic, 285-302, 326-329, 337-349, 479, 480-481.

clearing of fractions, 155-161.

consistent, 177.

defined, 8, 46, 153.

dependent, 176.

equivalent, 153.

exponential, 437-438.

fractional, 152, 155-161, 182-187, 195, 196, 264-265, 292, 293, 294, 297, 307, 308.

graphic solution of, 198-206, 326-338.

homogeneous, 312.

identical, 152.

impossible, 266-267.

in quadratic form, 303-308.

inconsistent, 177, 205.

independent, 177.

indeterminate, 176, 204.

integral, 152.

irrational, 262.

linear, 153, 203.

Equations, literal, 152, 160-161, 186- 187, 293-294.

members of, 41.

numerical, 152.

of condition, 1.53.

of first degree, 153.

of second degree, 279.

pure quadratic, 279-284, 478.

quadratic, 279-349, 478^82.

radical, 262-267, 295-297.

roots of, 153.

satisfied, 153.

simple, 41-44, 70, 85, 152-175, 198-203, 467^70.

simultaneous, involving quadratics, 309-325, 334-337, 482.

simultaneous simple, 176-197, 204-206, 471-476.

solved by factoring, 115-118, 285, 479.

solving, 153.

symmetrical, 310.

system of simultaneous, 177.

transposition in, 41^4. Equivalent equations, 153. Even root, 215.

Evolution, 214-231, 239, 240, 241, 254- 255, 259-261, 435-436, 445-446.

by logarithms, 435-436.

defined, 214.

of arithmetical numbers, 221-224, 227- 230, 231.

of complex numbers, 445-446.

of monomials, 216-217.

of polynomials, 218-221, 224-227, 231, 239, 240, 241.

of radicals, 254-255, 259-261. Exponential equations, 437-438. Exponents, defined, 18.

fractional, 235-241.

law of, for division, 73, 82, 232.

law of, for evolution, 216, 232.

law of, for involution, 208, 232.

law of, for multiplication, 53, 232.

negative, 233-241.

theory of, 232-241.

zero, 233, 234, 237-241. Expressions, algebraic, 19-20.

compound, 19.

degree of, 119.

fractional, 19.

homogeneous, 60.

integral, 20, 92.

irrational, 242.

mixed, 126.

radical, 242.

rational, 92, 242.

simple, 19.

symmetrical, 60. Extremes, of proportion, 373.

of series, 394.

Factor, common, 92, 119.

defined, 17.

rational, 244.

rationalizing, 256. Factor theorem, 106.

Factoring, 92-118, 240, 345-347, 462^64, 479.

by completing square, 345-347.

by factor theorem, 106-108.

492

INDEX

Factoring, defined, 92.

difference of cubes, 104, 463. difference of even powers, 105-106. difference of odd powers, 104-105. difference of squares, 97-99, 109-110,

462. equations solved by, 115-118, 285, 479. monomials, 92. polynomial squares, 109. polynomials, general directions for, 112. polynomials grouped to show common

polynomial factor, 93-94, 462. polynomials, whose terms have com- mon factor, 93, 462. review of, 111-114, 464. roots by, 230.

special devices for, 109-110. sum of two cubes, 104, 463. sum of two odd powers, 104-105. summary of cases. 111. trinomials like ox^ + bx + c, 101-103,

463. trinomials like x^ + px -{- q, 100-101,

463. trinomial squares, 95-97, 462. Finite geometrical series, sum of, 404-

405. Finite number, 412, Finite series, 404. First degree, equation of, 153. First member, of equation, 41.

of inequality, 361. First method of completing square, 286-

288, 479. Formation of quadratic equations, 342-

344. Formula, binomial, 211-213, 416-422. Formulae, 13, 22, 172-175, 187, 283-284, 290, 302, 358, 375, 384, 392-393, 395, 396, 400, 402, 404, 406, 407, 416, 417, 418, 419, 436, 439, 441, 442. Fourth proportional, 375. Fractional equations, 152, 155-161, 182- 187, 195, 196, 264-265, 292, 293, 294, 297, 307, 308. Fractional exponents, 235-241. Fractional expression, 19. Fractional number, 13. Fractions, 126-151, 465-466. addition of, 135-140, 465. complex, 145-148. continued, 147-148. defined, 15, 126, 268. dissimilar, 134. division of, 143-148, 466. indeterminate in form, 415. multiplication of, 141-143, 466. reduction of, 129-135, 465. signs in, 127-129. similar, 134.

