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HIGH SCHOOL ALGEBEA
Digitized by the Internet Archive
in 2010 with funding from
Ontario Council of University Libraries
http://www.archive.org/details/highschalgebracOOcraw
HIGH SCHOOL ALGEBRA
^
J. T. CRAWFORD, B.A.
CHIEF INSTRUCTOR IX MATHEMATICS, UNIVERSITY SCHOOLS,
LECTURER IN MATHEMATICS, FACULTY OF EDUCATION,
UNIVERSITY OF TORONTO
AUTHOEIZED BY
THE MINISTER OF EDUCATION FOR ONTARIO
TORONTO
THE MACMILLAN COMPANY OF CANADA, LIMITED
1916
Copyright, Canada, 1915
By the MACMILLAN COMPANY OF CANADA, Limited
Reprinteh 1916
PREFACE
This text covers the work prescribed for entrance to the
Universities and Normal Schools.
The book is written from the standpoint of the pupil, and
in such a form that he will be able to understand it with a
minimum of assistance from the teacher. The question
method is frequently used in developing the theory. The
purpose of this is to lead the pupil to think for himself.
The close connection between algebra and arithmetic is
constantly kept in view, and in many cases the arithmetical
and algebraic processes are shown in parallel columns.
There are numerous diagrams for the purpose of illus-
trating the theory, and algebraic methods are applied to many
of the theorems which the pupil meets in elementary geometry.
Special emphasis is placed upon the verification of results.
In the past, sufficient attention has not been given to this
important part of mathematical work.
Provision is made for oral work, many of the exercises
being introduced by a number of oral examples for use in
class.
The equation and the solution of simple problems are
introduced in the second Chapter. It is ho})ed that the
pupil will thus become interested much earlier in the work.
Long multiplications and divisions are not included in the
work of the first year. They are difficult for the beginner
and of little interest, as there is not much to offer in the
way of practical illustrations.
vi PREFACE
Chapter X., with which the pupil would begin the second
year's work, contains a thorough review of the simple rules.
Here the more complicated processes are dealt with.
The graphical work is introduced naturally in illustrating
the negative quantity and in the solution of equations.
Only graphs which can be drawn with the ruler and
compasses are included in the book.
More attention is given to methods of inspection in the
extraction of roots. The long process for cube root is
eliminated, as cube root is not now required in arithmetic.
The work on ratio and proportion is presented in as simple
a form as possible, and is intended only as an introduction to
the senior work in this subject. The geometrical illustrations
which are given should make it more interesting.
The division method of finding highest common factor has
been discarded, as it is usually performed mechanically and
not understood by pupils. The elimination method which
is used will be found easj' to apply with expressions which
are not too complicated. Finding the highest common
factor of expressions of the fovirth or higher degrees is of
little algebraic value, and few examples of such problems
will be found in the book.
The review exercises at the end of each Chapter will be
found useful, particularly for the purpose of reviewing the
work of a previous term.
On the recommendation of experienced teachers the answers
are not given to simple examples, or to such examples as the
pupil can verify without difficulty.
CONTENTS
CHAPTER I
PAGE
Algebraic Notation 1
Arithmetical and Algebraic Signs and Sj'mbols (1).
Fmidamental Laws (3). Factor and Pro-
duct (6). Power and Index (7). Terms (8).
Coefficient (9). Addition and Subtraction of
like Terms (9). Use of Brackets (11). Review
Exercise (13).
CHAPTER II
Simple Equations ■ . . . 16
Meaning of Equation, Solving an Equation, Root
of an Equation (16). Axioms used (18). Veri-
fjnng Results (19). Problems solved by
Equations (21). Review Exercise (25).
CHAPTER III
Positive and Negative Numbers 28
Graphical Representation of Positive and Negative
Numbers (28). Concrete Examples of Negative
Numbers (30). Signs of Operation and Signs
of Quality (32). Absolute Value (32). Review
Exercise (34).
CHAPTER IV
Addition and Subtraction 36
Addition of Quantities with hke Signs (36). Com-
pound Addition (37). Addition with unhke
Signs (39). Indicated Additions (41). Sub-
traction the Inverse of Addition (42). Rule for
Subtraction (43). Removal of Brackets (45).
Review Exercise (47).
viii CONTENTS
CHAPTER V
PAOK
Multiplication and Division 49
Multiplication of Simple Positive Quantities (49).
Index Law for Multiplication (49). Rule of
Signs (50). Compound Multiplication (53).
Verifications (55). Division by a Simple
Quantity (57). Index Law for Division (58).
Review Exercise (60).
CHAPTER VI
Simple Equations (continued) • . 62
Equation and Identity (62). Transposing Terms
(63). Simple Fractional Equations (67). Steps
in the Solution of an Equation (67). Problems
(69). Algebraic statements of Arithmetical
Theorems (74). Review Exercise (75).
CHAPTER VII
Simultaneous Equations 78
Equations with two Unknowns (78). Method of
Solution (79). Elimination (80). Fractional
Simultaneous Equations (82). Problems (83).
Review Exercise (85).
CHAPTER VIII
Type Products and Simple Factoring 88
Monomial Factors (88). Product of two Binomials
(89). Factors of Trinomials (90). Radical
Sign (92). Square of a Binomial (94). Square
Root of a Trinomial (95). Difference of two
Squares (97). Numerical Applications (99).
Review Exercise (101).
CONTENTS
CHAPTER IX
Simple Applications of Factoring . . .103
Highest Common Factor (103). Algebraic Fractions
(104). Lowest Terms (105). Multiplication and
Division of Fractions (106). Lowest Common
Multiple (107). Addition and Subtraction of
Fractions (108). Mixed Expressions (HO).
Review Exercise (111).
CHAPTER X
Review of the Simple Rules 114
Brackets (114). Collecting Coefficients (117).
Multiplication with Detached Coefficients (118).
Division by a Compound Quantity (121). Veri-
fjdng Division (122). Inexact Division (124).
Review Exercise (125).
CHAPTER XI
Factoring (continued) 128
Factors by Grouping (128). Complete Squares (130).
Difference of Squares (132). Incomplete
Squares (135). Trinomials (137). Sum and
Difference of Cubes ( 140) . The Factor Theorem
(141). Easy Quadratic Equations (144).
Review Exercise (146).
CHAPTER XII
Simultaneous Equations {continued) 149
Elimination by Substitution and by Comparison
(149). Equations with three Unknowns (152).
Special Forms of Equations (154). Solution of
Problems (156). Review Exercise (161).
X CONTENTS
CHAPTER XTII
PAOK
Geometrical Representation of Number .... 165
Function of x (165). Variables and Constants (165).
Arithmetical Graphs (166). The Axes (169).
Graph of an Equation (170). Coordinates (170).
Plotting Points (171). Linear Equation (173).
Graphical Solution of Simultaneous Equations
(175). Special Forms of Equations (176).
Review Exercise (177).
CHAPTER XIV
Highest Common Factor and Lowest Common Multiple 180
By Factoring (180). By Ehmination (183). Product
of the H.C.F. and L.C.M. (185). Review
Exercise (187).
CHAPTER XV
Fractions 188
Changes in the Form of a Fraction (188). Lowest
Terms (190). Addition and Subtraction (191).
Special Types (193). Cyclic Order (194).
Multiplication and Division (197). Complex
Fractions (199). Review Exercise (201).
CHAPTER XVI
Fractional Equations ....... 204
Cross Multiplication (204). Special Forms of
Fractional Equations (206). Literal Equations
with one Unknown (209), with two Unknowns
(212). Review Exercise (214).
CONTENTS XI
CHAPTER XVII
Extraction of Roots • . . 216
Square Root by Inspection (216), by the Formal
Method (217). Verifying Results (218). Cube
of a Binomial (222). Cube Root (223). Higher
Roots (224). Review Exercise (226).
CHAPTER XVIII
Quadratic Surds 228
Surd and Rational Quantities (228). Mixed and
Entire Surds (229). Like Surds (231). Addition
and Subtraction (231 ). Square Root Table (232)
Conjugate Surds (233). Rationalizing a De-
nominator (235). Surd Equations (237).
Review Exercise (238).
CHAPTER XIX
Quadratic Equations 240
Standard Form (241). Solution by Factoring (242),
by Completing the Square (244). Irrational
Roots (247). Inadmissible Solutions (248).
Review Exercise (250).
CHAPTER XX
Ratio and Proportion . . .... 253
Methods of Comparing Magnitudes (253). Com-
parison of Ratios (254). Proportion (256).
Finding a Ratio by Solving an Equation (257).
Mean Proportional (261). Ratio Theorems (263).
Review Exercise (266).
xii CONTENTS
CHAPTER XXI
PAGE
The General Quadratic Equation 268
Solution of Literal Quadratics (268). Solution by-
Formula (270). Imaginary Roots (271). Equa-
tions Solved like Quadratics (273). Review
Exercise (276).
CHAPTER XXII
Simultaneous Quadratics 279
Three Types of Simultaneous Quadratics (279).
Special Methods (284). Graphical Solutions
(288). Review Exercise (290).
CHAPTER XXIII
Indices 293
The Index Laws (294). Fractional, Zero and
Negative Indices (297). Operations with
Fractional and Negative Indices (301). Con-
tracted Methods (303). Review Exercise (305).
CHAPTER XXIV
Surds and Surd Equations 308
Surds of different Orders (308). Varying Forms of
Surds (309). Surd Equations (312). Extraneous
Roots (313). Square Root of a Binomial Surd
(317). Imaginary Surds (320). Impossible
Problems (323). Review Exercise (324).
CHAPTER XXV
Theory of Quadratic Equations 327
Sum and Product of the Roots (327). Reciprocal
Roots (328). Functions of the Roots (331).
Character of the Roots (335). The Discriminant
(337). Factors of the Quadratic Expression
(338). Review Exercise (341).
CONTENTS xiii-
CHAPTER XXVI
Supplementary Chapter . 344
Factors of the Product of two Trinomials (344).
Sum and Difference of Cubes (346). Factors
by Grouping (349). The Factor Theorem
(351). Symmetry (354). Factoring by Sym-
metry (355). Identities (359). Review Exercise
(362).
Answers 365
HIGH SCHOOL ALGEBRA
HIGH SCHOOL ALGEBRA
CHAPTER I
ALGEBRAIC NOTATION
I. Use of Arithmetical Signs. In arithmetic, signs are
used to abbreviate the work. In algebra the same signs are
used, with the same meanings and for the same purpose.
EXERCISE 1
Write the following statements in the shortest way you can, using
the signs and symbols with which you are famihar in arithmetic.
1. Two and two make iour.
2. The sum of five, ten and twenty is thirt^y five.
3. Six and four is the same as four and six.
4. Seven times eight is the same as eight times seven.
5. The diiference between twelve and five is seven.
6. Ten exceeds six by four.
7. The excess of twenty over fifteen is five.
8. The defect of thirty from a hundred is seventy.
9. Thirty-six divided by four is nine.
10. Three score and ten is seventy.
II. One half of the sum of seven and five is six.
12. The sum or the product of three, five and seven is the same in
whatever order they are written.