subtraction of, 135-140, 465. Fulcrum, 174. Function of x, 347. Fundamental property of imaginaries, 268.

General number, 14.

General term of binomial formula, 417,

420-421. Geometrical means, 407. Geometrical progressions, 402-410.

Geometrical series, 402.

last term of, 402^04.

sum of, 404-406. Graphic solutions, 198-206, 326-338.

of quadratics in x, 326-329, 337-338.

of quadratics in x and y, 330-337.

of simple equations, 198-206.

of simultaneous simple equations. 204- 206. Graphical representation of quadratic surd.

243. Graphs, 198-206, 326-338. Grouping, law of, for addition, 30.

law of, for multiplication, 52.

Highest common factor, 119-122. Hindoo method of completing square,

289-290. Homogeneous equation, 312. Homogeneous expression, 60. Homogeneous in unknown terms, 312. Hyperbola, 332, 333.

Identical equation, 152.

Identity, 152.

Imaginary numbers, 215, 268-271, 339,

443-446. Impossible equation, 266-267. Incommensurable numbers, 370. Incommensurable ratio, 370. Incomplete quadratic, 279. Inconsistent equations, 177, 205. Independent equations, 177. Indeterminate equation, 176, 204. Index, of power, 18.

of root, 18. Index law, for division, 73.

for multiplication, 53. Induction, mathematical, 417-419. Inequalities, 361-368. Inequality, defined, 361. Infinite geometrical series, sum of, 405-

406. Infinite number, 347, 411. Infinite series, 404. Infinitesimal, 412. Infinity, 347. Inspection, roots by, 230. Integer, 13.

Integral equation, 152. Integral expression, 20, 92.

reduction of fraction to, 133-134. Interpretation, of results, 411-415.

offormsaXO,^.^. A ^.-^.412-414.

Introducing roots, 154, 266-267, 291, 295. Inverse ratio, 370. Inversion, proportion by, 376. Involution, 207-213, 254-255, 269, 434-435.

by binomial theorem, 211-213.

by logarithms, 434-435.

defined, 207.

of imaginaries, 269.

of monomials, 208-210.

of polynomials, 210-213.

of radicals, 254-255. Irrational equation, 262. Irrational expression, 242. Irrational number, 242, 339.

INDEX

493

Known number, 14. .

Last term, of arithmetical series, 395-396.

of geometrical series, 402-404. Law, associative, for addition, 30.

associative, for multiplication, 52.

commutative, for addition, 30.

commutative, for multiplication, 52.

distributive, for division, 74.

distributive, for evolution, 216.

distributive, for involution, 208.

distributive, for multiplication, 55.

index, for division, 73.

index, for multiplication, 53.

of coefficients, for division, 73.

of coefficients, foi* multiplication, 53.

of exponents, for division, 73, 82, 232.

of exponents, for evolution, 216, 232.

of exponents, for involution, 208, 232.

of exponents, for multiplication, 53, 232. Law of grouping, for addition, 30.

of grouping, for multiplication, 52.

of order, for addition, 30.

of order, for multiplication, 52.

of signs, for division, 72, 82.

of signs, for involution, 208.

of signs, for multiplication, 52.

of signs, for real roots, 215. Lever, 174, 358, 390. Limit of variable, 411. Linear equation, 153, 203. Literal coefficient, 17. Literal equations, 152, 160-161, 186-187, 293-294.

defined, 152. Literal number, 8. Logarithms, 423-442.