13. Three multipUed by four is twelve, therefore twelve divided
by three is four.
14. The square of four is sixteen, therefore the square root of
sixteen is four.
B
2 ALGEBRA
2. Algebraic Symbols. In the preceding exercise you have
used symbols to represent the numbers stated and signs to
show the operations performed on those numbers.
In algebra, symbols are used more extensively than in
arithmetic.
A B
I 1
If the length of this line be measured it will be found to
be two inches. But without measuring it, we may say that
the measure of its length is some definite number which
might be represented by the letter a.
The measure of the length of another line might be repre-
sented by 6. The cost of an article might be c cents, or the
cost of a farm might be x dollars, or the weight of a stone
might be m pounds.
Here a, b, c, x, m are algebraic number-symbols, or briefly
algebraic numbers.
The symbols 1, 2, 3, etc., used to represent numbers in
arithmetic are called arithmetical number-symbols or arith-
metical numbers.
In algebra the number symbols of arithmetic are also used.
For the present, when letters are used to represent numbers,
it will be understood that each letter represents some integral
or fractional number.
3. Signs of Multiplication. In this square the measure of
D C the length of the side AB is a. What is the
measure of the length of BC ; of CD ; of
AB+BC ; of AB+BC^CD ?
The measure of the perimeter (sum of all the
sides) is a-f-a+a+a or 4 times a or 4xa.
In algebra, 4xa or ax4 is usually written 4a, the sign of
multiplication being understood. It is also written 4. a, the
dot representing multiplication.
Thus, 4 X a = 4 . a = 4a, and as in arithmetic, is a short way of writing
a-\-a-\-a-\-n.
Thvjs, if o=6, the measure of the perimeter of the square is
6+6+6+6 = 4x6 = 24.
ALGEBRAIC NOTATION 3
It will be observed that in algebra the multiplication of a
and 4 is only indicated in the form 4a, while in arithmetic
it may be actnallj^ performed as in the result 24.
The pupil must recognize the difference between 24
{twenty -four) and the product of 2 and 4 or 2 x 4 or 2 . 4. When
two numerical quantities are to be multiplied, the sign of
multiplication must be used, so that as stated, 24 may be
distinguished from 2x4. When both factors are not numerical
as 4xa or axh, the sign is omitted and these are written in
the form 4a, ah.
4. Signs of Division. As in arithmetic, the quotient ob-
tained by dividing one number by another ma}^ be written
in the fractional form.
In arithmetic the division may be actually performed, as
in 6-i-3, which may be written f or 2, but it is frequently
only indicated as in 6-^7, which is written ^.
So in algebra, the quotient obtained on dividing a by h,
or a^b, is written j, and here, as in multiplication, the division
can only be indicated unless the numerical values of a and
b are known.
5. Some Fundamental Laws. Since the letters used in
algebra represent arithmetical numbers, all the laws of
arithmetic must be true also in algebra.
In arithmetic. I In algebra,
(1) 7 + 3 = 3 + 7. i (1) a+b = b+a,
6+2+5 = 6+5+2 = 2+5+6. a+6+c = o+c+6 = 6+c+a.
(2) 3x5 = 5x3. (2) ab = ba.
2x4x3 = 2x3x4 = 3x4x2. abc = acb = cba.
(3) 10+5-2=10-2 + 5. (3) a + b-c=a-c+b.
10-5-2 = 10-2-5. a-b-c = a-c-b.
(4) 3x10-:- 5 = 3 + 5x10=10-1-5x3 (4) axb^c = a-^cxb = b^cxa.
From (1) and (2) it follows that the sum or the product of
several numbers is independent of the order in ivhich they are
written.
B 2
4 ALGEBRA
From (3) and (4) it follows that a series of additions and
std)tractions, or of multiplications and divisions, may be mxide
in any order.
In finding the numerical value of an expression hke
3a+4&— 2c for given values of a, h and c, the operations are
performed in the same order as in arithmetic, the multi-
plications being performed first and then the additions and
subtractions in any order.
Thus, when a = 2, 6 = 3, c = l,
3a+46-2c = 3x 2+4x3-2x1 = 6+12-2=16.
Similarly, for the same values of a, b, c,
ab + bc 2x3+3x1 _ 6+3_9_
a + b ~ 2 + 3 ~ 5 ~ 5"
Note. — Many of the examples in the following exercise may be
taken orally. The pupil, however, is advised to write the algebraic
forms so that he may thereby become familiar with them.
EXERCISE 2
1. When a =6, what are the numerical values of :
„ 1 a 5 12 2 5a «
Sa, -a, -, -a, — , -, — ?
2 3 6 a a S
2. When a:=5 and 2/=3, what are the vahies of;
x~\-y, x—y, xy, 3x-\-2y, 2x—3y, \xy ?
3. When m=4, w=6, r=2, find the values of :
m^n-\-r, m-^-r—n, mn-\-mr, mr—n, 4w— 3m— 6r.
4. Express algebraically the sum, 'the difference and the product
of a and b. What are their values when a =8 and 6==3 ?
5. The quantities a, b and c are to be added together. Express
the sum algebraically. What is its value when a=6, 6=4, c=12 ?
6. When a is divided by 6 the quotient is expressed in the form --.
When c is added to the quotient of x by y, how is the result expressed ?
What is its value when a;=12, y=4t, c=10 ?
7 . A boy has p marbles ; he wins q marbles and then loses r marbles.
How many has he now ? How many if ^=5, 5'=11, r=4?
ALGEBRAIC NOTATION 6
8. When a=4 and 6=5, find the numerical value of
12a— 56+6«— 76+10.
9, The sides of a triangle are a, 6 and c ; express algebraically the
perimeter and the semi-perimeter. What do they become if a=13,
6=14, c=15?
10. Find the cost of 8 articles at 5 cents each ; of 7 articles at x
cents each ; of x yards of cloth at 6 cents a yard ; of m tons of coal at
n dollars a ton.
11. How many cents are there in 4 dollars ; in x dollars ; in x
dollars and y cents ; in a quarters and 6 ten-cent pieces ?
12. Find the number of inches in 2 yards ; in 3 feet and 7 mches ;
in a yards ; in 6 feet ; in x feet and y inches ; in m yards n feet and
p inches.
13. What operations are to be performed to find the numerical
value of vm-\-nb, when a=2, 6=5, m='i, ?i=6 ? What is the value ?
14. What operations are to be performed to find the value of ^Zl^,
a+6
when a=5, 6=6, x=15, y— 7 ? What is the value ?
15. By varying the order of the letters, in liow many ways can
you write a-\-h-\-c ?
16. In how many different ways can you ^vrite xyz ? \_
17. In the figure, BC is twice as long as AB. li A Q C
AB \s I units in length, what is the length of BC ? ' '
of ^C?
18. In the figure, BC is three times as long as AB and CD is twice
as long as AB. If AB is x units in ^ B C D
length, what are the lengths of BC 1 ' 1 1 '
CD? BDt ADt
19. In the following statements c represents the cost of an article,
s the selling price, and g the gain :
(1) s-c=g, (2) c^g=s, (3) s-g=c.
Read them and explain their meanings.
20. What is the next integer above 27 ? The next below 27 ?
What is the next integer above n ? The next below n ?
21. If w is an even integer, what is the next even integer above it
and the next even integer below it ?
6 ALGEBRA
22. If a; is any number, what is the number which is 5 greater than
X 1 5 less than x ?
23. A boy is 10 years old. How old will he be in 6 years ? In
m years ? How old was he 4 years ago ? n years ago ?
24. A man is x years old. How old wUl he be in n years ? How
old was he m years ago ? In how many years wUl he be three times
as old as he is now ?
25. A boy was p years old 3 years ago. How old wQl he be 15 years
from now ?
26. Explain the difference between , a + - and a . -. What
c c c
are their values when a =6, b=9, c=3 ?
27. The side of one square is a and of a smaller one is b. Indicate
the difference in their perimeters. What is the difference if a=10 and
6=6?
28. The sides of one rectangle are a and b, and of another are c
and d. Indicate the difference in their areas, (1) when the first rect-
angle is the larger, (2) when the second is the larger.
29. What arithmetical number does 10x-\-y represent when x=5,
y=3 1 When x^l, y=9 ?
30. When a=3, 6=4, c=5, d=0, find the values of :
(1) I0a+4b-5c + 3d. (2) 5ab+2cd—3ac.
6a—2b+3c—d
(3) ^ac^lbc-lad. (4)
2a+b—c+d
6. Factor and Product. When numbers are multiplied
together the result is called the product, and the numbers
which were multiphed are called the factors of the product.
Thus, 3x5=15, therefore the factors of 15 are 3 and 5, so a xb = ab,
therefore the factors of ab are a and b.
The factors of ,3.t are 3 and x. The 3 is called a numerical
factor and the x, a literal factor.
Just as 12 may have different sets of factors as 3x4, 2x6,
2 X 2 X 3, so 3x1/ has the factors 3 X xy, 3x x //, a: x 3y or 3 X a; x ?/.
The prime factors of 12 are 2, 2 and 3> and the simplest
factors of 3xy are 3, x and y.
ALGEBRAIC NOTATION 7
In whatever order the factors are written the product is
the same, but it is usual to write the numerical factor first
and the hteral factors in alphabetical order.
7. Power and Index. What is the area of a square whose
side is 7 inches in length ? The measure of the area of the
square in art. 3 is axa, which is written a-, and is read " a
square," or " a to the second power."
The product when 2 a's are multiphed together is called
the power, and the 2 is called the index or exponent of the
power.
If the edge of a cube is 6 inches, what is the sum of all the
edges ? What is the area of each face of the cube ? What
is the area of all the faces ? What is the volume of the
cube ?
If the edge of a cube is a, the sum of all the edges is 12a,
The area of each face is a^, and of all the faces is Qa^.
The volume is ax ax a or a^, which is read "a cube," or
" a to the third power."
The pupil must distinguish between 3a and a^. The former
means 3 X a, and the latter ax ax a.
Thus, if a=5, 3a=3x5=15,
but 0^ = 5x5x5=125.
EXERCISE 3 (1-14, Oral)
1. Whai are the prime faotors of 35, of 42, of 75 ?
2. What are the simplest factors of 5xy, of 6mn ?
3. Express 3abc as the product of two factors in four different ways.
4. Give two common factors of 15ab and 256c.
5. Find the values of 3~ X 2", 10^ x 5^, 2* x 5, 3^ x 2^ x 5.
6. Using an index, express 100 as a power of 10, 16 as a power of
2, 27 as a power of 3, 625 as a power of 5.
7. What is a short way of writing
a+a ? a+a+a ? axal axaXat aaxtat
8. What is the area of a square whose side is 6 inches ? whose
side is x inches ?
8 ALGEBRA
9. What is the volume of a cube whose edge is 3 inches ? whose
edge is m inches ?
10. When a=4, what is the vaUie of a- ? of 2a ? What is their
difference ?
11. When x=2, what is the difference between x^ and 3,r ?
12. What is the difference between " x square " and " twice x "
when a;=ll ?
13. If TO=10, what is the difference between the square of 3m and
three times the square of m ?
14. The side of a square is x inches and of a smaller one is y inches.
What is the sum of their areas ? What is the difference ? What do
these results become when a;=10 and y=G ?
15.* If x=6 and y=2, find the numerical values of
Sx^, x^-\-y, x-\-y'^, x"—y'^, 2x"^ — 3?/^.
16. Find the values of x^-\-x'^^x for the following values of x :
x=\, 2, 3,0.
17. If 7/=4.r--7, find the value of y if x=2, if x^3, if .r=2|.
18. The unit of work is " a day's work," that is, the work which
one man can do in one day. How many units of work can 3 men do
in 5 days ? 6 men in x days ? m men in n days ? a men in a days ?
19. If a=3, 6=2, c~ I, find the quotient when a^-irb^+c^ is divided
by 2a-\-b—c.
20. Show that a;^4-26.r has the same value as 9x^+24 when x=2
or 3 or 4.
21. If a;=10 and y=5, how much greater is x"-\-y^ than 2xy ?
22. li d represents the diameter of a circle and c the circumference,
we know that c=3}d. Find c when (^=14. Find d when c=22.
23. If ^ represents the area of a circle and r the radius, A^==B]r^.
Find A when r—7 ; when r=14.
24. By arranging the factors m the most suitable order, find the
values of 2* . 5^, 25^ . 4^, 125 . 2^.
8. Terms of an Expression. The parts of an algebraic
expression which are connected by the signs of addition or
subtraction are called the terms of the expression.
ALGEBRAIC NOTATION 9
Thus, the expression 2a-\-3b has two terms, and the expression
ix'^ — Sxy — y^ has three terms.
Quantities which are connected b}^ the signs of multiplication
or division are not different terms.
Thus, iax is only one term, so is -7- •
9. Coefficient. In the product 4.r, 4 is called the coefficient,
or co-factor, of .r. In ab, a is the coefficient of 6 and b is
the coefficient of a.
The 4 is a numerical coefficient, and the a or h is a literal
coefficient.
In any product, any factor is called the coefficient of the
rest of the product.
Thus, in 5abx, 5 is the coefficient of abx, 5a is the coefficient of bx,
and 5ax is the coefficient of b.
In any term where the numerical coefficient is not stated,
the coefficient 1 is understood.
Thus, in xy the numerical coefficient is 1.
10. Addition and Subtraction of Like Terms. When terms do
not differ or differ only in their numerical coefficients, they
are called like terms.
Thus, 2ab, 5ab, ^ah are like terms, but 3a, 46, 6ab are unlike terms.
In arithmetic, quantities which have the same denominations
may be added or subtracted.
Thus, 3ft. + 4ft.- 2ft. = 5ft.
.S12-.$10 + $8-$3 = $7.
We cannot add or subtract quantities of different denomina-
tions, unless we can first reduce them to the same denomination.
Similarly, in algebra, like terms may be added or subtracted.
Thus, 5a+2a=la,5a—2a = 3a,
6afe4-5a&- 3a6= 1 \ab-3ab = 8ab,
8x2 - 6a;2+ a-c* - 2a;2 = 1 7a;2 - 8.r2 = 9a;2.
In the last example we may, of course, perform the operations in the
order in which they occur and obtain the same result.
10 ALGEBRA
Unlike terms can not be added or subtracted.
Thus, the stun of 3a and 56 can be indicated in the form 3a-\-5b,
but they can not be combined into a single term unless the numerical
values of a and b are given.
EXERCISE 4 (1-8, Oral)
1. What is the numerical coefficient of each term in the expression
2. What is the sum of the numerical coefficients in
2x^+3xy+x+y ?
3. Which are like terms in the expression
5a2+26— 3a+76— 4a2 ?
4. In 6bcy, what is the coefficient of bci/ 1 oi cy 1 oi by 1 of 6 ?
5. What is the sum of :
(1) 2a, 3a, 4a. (2) 5m, |m, fm.
(3) 4a2, 1a% 5a\ (4) 3xy,\xy, 2xy.
6. If a; =2, find the numerical value of the sum of 3.T- and 4a;2
in two different waj'S and comj^are the results.
7. Simplify the expression 3a+86-l-2a+6+a+36 by combining
like terms.
8. Express in as simple a form as possible :
(1) 5m+7m— 3mi— 2to. (2) 6a6— 3rt6-f 2a6— a6.
(3) 3a;+a+2a;+a. (4) 15a+106— 7a+46.
9.* Combine the like terms in the expression :
2x-^ly+bz—X'\-2y~~3z-\-Zx—4:y—z
and find its value when .t=3, ^=5, z=10.
10. If a = 6, find the value of /
15a2— 10a2_3a2+8rt— 5a— 20.
11. What arithmetical number does lOOa+lOft+c represent when
a=2, 6=3. c=4 ? When a^Q, b=5, c=7 ?
12. Simplify 2a:2+3.T+7— a;2+lla:— 2— a-^— 4a:-|-5.
13. A man walks 4.f feet East, then x feet West, then 3.i: feet East
then 5a; feet West. How far is he now from the starting point and in
what direction from it ?
ALGEBRAIC NOTATION 11
14. A man began to work for a firm on a salary of x dollars a year.
If his salary for each year was double the salary for the preceding year,
how much did he earn in four years ?
15. If x-i-3x-\-5x is equal to 72, what is the value of x ? How do
you know that j^our answer is correct ?
16. Write in the shortest form you can
17. Find the average of (1) 10, 8, 15, (2) 3a,-, 7.r, 5x.
11. Use of Brackets. In algebra brackets are used for the
same purpose and with the same meanings as m arithmetic.
In finding the value of 10+8+5, we may perform the
additions in any order, but if we write it 10+ (8+5), it is
understood that the 8 and 5 are first to be added and the
sum of 10 and the result is to be taken.
Similarly, a+(6+c) means that the sum of the numbers
represented by b and c is to be added to the number represented
by a.
In the expression 7+5x2, the multiplication is to be
performed first, and then the addition. If, however, we
wish the value of (7+5) X 2, we must add the 7 and 5 before
multiplying by 2.
Although 10+(8+5) is equal to 10+8+5, it is clear that
(7+5) X 2 is not equal to 7+5x2, the former being equal to
24 and the latter 17.
When a is to be multiplied b}^ b, the sign of multiphcation
is omitted in the indicated product ; so when (7+5) is to be
multiplied by 2 we may write 2(7+5) or (7+5)2, the sign
of multiphcation being understood.
It is thus seen that one of the Kses of brackets is to indicate
the order in which operations are to be 'performed.
Thus, 10— (7 — 3) means that 3 is to be subtracted from 7 and the
result is to be subtracted from 10.
If the values of the letters were given, what operations would you
perform to find the values of :
a+(6+c), a-(ft+c), a-{b-c), (a-b) -{c~d)'i
The pupil should recognize that 3a^ is not the same as
12 ALGEBRA
(3a) 2. The latter means that a is first to be multipHed by
3 and the product is to be squared.
Thus, ifa = 2, 3a2 = 3x4=12,
and (3a)2 = 3«x3a=6x6=36.
Brackets also indicate that the numbers within the brackets
are to be considered as a single quantity, thai is, they are used
for the purpose of grouping.
The dividing Una between the numerator and denominator
of a fraction has the same value as a pair of brackets.
Thus, in j, a-\-h is a single quantity and so is c + d. The fractional
form is another way of writing {a-\-b)-^{c-\-d).
EXERCISE 5 (1-18, Oral)
Perform the operations indicated :
1.
10-(6+3).
2.
8-(4-2).
3.
15-6^3.
4.
(15-6)^3.
5.
3(4 + 7-5).
6.
(10+2)(5-l).
7.
10+2x5-1.
8.
(16+12)^(6-2).
9.
7.r-(8.c— 4x).
10.
(6a— 2a)— (7a— 4a).
1.
(3.r+4x)-^7.
12.
(10a— 6a)^2a.
3.
(3a;+9a;)^(6a:-3a;).
14.
3(7-5)-2(8-6).
15.
43a;— (7.T— 4x)+2a;.
16.
x-{4:y^Zy-ly).
-7.
(56-46)(3z-22).
18.
6a— {la— 3a)
8a— (6a+a)
Indicate, using brackets :
19. That X is to be added to the sum of p and q.
20. That the sum of x and y is to be added to w.
21. That the sum of a and b is to be multipUed by 2.
22. That the difference of m and n, where m is greater than n, is
to be subtracted from a.
23. If p is greater than q, that the difference of p and q is to be
divided by the sum of 7n and n.
If a=10, 6=3, c=2, find the value of :
24.* 8a-(26+c)-5(a-6). 25. 7(a-6-c)-3(a-26+c).
ALGEBRAIC NOTATION 13
26. (3a+2b-c){a-3b). 27. a^--{-f^+c^-2{ab+bc+ca).
a+36— c 2a— 36— 3 ^_ a—b b—c _ c~\-a
2a—5b+2c ~ a+6— 2c ' " * be "^ c+a a+26 "
30. When a— 6 and 6=3, show that
5{a-6)+3(a+6)=2(4a-6).
EXERCISE 6 (Review of Chapter I)
1. If a; represents a certain number, what does 4x represent ?
Jx ? a;2 ? 3a;2 ?
2. If a represents a nvimber, what wiU represent 5 times the
number ? y times the number ?
3. How do you indicate that y is to be added to a; ? That x is
to be subtracted from y ?
4. Indicate the sum of x and y diminished by a.
5. If one yard of cloth costs x cents, how many cents will 10 yards
cost ? How many dollars ?
6. If a yard of ribbon is worth y cents, how much is a foot worth ?
7. A man bought an article for x dollars and sold it at a loss of
y dollars. What did he sell it for ?
8. If I paid a dollars for b articles, how much did I pay for each ?
What would c articles cost at the same price ?
9. A boy has a dollars. He earns b cents and then spends c cents.
How many cents has he left ?
10. I have x dollars. If I pay two debts of a dollars and b dollars,
how much shall I have left ?
11. If one number is x and another is 5 times as large, what is the
sum of the numbers ?
12. If one part of 10 is x, what is the other part ?
13. A man worked m hours a day for 6 days. If he was paid $2
per Ifour, how much did he earn ?
14. How far can a man walk in 5 hours at 4 miles per hour ? In a
hours at 6 miles per hour ?
15. A man bought x acres of land at a dollars per acre and sold it
at a loss of b dollars per acre. What did he sell it for ?
16. What number is 15 greater than a; ? 15 less than x ?
17. By how much does a^ exceed 6' when a — I, 6 = 3 ?
14 ALGEBRA
18. When x=\, y=2, z = 3, what are the values of x-{-y — z,
2x+5y-3z, lx-3y+z ?