Briggs system of, 424.

characteristic of, 424.

common system of, 424.

defined, 423.

division by, 431-434.

evolution by, 435-436.

finding, 425-429.

in interest problems, 439H142.

involution by, 434-435.

mantissa of, 424.

multiplication by, 430-431, 432^34.

table of, 426^127. Lower terms, reduction of fractions to,

130-132. Lowest common denominator, 134. Lowest common multiple, 123-125. Lowest terms, 129.

Mantissa, 424.

Mathematical induction, 417—419. Mean proportional, 374. Means, arithmetical, 399-400.

geometrical, 407.

of proportion, 373.

of series, 394. Member, of equation, 41.

of inequality, 361. Minimum points, 328. Minuend, 25. Mixed coefficient, 17. Mixed expression, 126.

Mixed number, 126. Mixed surd, 244.

reduction to entire surd, 247. Monomials, addition of, 30-31, 448.

defined, 19.

division of, 73-74, 238, 241, 447, 459.

evolution of, 216-217, 239, 241.

factoring, 92.

involution of, 208-210, 239.

multiplication of, 53-55, 237, 241, 447, 454. Multiple proportion, 378. Multiplicand, 51.

Multiplication, 51-71, 141-143, 237-238, 241, 250-252, 270-271, 430-431, 432-434, 444-445, 447, 454, 455, 477.

by logarithms, 430-431, 432H134.

defined, 51.

of complex numbers, 444-445.

of fractions, 141-143, 466.

of imaginaries, 270-271, 444-445.

of monomials, 53-55, 237, 241, 447, 454.

of polynomial by monomial, 55-56, 454.

of polynomial by polynomial, 56-61, 62, 454.

of radicals, 250-252, 477.

special cases in, 63-68, 455. Multiplier, 51.

Nature of roots of quadratic equation,

339-341. Negative exponent, 233-241. Negative numbers, 23-29, 268, 447. Negative term, 27. Negative unit, 24. Notation, 13-22. Number, absolute, 13.

algebraic, 25.

arithmetical, 13.

cologarithm of, 432.

finite, 412.

fractional, 13.

general, 14.

infinite, 347, 411.

infinitesimal, 412.

irrational, 242.

jcnown, 14.

literal, 8.

logarithm of, 423.

mixed, 126.

negative, 24.

of roots of quadratic equation, 344.

positive, 24.

prime, 92.

rational, 242, 339.

root of, 18.

unknown, 14.

whole, 13. Numbers, commensurable, 370.

complex, 443-446.

imaginary, 215, 268-271, 443-446.

incommensurable, 370.

positive and negative, 23-29, 447.

real, 215, 268. Numerals, 13. Numerator, 126. Numerical coefficient, 17. Numerical equation, 152. Numerical substitution, 20-22. 62, 192.

494

INDEX

Odd root, 215.

Order, law of, for addition, 30.

law of, for multiplication, 52.

of operations, 15.

of radical, 243.

of surd, 243. Ordinate, 201. Origin, 201.

Parabola, 327, 331. Parentheses, 15.

grouping by, 39-40, 450.

removal of, 37-38, 450. Pendulum, 384, 391, 393. Physical law, 22.

algebraic expression of, 392-393. Plotting point? and constructing graphs,

202-203. Polynomial squares, factoring, 109. Polynomials, addition of, 32-33, 448.

cube root of, 224-227, 240.

defined, 19.

division of, 74-80, 238, 239, 459.

evolution of, 218-221, 224-227, 231, 239-241.

general directions for factoring, 112.

involution of, 210-213.

multiplication of, 55-61, 454.

square of, 64-65.

square root of, 218-221. Positive and negative numbers, 23-29,

447. Positive number, 24. Positive term, 27. Positive unit, 24.

Powers, 17-18, 207-213, 254-255, 269. 377, 434^35.

by binomial formula, 420-422.

by logarithms, 434-435.

defined, 17.

index of, 18.

of V^T, 269.

Present value of annuity, 442.

Prime number, 92.

Prime to each other, 119.

Principal root, 215.

Problems, 8-12, 22, 46-48, 71, 86-87, 163-175, 188-192, 197, 282-284, 298-302, 323-325, 356-358, 359, 360, 368, 383-384, 389-391, 397, 400-402, 403^04, 408^10, 436, 439-440, 441, 442, 450-453, 456-458, 460-461, 467-470, 472-475, 476, 478, 480-481, 482, 483-484, 485, 486, 487, 488, 489. area, 71, 164, 172, 190, 282, 283, 284,

298, 299, 302, 323, 324, 325, 357, 358, 360, 389, 390, 457, 468, 478, 480, 481, 482, 483, 484.

clock, 168, 469.

commercial, 9, 10, 11, 48, 71, 87, 163, 164, 167, 173, 188, 189, 192, 197, 283, 298,

299, 300, 301, 324, 325, 357, 373, 383, 402, 403, 404, 410, 439^42, 451, 452, 456, 457, 458, 461, 467, 468, 469, 470, 473, 474, 475, 476, 481, 482, 484, 486.

defined, 9.

digit, 166, 190, 197, 323, 356, 470, 474,

475, 476, 482, 487, general directions for solving, 46.

Problems, geometrical, 11, 22, 172, 284, 302, 324, 384, 389, 390, 391, 436, 482.

in affected quadratics, 298-302, 480-481.

in annuities, 440-442.

in arithmetical progressions, 397, 400- 402.

in compound interest, 439-442.

in geometrical progressions, 403^04, 408^10.

in inequalities, 368.

in physics, 22, 171, 173-175, 191, 283- 284, 302, 358, 384, 389-391, 475.

in proportion, 383-384.

in pure quadratics, 282-284, 478.

in quadratics, 282-284, 298-302, 323- 325, 478, 480-481, 482.

in review, 356-358, 359, 360, 483-484, 485, 486, 487, 488, 489.

in simple equations, 8-12, 22, 46-48, 71, 86-87, 163-175, 450-453, 456-458, 460-461, 467-470.

in simultaneous quadratics, 323-325, 482.

in simultaneous simple equations, 188- 192, 197, 472-475, 476.

in variation, 389-391.

interest, 167, 173, 192, 325, 410, 439- 442, 469, 470, 475, 482.

miscellaneous, 9-12, 48, 71, 86-87, 163-164, 172-173, 188-192, 197, 282- 283, 298-301, 323-325, 356-358, 368, 383, 389-391, 397, 401-402, 403^04, 409, 450-453, 456-458, 460- 461, 467-470, 472-475, 476, 478, 480-481, 482, 483-484.

mixture, 169-170, 356, 359, 469, 470, 475.

number relations, 9, 10, 11, 12, 46-47, 71, 87, 163, 164, 188, 190, 197, 282, 298, 301, 323, 324, 356, 357, 358, 368, 383, 401, 450, 451, 453, 456, 457, 458, 460, 467, 472, 474, 476, 478, 480, 481, 482, 483, 485, 488, 489.

percentage, 167, 170, 173, 192, 301, 325, 359, 403, 404, 410, 439-442, 457, 467, 468, 469, 470, 473, 475, 482, 483.

rate, 22, 165, 168-169, 173, 190, 191, 192, 197, 283, 284, 300-301, 302, 324, 325, 356, 357, 358, 389, 397, 408, 409, 469, 470, 473, 475, 481, 484, 485, 486, 487, 488.

volume, 284, 299, 358, 360, 389, 390, 391, 436, 483, 484.

work, 165, 191, 192, 197, 325, 357, 358, 389, 470, 484, 485. Product, defined, 51.

of sum and difference of two numbers, 65-66, 455.

of two binomials, 67-68, 455. Progressions, 394-410. Properties, of complex numbers, 446.

of inequalities, 362-365.

of proportions, 374-379.

of quadratic equations, 339-349.

of quadratic surds, 260-261.

of ratios, 370-373. Proportion, 373-384.

by alternation, 375.

by composition, 376.

by composition and division, 377.