19. If a =3, b = 4, c = 0, find the values of 2ab, 4ac, a^+b^+c^,
5a2-262+4c2.
20. If x=5 and y=\, what are the values of Zx—2y, 6xy, 2x^—2y^,
8x3-272/3?
21. If a= 10, 6 = 5, c = 3, find the values of a{b+c), a{b-c), a{b^~c^)'
c{a^-b^).
22. What is the sum of 2x, 5x, Ix and 3a; ?
23. Simplify 5a— 3a+lla+«— 10a.
24. What is the average of 20, 15, 0, 8, 12 ? Of 2a, 3a, la ? Of
a, b, c, d ?
25. In 8 years a man will be x years old. How old was he 8 years
ago ?
26. B has $20 more than A, C has $20 more than B. If A has
$x, how much has C ?
27. What is the sum of the numerical coefficients in the expression
3a+^ab+ac+^ad 1
28. Express 1000 as a power of 10 ; 32 as a power of 2 ; 81 as a
power of 3 ; 64 as a power of 4.
29. Express 15, 105, 3a6, 35x^y, as the products of simple factors.
30.* How long will it take me to walk a miles at 3 miles per hour
and ride b miles at 1 2 miles per hour ?
31. A farmer buys 5 lb. of tea at x cents per lb. and 20 lb. of sugar
at y cents per lb. He gives in exchange 7 lb. of butter at z cents
per lb. If he still owes something, how much is it ?
32. If I buy 100 lb. of nails at a cents per lb. and 200 lb. at 6 cents
per lb., what is the average cost per lb. ?
33. What is the total number of cents in x five-cent pieces, y. ten-
cent pieces and z half-dollars ?
34. What number is represented by lOOOa+lOOfe-f lOc-f d, when
a=l, 6 = 2, c = 3, d = 4 ? When a = 4, 6 = 0, c=l, d = 9 ?
35. When a = -2 and 6 = -l, what are the values of a-\-b, o6, —
"+i', a^+b^ a^-b^l
ab
ALGEBRAIC NOTATION 15
36. If A can do a piece of work in 10 days and B in 15 days, what
fraction of the work can they together do in 1 day ? Wliat fraction
if A could do it in x days and B in y days ?
37. If a = 2p, 6 = 15, c=10, d = 5, find the difference between
(a+6) — (c+d) and (a — b) — (c — d), also between 3(a+6) — 5(c — rf) and
5(a-rf)-3(6-c).
38. Wlien x — 7 and y=l, the product of x+y, x-\-2y, x — 5y is how
much greater than the product oi x—y, x—2y, x—Sy"!
1
39. If a= 3, find tlie value of 1 +
^ a+1
CHAPTER II
SIMPLE EQUATIONS
12. Idea of Equality. In weighing an article, when you
see that the scales are balanced, what conclusion do you
draw ? If a 5 lb. bag of salt is placed in one scale pan, what
weight (w) must be placed in the other pan to restore the
balance ? What must w be to balance a 3 lb. bag a-nd a
4 lb. bag ?
If the scales are balanced in each of the following figures,
what must w be equal to ?
Fio. 1.
Pio. 2.
Fio. 3.
If w-^4: = 9, as in fig. 1, what is w equal to ?
If w-\-w= 12, as in fig. 2, what is w equal to ?
If w-\-3 = 5-\-2, as in fig. 3, what is iv equal to ?
If the scales are balanced and I add 2 lb. to one side, what else must
I do to preserve the balance ? What, if I take away 3 lb. from one side ?
If I double the weights on one side ? If I halve the weights on one side ?
13. The Equation, When a certain number is added to
10 the result is 27. What is the number ?
The condition expressed in this problem might be more
briefly shown in the form :
10+ a certain number =27,
or in the form 10+? =27, where the question mark stands for
the required number.
Any other symbol would answer the same purpose as the
SIMPLE EQUATIONS 17
question mark. Thus, if x represents the required number,
then the in-oblem states that
10+a:=27.
This statement is called an equation and is merely a short
way of stating what is given in the arithmetical problem
preceding. In order that the statement may be true, it is
easil}^ seen that the symbol x must stand for the number 17.
Ex. — When a number is multiplied by 3, and 5 is sub-
tracted from the product, the result is 19 What is the
number ?
Here, if x standi^ for the number, the problem states that
3a;-5=19.
Before the 5 was subtracted the product was evidently 5 more than
19 or 19+5 or 24.
If 3 times the number is 24, then the number must be J- of 24 or 8.
The solution may be written more briefly thus :
If 3a:- 5= 19,
.-. 3a;= 19+5 = 24,
a;=iof24 = 8.
That 8 is the correct value for the number is shown by the
fact, that when it is multiplied by 3 and 5 is subtracted
from the product, the result is 19.
14. Solving an Equation. The process of finding the value
of X, such that 3.r~5=19, is called "solving the equation,"
and the value found for x is called the root of the equation.
EXERCISE 7 (Oral)
1. State the number for which the question mark stands in each
of the following :
(1) 5 + ?=]2. (2) ? + 12=20. (3) 10-?=2.
(4) 15=^8 + ?. (5) 40=62-?. (6) ?-8=42.
2. What is the number for which x stands in each of the following :
(1) x+6=20. (2) 8 + a:=32. (3) 25=a;+6.
(4) a;-15=7. (5) 10-a;=8. (6) 12=17-a;.
C
18 ALGEBRA
3. The first equation in Ex. 2 states that when a number is in-
creased by 6 the result is 20. What does each of the other equations
say?
4. If 3 times a number is 45, what is the number ? If one-half
of a number is 16, what is the number ? If n stands for a given number,
what would represent ^ of the number ? 4 of the number ?
5. For what number does n stand in each of the following equations :
(1) 4n=24. (2) 4n=10. (3) fw=36. (4) |w=14.
6. If 2x+5=ll, what is the value of 2x ? of .r ?
7. If 3to — 2=13, what is the value of 3»i ? of m ?
8. If 125+3=10, what is the value of i?) ? of ;p ?
9. If fa:— 11 = 7, what is the value of 'ix ? of a; ?
10. If 2(a;+4) = 14, what is the value of a;+4 ? oi x 1
Solve the equations :
11. a;+10=30. 12. 3,t— 2=16. 13. 5y+2=n.
14. 4<— 5=27. 15. 2h=11. 1«. In— 4=24..
17, 3w+2=38. 18. ix— 1=4. 19, 2«+l=4.
20. 3n-|=5|. 21. Jw+2=5. 22. |a;-5=15.
23. 3(.T+1) = 30. 24. 5(a:-2)=45. 25. A(a;-1)=3.
15. Axioms used in Solving Equations. If two numbers are
equal, what is the result when the same number is added
to each ?
Thus, if x=G, what is x-\-2 equal to ?
What is the result when the same number is subtracted
from two equal numbers ; or when each is multiphed by the
same number ; or when each is divided by the same number ?
Thus, if a; = 10, what is cc — 4 equal to ? What is 3a; equal to ? What
is ^x equal to ?
The preceding conclusions may be stated thus :
(1) // the same number be added to equal numbers, the sums
are equal.
(2) // the same number be subtracted from equal numbers,
the remainders are equal.
SIMPLE EQUATIONS 19
(3) // equal numbers be multiplied by the same number, the
products are equal.
(4) // equal numbers be divided by the same number, the
quotients are equal.
These statements are called axioms, or self-evident truths,
and are used in solving equations. The method is illustrated
by the following examples :
Ex. 1.— Solve 3a;— 7=35.
Add 7 to each side, .-. Sx — 7+7 = 35 + 7, axiom (1),
.-. 3a; =42.
Divide each side by 3, .". x = ^ = \4:, axiom (4).
Ex. 2.— Solve \x+2=M.
Subtract 2 from each side, .-. ix+2 — 2 = 34— 2, axiom (2),
ia;=32.
Mviltiply each side by 2, .■. a;=64, axiom (3).
Ex. 3.— Solve 5a:— 3=2x+12.
Add 3 to each side, .'. 5x=2x-\-\5.
Subtract 2x from each side, .'. 5x — 2x=\5,
3x=15.
Divide each side by 3, .'. x=5.
The object of the changes which hav^e been made in these
equations is to get the quantities containing the unknown
{x) to one side and the remaining quantities to the other side.
The unknown quantities are usually transferred to the left
side, but sometimes it is better to transfer them to the right.
Ex. 4.— Solve 3m+20=5m— 16.
Add 16 to each side, .'. 3m+36 = 5m,
Subtract 3?^ from each side, .". 3m+36 — 3m=5m — 3m,
36 = 2m,
18 = m or »n.= 18.
16. Verifying the Result. If we substitute 18 for m in the
first side of the last equation we get
3m+20=3 X 18+20=74.
If we substitute in the second side we get
5m- 16=5x18-16=74.
c 2
20 ALGEBRA
This process is called verifying or testing the correctness
of the result. If the root obtained is the correct one, the
two sides of the equation should be equal to the same number
when the value found for the unknown is substituted.
The equation is then said to be satisfied.
The beginner is advised to verify the result in every case.
Verify the results obtained in Ex.'s 1, 2 and 3.
EXBRCISB 8 (Oral)
1. If 3a;=15, what does x equal ? What axiom is used ?
2. If 5a:-|-2:=17, what does 5x equal ? What axiom is used ?
What does x equal ? What axiom is used ?
3. If 2^— 3 = 13, what does 2/ equal ? What two axioms are used ?
4. If hx-A—Q, what does hx equal ? What does x equal ? What
two axioms are used ?
5. If ix=6, what does }x equal ? What does x equal ? What
two axioms are used ?
What is the value of x in the following equations :
6. 2a;=18. 7. 6.r=72. 8. 5.i=-16. 9. 3,r=6-9.
10. a;+20=-25. 11. 2x+l = 15. 12. 3x-l=20. 13. 6.x+5==29.
14. ^x=8. 15. |a;=-I2. 16. ix^2h. 17. ^x-==15.
EXBRCISB 9
Solve the following equations, giving full statements of the methods.
In each case verify the result :
2a;+5=27. 3. 4a;-5=51.
4a;=a:+21. 6. ^y=2y+80.
|a;+5=50. 9. 6.x- +42= 9a;.
Qa-3a=a-\-5. 12. 10.r+20=20.
14. 20+6.r+5=50-3a;+ll.
IG. 764.r-9=680;r+12.
;hts (iv) together with a 20-gram weight
are balanced by weights of 50 grams and 10 grams. Express this by
an equation and find the weight of each block.
18. If 17a:;— 11 is equal in value to 5a;+121, what is the value of x ?
19. What value of y will make 11?/+ 60 equal to 20^-30 ? J
1.
3a;+ 11=47. 2
4.
3a;- 10=65. 5
7.
7a-=60+3a;. 8
10.
10a;+3 = 3a;+66. 11,
^13.
8to=36— 4>n.