INDEX

495

Proportion, by division, 376.

by inversion, 376.

continued, 379.

defined, 373.

extremes of, 373.

means of, 373.

multiple, 378.

properties of, 374-379. Proportional, 374, 375. Pure quadratics, 279-284, 478.

Quadratic equations, 279-349, 478-482

affected, 285-302, 326-329, 337-349 479, 480-481.

defined, 279.

formation of, 342-344.

general directions for solving, 291.

graphic solutions of, 326-338.

nature of roots, 339-341.

number of roots, 344.

properties of, 339-349.

pure, 279-284, 478.

relation of roots and coefficients, 342.

simultaneous, 309-325, 334-337, 482.

solved by completing square, 286-290, 479.

solved by factoring, 115-118, 285, 479.

solved by formula, 290, 479. Quadratic expression, values of, 347-349. Quadratic form, 303.

equations in, 303-308. Quadratic surd, 243.

graphical representation of, 243. Quality, signs of, 24. Quotient, 72.

Radical equations, 262-267, 295-297,

defined, 262.

general directions for solving, 263. Radical expression, 242. Radical sign, 18. Radicals, 242-267, 477.

addition of, 249-250, 477.

defined, 242.

division of, 253-254, 477.

evolution of, 254-255.

in simplest form, 244.

involution of, 254-255.

multiplication of, 250-252, 477,

order of, 243.

reduction to same order, 248.

reduction to simplest form, 244-247,

similar, 249,

subtraction of, 249-250, 477 Radicand, 242, Ratio, 369-373.

commensurable, 370.

defined, 369.

duplicate, 370.

incommensurable, 370,

inverse, 370.

of equality, 370.

of geometrical series, 402.

of greater inequality, 370.

of less inequality, 370.

reciprocal, 370.

sign of, 369.

triplicate, 370. Ratio and proportion, 369-384.

Rational expression, 92, 242. Rational factor, 244. Rational number, 242. Rationalization, 255-258. Ratios, properties of, 370-373. Real numbers, 215, 268. Reciprocal, 143. Reciprocal ratio, 370. Rectangular coordinates, 201. Reduction, 129. Reduction of fractions, 129-135, 465,

to higher or lower terms, 130-132.

to integral or mixed expressions, 133- 134.

to similar fractions, 134-135, 465. Reduction of mixed surd to entire surd,

Reduction of radicals, 244-248. to same order, 248.

to simplest form, 244-247. Relation of roots to coefficients in quad- ratic equations, 342, Remainder, 25.

Removing roots, 154, 291, 295, Representation, algebraic, 16-17. 44-45.

69, 84, 161-162. Results, interpretation of, 411-415. Review, 49-50, 88-91, 111-114, 149-151.

272-278, 350-360, 464, 483-489. Root, cube, 18, 224-231.

even, 215.

index of, 18.

odd. 215.

of equation, 153.

of number, 18.

principal, 215.

square, 18, 218-224, 239, 241. Root sign, 18.

Roots, 18, 214-231, 239-271, 435-436, 477.

by factoring, 230.

by inspection and trial, 230.

by logarithms, 435-436.

introduced, 154, 266-267, 291, 295.

nature of, 339-341.

number of, 341.

of quadratic equation, 280, 339-344.

relation to coefficients, 342.

removed, 154, 291, 295.

successive extraction of, 231,

Satisfying an equation, 153. Scale of algebraic numbers, 24. Second member, of equation, 41.

of inequality, 361, Series, arithmetical, 394.

defined, 394.

extremes of, 394.

finite, 404.

geometrical, 402.

infinite, 404.

means of, 394,

terms of, 394, Sign, of addition, 14.

of continuation, 16.

of deduction, 16,

of division, 14,

of equality, 15.

of inequality, 361.

of infinity, 347, 411.

496

INDEX

Sign, of infinitesimal, 412.

of multiplication, 14.

of product, 52.

of ratio, 369,

of subtraction, 14.

of variation, 385.

radical, 18.

root, 18. Signs, algebraic, 14.

direction, 24.

in fractions, 127-129.

law of, for division, 72, 82.

law of, for involution, 208.

law of, for multiplication, 52.

law of, for real roots, 215.

of aggregation, 15.

of quality, 24. Similar fractions, defined, 134.

reduction to, 134-135, 465. Similar radicals, 249. Similar terms, 19.

Simple equations, 41-i4, 70, 85, 152-175, 198-203, 467-470.

defined, 153.

graphic solution of, 198-203. Simple expression, 19. Simplest form, of radical, 244.

reduction of radicals to, 244-247. Simultaneous equations, defined, 177. Simultaneous equations involving quad- ratics, 309 325, 334-337, 482.

both quadratic and homogeneous in unknown terms, 313-315.

both quadratic, one homogeneous, 312- 313

both symmetrical, 310-312.

division of one by other, 318.

elimination of similar terms, 319.

graphic solution of, 334-337.

one simple, other higher, 309-310.

special devices, 315-320.

symmetrical except as to sign, 316-317. Simultaneous simple equations, 176-197, 204-206, 471^76.

graphic solution of, 204-206. Solution of exponential equations, 437-438. Solution of problems, 45-48. defined, 9.

general directions for, 46. Solutions, graphic, 198-206, 326-338. Solving the equation, 153. Square, 17.

of any polynomial, 64-65.

of difference of numbers, 63-64, 455.

of sum of two numbers, 63, 455. Square root, 218-224, 239, 240, 255, 259- 261, 445^46.

defined, 18.

of arithmetical numbers, 221-224.

of binomial quadratic surds, 259-261.

of complex numbers, 445-446.

of polynomials, 218-221, 239. Substitution, defined, 20.

elimination by, 181, 471.

numerical, 20-22, 62, 192. Subtraction, 25, 28-29, 34-48, 61-62, 135-140, 249-250, 269-270, 444, 447, 449, 465, 477.

defined, 25.

elimination by, 178-179, 471.

Subtraction, of complex numbers, 444.

of fractions, 135-140, 465.

of imaginaries, 269-270, 444.

of radicals, 249-250, 477. Subtrahend, 25.

Successive extraction of roots, 231. Sumj of arithmetical series, 396-397.

of finite geometrical series, 404-405.

of infinite geometrical series, 405-406.

of two cubes, factoring, 104, 463.

of two numbers, square of, 63, 455.

of two odd powers, factoring, 104-105.

of two or more numbers, 25. Sum and difference of two numbers, prod- uct of, 65-66, 455. Surd, 242, 268.

binomial, 256.

binomial quadratic, 256.

biquadratic, 243.

cubic, 243.

entire, 244.

mixed, 244.

order of, 243.

quadratic, 243. Surds, conjugate, 256. Symmetrical equation, 310. Symmetrical expression, 60. System of equations, 177.

Table of logarithms, 426^27. Tabular difference, 428. Term, absolute, 68, 285.

defined, 19, 394.

degree of, 119.

negative, 27.

positive, 27. Terms, dissimilar, 19.

lowest, 129.

of fraction, 126.

of series, 394.

similar, 19. Theory of exponents, 232-241. Third proportional, 375. Transposition in equations, 41-44. Trinomial, 19. Triplicate ratio, 370.

Unit, 24.

Unknown number, 14.

Value, absolute, 25.

of fraction indeterminate in form Values of a quadratic expression, 347 Variable, 385, 411. Variation, 385-393. Vary, 385, 386. Velocity, 22, 173, 393. Vertical bar, 15. Vinculum, 15.

Whole number, 13.

X-axis, 200.

41/

-34'J

/

y-axis, 200.

Zero, 24.

Zero exponent, 233, 234, 237-241.

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