15.
12a; -652= 7a; +428.
17.
Nine blocks of equal w(
SIMPLE EQUATIONS 21
17. As we have already sliowii, an equation is merely the
statement in algebraic form of the condition given in an
arithmetical problem.
The solution of the problem is thus obtained hy solving the
equation.
E3XERCISB 10
State the condition in each of the following problems in the form of
an equation :
1. What must be added to 33 to make 50 ?
2. What must be taken from 90 to leave 40 ?
3. What is the number which when doubled is 36 ?
4. Five times a certain number is 45. What is the number ?
5. If a number is doubled and 3 added, the result is 25. What
is the number ?
6. What number is doubled by adding 27 ?
7. What number is halved by subtracting 20 ?
8. If 8 is subtracted from | of a certain number, the result is 7.
What is the number ?
9. Solve the equation in each of the preceding examples.
18. Problems Solved by Equations. The following examples
will illustrate the method of solving problems by means of
equations :
Ex. 1. — When I double a certain number and add 16, the
result is 40. What is the number ?
Let X represent the required number.
Then 2x is the double of the number.
Then 2x+16 is the double with 16 added.
But the problem states that this is 40,
.-. 2a;+ 16 = 40,
2a; = 24,
3^ = 12.
Therefore the required lumber is 12.
The result should be verified by showing that the number
obtained satisfies the given problem.
Verifi6ation : When 12 is doubled I get 24 and when 16 is added
I get 40. Therefore tbc result is correct.
22 ALGEBRA
Note til at the substitution is made in the original problem,
not in the equation. There might be an error in writing down
the equation and then the solution obtained might satisfy
the equation, but would not necessarily satisfy the given
problem.
Ex. 2. — The number of pupils in a class is 33, and the
number of boys is 7 greater than the number of girls. J'ind
the number of each.
Let a; = the number of girls,
a;+7 = the number of boys,
.'. x+x-i-l = the total number,
.-. a;+x+7 = 33,
2x=33-7 = 26,
x=13, .-. a;+7 = 20,
.'. the number of girls is 13 and the number of boys is 20.
Verification : 20+ 13 = 33, 20- 13 = 7.
Ex. 3. — Divide $100 among A, B and C, so that B may
receive 3 times as much as A, and C $30 more than B.
Let x= the number of dollars A receives,
3x= „ „ „ B
:. 3a;+30= „ „ „ C „
.'. they all receive (rc+3a;+3a;+30) dollars,
.-. a;+3a;+3a;+30=100,
7a;+30=100,
7ic=70,
.-c=10,
.-. A receives $10, B $30 and C $60.
Verify this result.
19. Steps in the Solution of a Problem. The examples
which have been given will show that in solving a problem
the steps in the work are usually in the following order :
(1) Read the problem carefully to see what quantity is to he
found.
(2) Represent this unknoivn by a letter.
(3) // there be more than one quantity to be found, represent
the others in terms of the same letter.
SIMPLE EQUATIONS 23
(4) Express the condUion stated in the problem in the form
of an equation.
(5) Solve the equation and draiv the conclusion.
(6) Verify the solution by substitution in the problem.
On referring to Ex. 1, we see that there was only one quantity to
be found, and therefore step (3) did not appear in the sohition. In
Ex. 2 there were two quantities to be found, and when we represented
the number of girls by x, we could represent the number of boys by
a;+7.
The pupil is advised to make full statements, in plain
English, as to what the unknown represents.
Thus, in Ex. 3 to say, let x=^A, or let a;=^'s monej', will
only lead to difficulties.
Note. — The examples in the following exercise are to be solved by
means of the equation and the results should be verified in every
case. Although the answers to many of them may be given mentally,
the pupil is advised to give complete solutions, so that he may become
familiar with algebraic methods.
EXERCISES 11
1. If 37 is added to a certain number, the sum is 53. What is
the number ?
2. If 27 is subtracted from a number, the result is 5. What is
the number ?
3. A number was doubled and the result was increased by 27.
If the sum is now 73, what was the number ?
4. When a number is multiplied by 7, and 25 subtracted from the
product, the result is 59. Find the number.
5. If five times a number be increased by 6, the sum is the same
as if twice the number were increased by 15. Find the number.
6. What number if trebled and the result diminished by 36 gives
twice the original number ?
7. If you add 19 to a certain number the sum is the same as if
you add 7 to twice the number. Find the number.
8. Five times a number, plus 19, equals nine times the number,
minus 41. What is the number ?
24 ALGEBRA
9. Two miinhers differ by 11 and their sum is 51. Find the
tuimbers.
10. The sum of two numbers is 47 and one exceeds the other by
15- What are the numbers ?
11. ^'s salary is three times 5's and the difference of their salaries
is §1500. Find the salary of each.
12. A horse and carriage are worth $.360. The carriage is worth
twice as much as the horse. Find the value of each.
13. Divide 93 into two parts so that one part wiU be 27 less than the
other.
14. The length of a rectangle is three times the width. The peri-
meter is 72 feet. Find the sides.
15. ^ is tmce as old as B. In 10 j^ears the sum of their ages will
be 41 years. What are their ages ?
16. Divide 8500 between A and B so that A wiU receive $20 more
than twice what B will receive.
17. The sum of two consecutive numbers is 59. What are the
numbers? (Let x be the smaller number, then .c+1 will be the
greater. )
18. Find three consecutive numbers whose sum is 150.
19. ^'s age is twice 5's and C is 7 years older than .4. The sum of
their ages is 67 }-ears. Find the age of each.
20. The difference betw-een the length and width of a rectangle is
10 feet and the perimeter is 68 feet. Find the sides.
21. Divide $468 among A, B and C, so that B may get twice as
much as A, and C three times as much as B.
22 . A railway train travels m miles per hour. If it goes from Toronto
to Montreal, a distance of 333 mUes, in 9 hours 15 minutes, what is the
value of m ?
23. A line 20 inches long is divided into two parts. The length
of the longer part is J inch more than double the shorter one. Find
the lengths of the parts.
24. What value of x will make 5.r+6 equal to 3x -{-40 ?
25. If 5% of a .sum is $48, what is the sum ?
26. An article sold for $2-61 the loss being 10%. What was the
cost ?
SIMPLE EQUATIONS 25
27. Divide S145)(i among .-I, B and C, so tliat /)' will get three times
A's share and C will get SlOO more than ^4 and B together.
28. A has five times as much money as B. After A has spent S63
he has only twice as much as B. How much has B ?
29. If S20 less than | of a sum of money is SIO more than \ of it,
what is the sum ?
30. Three bojs sold 42 papers. The first sold \ as many as the third
and the second sold \ as many as the third. How many did each sell T
31. The sum of J- of a number and i of the same number is 55.
What is the number ?
32. A man paid a debt of S4500 m 4 months, paying each month
twice as much as the month before. How much did he pay the first
month ?
33. The half, third and fourth parts of a certain number together
make 52. Find the numbei-.
34. Divide 72 into three parts so that the first part is h of the
second and ^ of the third.
35. What number is that to which if you add its half and take
away its third, the remainder will be 98 ?
36. If 3a=46c,
(1) Find a, when 6=10, c=15.
(2) Find b, when a=12, c= 3.
(3) Find c, when a= 8, b= \.
EXERCISE 12 (Review of Chapter II)
1. State the four axioms which are used in solving equations.
2. Show that x^^lS is the correct solution of the equation
3x-7 = 2x+ll.
3. Determine if 8 is a root of 3(x+6) = .5(a;— 1).
4. Solve (a) 5a;+3 = 2a;+9; (fo) l+2x = 9-2x ; (c) 3a;-7 = 8-2a; ;
{d) 7a;+l = 9x-9; (e) lla;-l = 5a;+l.
5. My hotise and lot cost 816,800, the house costing five times as
much as the lot. Find the cost of each.
6. A horse and carriage cost $520. If the carriage cost $60 more
than the horse, what did the horse cost ?
26 ALGEBRA
7. Three farmers together raised 2700 bushels of wheat. A raised
three times as much as B, and G raised twice as much as A. How
much did each raise ?
8. What value of x will make 136— 3x equal to 172 — 9a; ?
9. Where r is the radius of a cii'cle and c is the circumference,
c = 27rr, where 7r = 3y.
(a) Find c, when r= 7 ; when r=42.
(6) Find r, when c = 88 ; when c=ll.
10. If s=\ft-, find s when < = 4 and/=32; when t=\Q and /= 32-2.
11. In a company of 98 persons, there are twice as many women
as men, and twice as many children as women. How many children
are there ?
12. Six boys and 15 men earn $264 a week. If each man earns
four times as much as each boy, how mucli does a boy earn in a week ?
13. Five times a certain number, increased by 47 is equal to eight
times the number, diminished by 43. What is the number ?
14. An agent charges 3 % commission for collecting an accoimt.
If his charge is $11-13, what was the amount of the account ?
15. Solve (a) •05x = 4 ; (b) x+-04a; = 208 ; (c) a;--06a; = 235 ;
(d) x+5%a; = 630.
16. If 6 is the base of a triangle and h is its height, the area (o) is
given by the formula a=^bh.
(i) Find a, when b= 8, /i= 4.
(ii) Find b, when a= 36, /i=12.
(iii) Find h, when a=176, 6 = 22.
17. The sum of the unequal sides of a rectangle is 65 feet and their
difference is 15 feet Find the area of the rectangle.
18. If 6x — y = 2x-\-y, what is the value of ^J if x = 6 ?
19. For what ninnber does the question mark stand, if
5x + i = 3a;+?
is satisfied when a; = 3 ?
20. If 4 % of a; together with 3 °;' of x is equal to 35, find x.
21. State a problem the condition of which is expressed by the
equation 3.1; — 20 = . r.
22. B has $10 more than A, and C has $20 more than B. Together
they have $190. How much has each ?
23. A turkey costs as much as three chickens. If 2 turkeys and
3 chickens cost $7-20, find the cost of a chicken.
SIMPLE EQUATIONS 27
24. What number increased by f of its(ilf is ocjual to 60 ?
25. Divide $6400 among A, B and C, so tliat B will get $120
more than A, and G $160 more than A.
26. The net income from an enterprise doubled each year for five
years. If the total net income for the five years was $7750, what was
the income for the first year ?
27. If 2ab = 37nn,
(1) Find a, when 6 = 15, m— 6, n= 5.
(2) Find b, when a=12, m= 2, n= 2.
(3) Find m, when a= ^, 6 = 6, n = J.
(4) Find n, when a= ■d,b=-6, m=-12.
28. Show that 6 is a root of the equation
2(a;-l)(a;+2) = 4(a;+3){x-5) + (a;-2)(.r + 5).
29. The area of the United States is 4000 square miles more than
seventy times the area of England. If the area of the United States
is 3,560,000 square miles, find the area of England.
30. Solve and verify :
(1) 6850+a; = 27a;+350.
(2) lx+lx+ix = Sd80.
(3) 1607a;+20= 1762a;- 11.
CHAPTER III
POSITIVE AND NEGATIVE NUMBERS
20. Arithmetical Numbers. In the diagram the hours from
12 noon to 12 midnight are represented on the horizontal
hne, and the temperature at each
hour is shown by the position
of a point on the corresponding
vertical Une.
Thus at 3 P.M. the tempera-
ture was 61°, at 7 P.M. 53° and at
11 P.M. 55-5°.
The points which show the tem-
perature for each hour are con-
nected by a curve. This curve
gives a picture of the changes in temperature during these
twelve hours.
These cnanges might be shown by a column of figures,
but the curve exhibits the variations in temperature more
readily to the eye. We can see at a glance when the tempera-
ture was rising and when falling, at what hours it was the
same, that it rose or fell more rapidly during certain hours
than during others.
Here we say that we have represented graphically the
changes in temperature, and the curve shown is called a graph.
12 1 2 3 4 5 6 7 8 9 10 11 12
BXE3RCISB 13 (1-8, Oral)
Using tlie diagram, answer questions 1-8.
1. What was the temperature at 1 p.m., at 4 p.m., at 10 p.m. ?
2. At what hours was the temperature the same ?
28
POSITIVE AND NEGATIVE NU31BENS
29
3. What was the highest temperature ? What the lowest ?
4. What was the range of temperature ?
5. Between what liours was it rising ?
6. How much did it rise between 10 and 11 ? How much did it
faU between 6 and 7 ?
7. When was it 60°, 58°, 55° ?
8. Between what hours did it rise most rapidly ? When did it
fall most rapidly ?
9. The percentage of games won by a baseball team, up to the
beginning of each month of the playing season, was as follows :
June, 66; July, 63 ; Aug., 60-5 ; Sept., 62 ; Oct., 61-5.
Draw a graph showing these changes.
10. A boy's height in inches, for each year from the age of 7 to thfe
age of 14, was 44, 47, 50, 51, 52-5, 54, 56-5, 58. Draw a graph to
illustrate the variations in his height.
21. Negative Numbers. This diagram shows the average
temperature for a week.
Thus on Monday it was 25°
above zero, while on Thursday it
was 15° below zero.
We might express this algebra-
ically by saying that on Monday
the temperature was +25°, and on
Thursday it was —15°.
The number ^-25 is called a
positive number and is read " posi-
tive " 25 or " plus " 25, while
number, and is read " negative " 25 or " minus " 25
A negative number is therefore one which is measured on
the opposite side of zero from a positive number.
25 is called a negative
BXBRCISB 14
1. Using algebraic signs, write down the temperature for each day
in the diagram. Also read the temperature.
2. On what days was the temperature negative ?
30 ALGEBRA
3. How much higher was it on Monday than' on Thursday ? How
much lower on Tuesday than on Saturday ?
4. Tf the temperature is —30° and it rises 40°, how much will it
be then ? If it had fallen 10°, how much would it have been then ?
5. The temperature at which mercury freezes is — 39°C. What does
that mean ? How much lower is it than the normal temperature
of the blood which is +37°C. ?
6. Tf the price at which a certain stock sells above par is positive
and the price below par is negative, make a diagram similar to the
preceding, showing the prices of a certain stock for a week, when the
record was as follows :
Mon., 4 above par ; Tues., 2 below ; Wed., 1 above,
Thurs., at par; Fri., 3 below; Sat., Ih below.
22. Distances measured on a Horizontal Line.
P B O AC
I \ 1 \ 1 1 \ \ 1 1 1
-5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5
On this diagram the distance between each successive
marking represents one foot.
What is the length of OA ? of OB ? In what respect does OA
differ from OB ? How might we use signs to show this difference ?
It is usual to consider measurements made to the right as
positive and to the left as negative.
What point is +5 feet from O ? What one is —5 feet from 0 ?
If a point moves from O, 4 feet to the right and then 7 feet to the left,
how far is it then from 0 1 Is the distance positive or negative ?
We thus see that in addition to the numbers of arithmetic
which begin with zero and extend indefinitely in one direction,
we now have another series of numbers which also begin
with zero and extend indefinitely in the opposite direction.
In each series all integral and fractional numbers are included.
23. Further Examples of Negative Numbers.
(1) A man has property worth $100, and debts amounting
POSITIVE AND NEGATIVE NUMBERS 31
to $60. When he has paid his debts he will have property
worth $40.
Thus, $100- $60= $40.
If, however, he has debts amounting to $100, when these
are paid he will have nothing left.
Thus, $100— $100= $0.
If he has debts amounting to $140, when he has paid all
he can he will still owe $40. We express this algebraically
thus: $100— $140= —$40.
In the first case we say that his net assets are $40, in the
second they are zero, and in the third they are minus $40.
When we say his assets are — $40, we mean he is $40 in debt.
It will be seen that the difference in meaning between
+40 and —40 when referring to dollars is practically the
same as the difference between +40 and —40, when
referring to degrees of temperature, as in art. 21, or to distances
measured in opposite directions on a horizontal hne, as in
art. 22.
(2) If a man gains $20 on one transaction and loses $15
on another, what is the net result ? If he had lost $25 on
the second transaction what would have been the net result ?
If we attach a joins sign to the result when it is a gain, how
may we indicate a loss ?
If O represents a sum gained and L a sum lost, state the result in
each of the following, attaching the proper sign :
1. $30 G+ S20 G. 2. $30 G+ $20 L. 3. $30 L+ $20 L.
4. $30i+$20G. 5. $40G+U0L. 6. $20 (7+ $60 L.
(3) If a game won is represented by +1, then —I would
represent a game lost.
In a series of games I find that my record is : won, lost,
lost, won, lost, won, lost, won, won.
This might be represented thus :
+l-l-l + l-l + l-l + l + l = +5-4= + l.
What does this result mean ?
Write in a similar way the following record : lost, lost, won, lost,
won, lost, lost, won. Also the following : won, lost, drawn, won, won.
32 ALGEBRA
(4) In locating points on the earth's surface, the distance
in degrees north of the equator (north latitude) is said to be
positive, and south of the equator negative.
Thus, the latitude of Toronto is -f 44° and of Rio de Janeiro is — 23°.
What is the distance in degrees of latitude between these two cities ?
The preceding illustrations show that a positive number
differs from a negative number in direction or quality.
Thus, if + 10 means 10 yards measured to the right ; or *10° east
longitude ; or 10 games won ; or 10 miles a boat goes up stream ;
or 10 minutes a clock is fast ; or SIO in my bank balance ; or 10
pounds lifted by a balloon ; what would — 10 mean in the corres-
ponding cases ?
24. Signs of Operation and Signs of Quality. The numbers
+25 and —25 are the same in magnitude, but differ in
direction or quality.
When a number is preceded by the sign +, it means that
the number is taken in the positive direction or sense, and
when preceded by the sign — , that it is taken in the
negative direction.
It will thus be seen that we use the signs + and — with
two different significations. When they are used to indicate
the operations of addition or subtraction, they are called
signs of operation. When they are used to indicate direction
or quality, they are sometimes called signs of quality.
The beginner might think that this ambiguity would lead
to confusion, but he will find that such is not the case.
When we read a quantity like —25, we should say " negative
25," but this is not followed in practice, as it is usually read
" minus 25."
When no sign precedes a number, it is understood to be a
positive number.
25. Absolute Value. The absolute value of a number is
its value without regard to sign.
Thus, + 8 and — 8 have the same absolute value.
POSITIVE AND NEGATIVE NUMBERS 33
EXERCISE 15 (1-15, Oral)
1. What is the net property of a man who, {a) has $60 and owes
$47, {b) has $40 and owes $50, (c) has $65 and owes $65 ?
2. What is the value of, (a) $40- $30, (b) $40- $60, (c) $30- $20,
(d) $20- $30, (e) $10- $0, (/) $0-S10 ?
3. The temperature was —10° at 6 p.m. and 4° at 10 p.m. How
many degrees did it rise in the interval ?
4. A Uquid whose temperature is 20° is cooled through 30°. What
is the final temperature ?
5. A vessel sailed on a meridian from latitude 15° to latitude —5°.
How many degrees did it sail and in what direction ?
6. What is the distance between a place 90 miles due east of
Toronto and another 60 miles due west ?
7. I am overdrawn at the bank $20. What must I deposit to
make my balance $100 ?
If — 20+x=100, what is x ?
8. What would a negative number mean in stating the height of
a tree above the window of a house ? The height above sea level of the
bottom of a weU ?
9. A man buys a horse for $100 and sells him for $80. What is
his gain and his gain % ?
10. A man travels 20 miles from A and his friend travels —10 miles
from A. How far are they apart ?
11. What is the rise in temperature from —30° to —10° ?
If — 30+x= — 10, what is xl
12. What is the distance between two places which are a mUes
and b miles west of Montreal, (1) if a is greater than b, (2) if b is
greater than a ?
13. Denoting a date a.b. by + and B.C. by — , state the number
of years between these pairs of dates :
(1) +1815 to +1915. (2) -20 to +75. (3) -65 to -37.
(4) -120 to +60. (5) -200 to +200. (6) +1900 to +1800.
14. Augustus was Roman Emperor from —31 to +14. How many
years was he Emperor ? What is the difference between 14 and — 31 ?
D
34
ALGEBRA
15. The First Punic War lasted from —264 to —241. How Jong
did it last ? What is the difference between —241 and —264 ?
16. A boy adds 15 marbles to his supply, gives away 10, buys 5
and gives away 12. How many has he thus added to his supply ?
17. I have $a in the bank. If I issue a cheque for $6. what is my
balance when the cheque is p^id ? If a =40 and 6=50, how do you
interpret the result ?
18. A has $50 and B has $20. A owes B $10 and B owes A $40.
How much will each have when his debts are paid ?
19. The weights of two pieces of iron are 65 lb. and 147 lb. If they
are attached to a balloon with an upward pull of 239 lb., how would
you represent the combined weight ?
20. Represent graphically the following changes in the price of
a stock :
Month.
July.
Aug.
Sep.
Oct.
Nov.
Dec.
Jan.
Feb.
Mar.
Apr.
Amount above par . . .
6
1
4
5
2
3
Amount below par . . .
2
3
1
4
EXERCISE 16 (Review of Chapter III)
1. Using signs, express the results of the following transactions :
(1) A gain of $10 followed by a loss of $15.
(2) A loss of $12 followed by a loss of $4
(3) A loss of $8 followed by a gain of $10.
2. What is the difference between 40° and -3° ?
3. If an vipward force or pull is positive and a downward force is
negative, what single force is equal in effect to these pairs of forces :
(1) 101b., -3 1b. (2) 8 1b., -12 1b. (3) -7 1b., -2 1b.
(4) -9 1b., 3 1b. (5) 61b., -6 1b. (6) 2a lb., -alb.?
4. In firing at a. target each hit counts 5 and each miss —3. If I
fire 10 times and make 0 hits, what is my score ? If I make only 2 hits
what is my score ?
5. What is the fall in temperature from 27° to - 1 1° ?
If 27— «= —11, what is a; ?
POSITIVE AND NEGATIVE NUMBERS 35
6. In a 100 yards handicap race A has 3 yards start and B has
— 3 yards start. What do these mean ? How far has each to run ?
7. In solving a problem in which it is required to find in how
many years A will be twice as old as B, I get the answer — 10. What
does this answer mean ?
8. Find the average noon tesoperature for a week in which the
noon temperatures were : 20°, 10°, 15°, 0°, 4°, —6", —15°.
9. A train was due at 10 minutes to 3. How many minutes before
three did it arrive if it was half an hour late ?
10. A man travels 8 miles, then —6 miles, then 4 miles, then
— II miles. How far has he travelled ? How far is he from the
starting point and in what direction (positive or negative) from it ?
11. Egjrpt was a Roman province from —30 to 616. How many
years was this ? What is the difference between 616 and — 30 ?
12. The daily average temperature for 14 days were : 6°, 5°, 0°,
-4°, 2°, -6°, -2°, 0°, 5°, 1°, -1°, -6°, -3°, 3°. Show these
variations by means of a graph.
13. If a gain of a dollar be the positive unit, what will represent
a loss of 83-50 ?
14. The record of a patient's temperature for each hour beginning
at 12 noon was : 100°, 100-5°, 102°, 101°, 104°, 101-5°, 99-5°, 98°,
97- 5"^, 97°. Represent these changes graphically, taking two spaces on
the vertical line to represent one degree.
15. If the normal temperature of the body is 98-5°, write the record
in the preceding question vising positive and negative signs.
16. The minimum temperatures for the first 15 days of December
were: 26°, 22°, 14°, 25°, 21°, 18-5°, 13°, 7-5°, 11°, 6°, -4°, -6°, -1°,
10° 12-5°. Make a chart to show these variations.
d2
CHAPTER IV
ADDITION AND SUBTRACTION
26. Addition of Positive Quantities. What is the result of
combining :
(1) A gain of $20 with another gain of $10 ?
(2) A measurement of 5 feet to the right with another of
3 feet to the right ?
(3) A rise in temperature of 10° with a rise of 8° ?
(4) 6 points won with 4 points won ?
As explained in Chapter III., we will consider all of these
to be positive quantities, and we might show this by attaching
the positive sign to each.
We might write these four questions as problems in
addition, thus :
+ $20 +5 feet +10° + 6 points
+ $10 +3 feet +8° + 4 points
+ $30 +8 feet +18° +10 points
Similarly, the sum of 6a and 4a is 10a, and the sum of 2x^, 5x^ and
6a;* is 13x*.
Here we have not prefixed any sign, and when that is the case the
positive sign is imderstood.
We see then that the sum. of any number of positive quantities
is always positive.
27. Addition of Negative Quantities. W^e might change the
data of the four qv;estions in the preceding article so that
all the qviantities would be negative.
ADDITION AND SUBTRACTION 37
Thus, the first might be changed to — . " What is the
result of combining a loss of S20 Avith a loss of $10 ?
Read the other three questions making similar changes.
What would now be the answer to each question ?
As problems in addition they would now appear thus :
-S20 -5 feet -10° - 6 points
-SIO -3 feet — 8° - 4 points
- S30 - 8 feet - 18° - 10 points
Similarly, the sum of —7x and —5x is —12a;, and the suna of —5a',
— 2a', —a' and — 6a' is —14a'.
Thus, the sum of any number of negative quantities is negative,
and is found by adding their absolute values and prefixing the
negative sign to the result.
BXEROISB 17 (Oral)
State the results of the following additions :
1.
+ $3
- 7
-13
Add 10a:,
Add ^m,
2. -SIO
-S 8
10.
12
3. -12°
-10°
4.
8.
— )n'^,
-Sy,
3 yd.
5 yd.
5.
6. 4a2
12x, I5z.
fm, |m, |to.
7. -lOx'-y
— 5x^y
—^abc
— 'labc
9.
11.
Add -3m\
Add —I2y,
-7m 2.
-ny.
28. Compound Expressions. An expression of one term is
frequently called a simple expression, while one of more than
one term is called a compound expression.
Thus, 2a, 3x^y, labc are simple expressions, and 2a + Zb, 5x — 3m + a*
are compound expressions.
29. Addition of Compound Expressions. In arithmetic if we
wish to add two or more com]:)ound expressions, we write
them under each other, with the like denominations in the
same column.
38 ALGEBRA
We proceed in a similar way in algebra, writing like terms
in the same column.
In arithmetic. In algebra.
2 yd. 1 ft. 6 in. 2a+ 6+ 6c
3 vd. 1 ft. 4 in. 3a+ 5+ 4c
5 yd. 2 ft. 10 in. 5a+26+10c.
If the like terms are not in the same order, they must be
properly arranged for addition.
Ex.— Add 5x+Sy-2z, 4y—5z-{-3x, —^z-\-4x+y.
Here the problem might be written thus :
5x+32/— 2z
3x+42/— 5z
4x+ y— 3z
Smn=12a;+82/— lOz.
EXERCISE 18 (1-6, Oral)
Add:
1. 3 ft. 2 in. 2. 3a;+22/ 3. 5a— 116
5 ft. 3 in. 5x+Zy 2a- 3b
6hr. lOmin. 11 sec. 5. 5a— 36+2c 6. — a—3b+7c
5 hr. 12 min. 3 sec. 2a— 46+3c —3a— 6+4c
2 hr. 15 min. 20 sec. 5a— 6+ c —5a— 26+ c
7.* a+b—c, 26 — 3c-fa, 36+5a — lie.
8. 5x^—lx+Q, 3-5x+x^-, -2x+ix-.
9. 2a— 36, 3a— 26, 4a, -6, a— 6.
10. a+26, 6+2c, c+2a, a+6+c.
11. a-26+c— 3d, c— 56— d+2a, -6+c— c^+a.
12. 5x—3y, —2y+z, iz—y+3x.
13. ia-lb, fa- 16, ^a— #6, |a— 16.
14. a2+262-3c2, 562-c2+2a2, Za^+b^-2c~.
15. a+6+c, 6+c+c?, c-\-d+a, d+a+6.
16. ^^x+ly-lz, §x+3y-|z, x+y-fz.
ADDITION AND SUBTRACTION
30. Addition of Quantities with Unlike Signs.
What is the result of combining :
(1) A gain of $20 with a loss of $10 ?
(2) A gain of $5 with a loss of ^15 ?
(3) A loss of $8 with a gain of $6 ?
(4) A loss of $7 with a gain of $12 ?
These might be written as problems in addition, thus .
+ $20 +$5 -$8 -$ 7
— $10 —$15 +$6 +$12
39
+ $10 -$10 -$2 +$5
It is thus seen, that when we add two quantities differing
in sign, the sum is sometimes positive and sometimes negative.
When is it positive and when is it negative ?
How is the numerical part of the sum found when the
signs are alike ? How is it found when the signs are different ?
The answers to these questions might be combined mto
the following rule :
When the signs are alike, the sum is found by arithmetical
addition, and the common sign is affixed ; when the signs are
different, the sum is found hy arithmetical subtraction, and the,
sign of the greater is affixed.
Ex. 1. — Find the sum of 6 and —8.
Here the result is —2, since the difference between 8 and 6 is 2 and
the one with the greater absolute value is negative.
If there is doubt in any case, it is advisable to make the problem
concrete by substituting for +6, a gain of $6 and for —8, a loss of $8,
when the result will at once be evident.
Ex. 2. — Find the sum of 5a, —8a, —7a, 6a, —2a,
The sum of the positive quantities is 11a.
The sum of the negative quantities is —17a.
The sum of 11a and —17a is —6a,
.'. the requii'ed sum is —6a.
They might also be added in the order in which they coma.
Thus, the sum of 5a and —8a is —3a, of —3a and —7a is —10a,
of — 10a and 6a is —4a, of —4a and —2a is —6a.
40
ALGEBRA
Ex. 3.— Add 3a— 116+5C, 66— 5a, 56— c+a.
Write the expressions in colunuis as already explained.
a=6=c=l
3&-116 + 5C I = -3
-■5a+ 66
a+ 56— c
= +1
= +5
-a +4c = +3
The sum is — a+4c or 4c— a, the sum of the second column being
zero.
We may check the result by substituting particular numbers
for the letters. Thus, if we substitute unity for each letter
the first quantity becomes 3—11+5 or —3, the second is +1,
the third is +5, and the sum (— a+4c) is +3. Since the sum
of —3, +1 and +5 is +3, we assume that the work is correct.
1.
+6 ft
-3 ft.
2.
6.
10.
EXERCISE
-$10
+ $27
19 (1-
3.
7.
11.
12, Oral)
+ 10 lb.
-151b.
4.
8.
12.
-50°
+40°
5.
-3
7
3a— 26
5a +36
5a2
-3a-
2x-ly
-^x+2y
- Ix
Ux
— 8a6
8a6
9.
a+66
a— 66
3a+26— c
2a— 26+c
13. 3j;+5t/, 2x—Zy, —^x-y, Qx-4.y. (Check.)
14. 5m, — Gw, —7m, 8m, —9m, 10m.
15. 2a+36— 5c, 6a— 46+c, 3a+26+4c. (Check.)
16. 3a— 56, 46-3c, 4c— 3a, a+6+c. (Check)
17. lx+y—\z, ^x—ly+\z, x-\-ly-{-z. (Check.)
18.* a— 26+3c, 6— 2c+3rf, c—M+2a, 6— 2c— 3a.
19. Zx+5y—2z, 2.r— 3y+42, 4.x+y—5z, 6.r+2?/+32.
20. 3a6— 4ac+56c, 5ac— 46c— 2a6, 36c— a6— oc.
21. 6a2-5a6+62, 3a2+7a6-262, a~-ab+b\
22. 2a;2-3?/2+422, 5y^-%z^^x^, 2z^+2y^-3x\
23. ^a-lb+lc U-^b-^c.
ADDITION AND SUBTRACTION 41
24. If the sum of 13x— 7, 2x-+5 and 6— 4x is 48, find the value of
X and verify.
25. If the sum of x— 6, 3x— 6 and 5x—Q is the same as the sum of
12— a;, 12— 3.r and 12—5.1% find x and verify.
31. Indicated Additions. If we wish to use the sign of
addition to indicate that 6 is to be added to a, we ^\Tite it
thus : a-\-b.
Similarly, if we wish to indicate the sum when —7 is added
to 11, we write it ll+( — 7), the negative quantity being
enclosed m brackets.
To find the value of ll+(— 7), we must add 11 and —7,
which is done by subtracting their absolute values and
prefixmg the positive sign.
Thus, ll + (-7) = ll-7 = 4.
Similarly, 6a+( — 3a) = 6a— 3a=3a,
5m-|-( — 3m)4-( — m) = 5m— 3m— m = m,
and a+{ — b)=a—b.
We thus see that, to add a negative quantity is the same as
to subtract a positive quantity of the same absolute value.
If we 'oish to simphfy a quantity like
(3a-2&)+(2a-36),
we may write 2a— 3b under 3a— 26, and add in the usual way,
or we may remove the brackets and say that the quantity
=3a— 26+2a— 36,
=5a— 56, when the like terms are collected.
EXERCISE 20 (1-12, Oral)
Simplify :
1. -3+4. 2. 10+(-6). 3. 3+(-4).
4. (-2)+(-3). 5. 5a4-(-4a). 6. 76+(-46).
7. — 8a+(+7a). 8. —5ab+{—2ab). 9. 9x-+{ — 3x^).
10. —p+{-3p). 11. (-3m)+(-8wi).
12. _a+(_2a) + (-3a).
13.* l0xy+(-Zxy) + {-4^y)-xy+i-5^y)-
42 ALGEBRA
14. -6 + (-26) + (-36) + (-46) + 106
15. -5/j2-|-3p2_^(_2^2)_^8^2,
16. (2/rt+3H.) + (5//i— ;0 + (3m-5«).
17. (6.i--f3(/-43)+(a;+2^-;:) + (^+,--7a:).
18. a + (-6)4-& + (-c)+c+(-«).
19. a;+(a— 26+c) + (6— 2c+a)+(c-2aH-6).
20. When —20 is subtracted from 10, the difference is 30. Show
that this is true by adding the difference to the quantity which was
subtracted.
21. Show by addition, that when 2a — &+5c is subtracted from
3a— 46 -|- 3c the remainder is a — 36— 2c.
22. Solve (2x+3) + (3a;-5)-|-(5.r-l)=57. (Verify.)
23. Solve (8x-7)+(-4x-3) = (-5.x--7)+(7a;-2). (Verify).
32. Subtraction is the Inverse of Addition. To subtract 4
from 7 is equivalent to finding the number which added to
4 will make 7.
Thus every problem in subtraction may be changed into a
corresponding problem in addition.
If we wish to subtract —4 from 7, we enquire what number
added to —4 will make 7. We might make the problem
concrete by finding what must be added to a loss of $4 to
result in a sain of S7, and the answer is evidently a gain of
$11.
.'. when —4 is subtracted from 7 the remainder is 11.
Thus, 7 less— 4=11, because —4+ 11 = 7.
- 10 less -3 = -7, because -3 + (-7)= —10.
66 less -46=106, because —46+106 = 66.
EXERCISE 21 (Oral)
What must be added to
1. A gain of SIO to give a gain of $15 ?
2. A gain of $8 to give a gain of $3 ?
3. A gain of $5 to give a loss of $4 ?
ADDITION AND SUBTRACTION 43
4. A loss of S6 to give a gain of $3 ?
5. A loss of $20 to give a loss of $15 ?
6. A loss of $5 to give a loss of $8 ?
7. 8 to give 12. 8. 10 to give 3.
9. —8 to give 2. 10. —8 to give —2.
11. 7x to give 3a:. 12. 3a to give —5a.
13. 5a;- to give —2x^. 14. —8a; to give —10a;.
15. 3abc to give 2abc. 16. — 6y- to give —9y-.
17. —5t to give —41. 18. a to give —a.
33. Rule for Subtraction. Examine the following sub-
tractions :
(1) 9 (2) 9 (3) - 9 (4) -9
3 -3 3 -3
6 12 -12 -6
In (I) the result 6 might have been found equally well by
adding —3 to 9, in (2) by adding +3 to 9,. in (3) by adding
— 3 to —9, and in (4) by adding -\-3 to —9.
Thus we see that the problems might have been re-written
as problems in addition by changing the sign of the quantity
to be subtracted.
We would then have these problems in addition :
(1) 9 (2) 9 (3) - 9 (4) -9
-3 +3 - 3 +3
6 12 -12 -6
We have therefore the following rule for subtraction :
Change the sign of the quantity to be subtracted and add.
To subtract compound expressions, apply the rule to the
like terms of the expressions.
Thus, to subtract 6a— 36 + 2c change to 6a — 36 + 2c and add
3a+46-6c -3a-46 + 6c
3a -76 + 8c.
44
ALGEBRA
EXERCISE 22
Re-write the following problems in subtraction as problems in
addition and find the result :
1.
5.
25
2.
-11
13
+ 3
16a2
6.
56
20a2
86
10a
■ 8a
4.
20a;
■ 4x
-3x^y
8. -23m2
9. 3a+76 10. 6x—Sy 11. 5TO+4yi 12. 6a;2— 5x+2
2a_26 — 2a;+4y 5m— An 3x2— 2x— 3
13. Subtract a— 26 + 3c from 3a— 46+c.
14. Subtract 5a;2-llx+4 from 3a;2-2j;+5.
As soon as possible the pupil should learn to subtract,
without actually changing the signs, but by making the
change mentally.
EXERCISE
23 (1-8, Oral)
Subtract :
1. 8
2.
-9
3.
-12
4.
-7a
-3
4
- 6
2a
5. -8a;
6.
—7m
7,
— 4a6c
8.
-9i
-3a;
10.
—2m
11.
lla6c
12.
-H
9. 3a;+42/
2a -36
7a:2-5
6m— 3n
x—2y
3a +26
3a;2-7
-3m+6Ji
13. 2m+4n— 3^ from 5m—3n+6p, and verify the work by addition.
14. Find the remainder when 6a— 46 — 5c is subtracted from
2a-}-36-llc. (Verify.)
15.* Subtract 5a — 36 from the sum of 2rt— 6 and 6a— 46.
16. Subtract a— 26+c from 6a— 6 + 5c and from the remainder
subtract 3a+6— c.
ADDITION AND SUBTRACTION 45
17. Subtract the sum of Sx-— 5a;+6and5a;2+4a;— 3 from 5x^—x-\-Z.
Check when x=l.
18. What must be added to 2??i+3n— 4p to give 5m — n—2p 1
(Verify.)
19. By how much is Ta^— 15a— 11 greater than Sa^— lla4-4?
20. By three subtractions simplify
(6a+106-c)-(3a+46-2c)-(a-36+4c)-(2a+76-3c).
21. Subtract the sum of 2p—5q—2r, —p-\-Zq—2r and 4^+65'— 4r
from the sum of 3^— 4g'+5r, 3g'— 4r4-5/> and 3r— 4p+55'.
22. Subtract 2a — 36+ 5c from zero.
23. What is the excess of 15 over 10 ? 8 over — 2 ? —4 over
— 11 ? a+6 over a— 26 ? 2x-—bx+2 over 2a;2— lla;+7 ?
24. Add a2+2a— 5 to the excess of 2a2_4a+3 over a^- 3a+10.
25. Subtract the sum of a— 36+c, 6— 3c+a, and c— 3a4-6 from
zero.
26. What must be added to the sum of x^—5x, 2x-— 3a;+4 and
6x— 3a;2 so that the result will be unity ?
27. From 2x—Zx^-^5—x'^ take 3-ll.r3— 5a;2— 6.r.
34. Indicated Subtractions. If we wish to indicate that
— 3 is to be subtracted from 9 we write it: 9— (—3). From
the preceding this is at once seen to be equal to 9+(+3) = 12.
Also, _8-(-5)= _8 + ( + 5)= -3.
— 3a— ( — 5a)= — 3a+5a=2a.
a-(-6)= a + { + h) = a-{-b.
We thus see tliat to subtract a negative quantity is the same
as to add a positive quantity of the same absolute value.
Similarly, (3a+46)-(2o-36) = (3a+46) + (-2o+36),
= 3a+46 — 2a + 3fe,
= a+7b.
Also a — (6 + c)=a + ( — 6— c)=a— 6— c,
and o — (6 — c) = a + ( — 6 + c) = a— 6 + c.
Thus, brackets which are preceded by a minus sign may be
removed if the signs of all the quantities within the brackets be
changed.
46 ALGEBRA
Ex.— Simplify 5x-3y+4z- {Sx—2y+2z).
5x — 3y-\- 42
We may consider this as an ordinary problem 3x — 2y4-2z
in subtraction and proceed in the usual way
2x- y + 2z
We may, however, remove the brackets using the rule and then
collect the like terms, thus :
The expression —5x—3y-\-4:Z—3x-\-2y — 2z,
= 2x-y + 2z.
EXERCISE 24 (1-10, Oral)
Simplify
1. io_(_3). 2. -5-(-6). 3. -la-i-ia).
4. 8x-{-3x). 5. {-2ni)-(-3m). 6. -(-6) + 6
7. 8— (—4) — ( — 2). 8. 8a6— 10a6— ( — 7a6).
9. m-(-3m)-(-5m). 10. -4x^+{-3x^)-{-lx^).
11.* {5x-2y)-{2x-4:y). 12. 3a-116-{5a-86).
13. 2a-36f5c-(a-46+5c).
14. (a+6) + (2a-36)-(4a-36).
15. a+6—c—(6+c— «) + («+&— c).
IG. {Qx~-3x+5) + {2x--5x-e)-{5x^-8x-}-2).
17. Find the value of 5a+&, when a=2, b= — 3.
18. Find the value of 2a-\-3b—c, when a=\, 6=2, c= — 3.
19. By two different methods find the value of a— (6— c), when
a=20, 6 = 10, ,c=7.
20. Solve and verify
(1) 2x-3-(a;-4)=8.
(2) 3a;-l-(a:-3)-(a;-|-7)=40.
(3) l-{4:-x)-{5-x)-(Q-x)=52.
21. What value of x will make 5x-— 6 exceed 3.r— 11 by 70 ?
22. Find the value of 3x^-2x+5-{2x^-\-x-l), whenx=0, 1, 2, 3,4.
ADDITION AND SUBTRACTION 47
23, Remove the brackets and simplify :
(1) (a+3b-Uc)-{b+3c-8a)-{c+5a-2b).
(2) {a-3b)~{b-3c) + {c-3d)-{d-3a).
(3) -{3x-y+2z)-{2x-Sy+4z)-i3y-6z-5x)
(4) -{a-b)-{b-c)-{c-d)-{d-a).
EXERCISE 25 (Review of Chapter IV)
1.* Find the sum of oa, —3a, la and —8a.
2. Find the sum of four consecutive integers of which x is the
least.
3. Find the svun of five consecutive integers of which n is the middle
one.
4. Add 3a-26 + 7c, 56-3c-2a and c-a-36.
5. Subtract —106 from —66, —3a from 5a.
6. From 4a— 36 + 5c subtract 2a — 56 — c.
7. Subtract 5a;— 3,v + 42 from 4:X—2y — z.
8. What must be added to 3a — 56 + 6c to give 6a — 76 + 4c ?
9. If cc + 2/=10 and x — y = 4:, what is the value of 2x ? What is
the value of 2y ?
10. What is the sum of the coefficients in 6a— 116 + c — 3f/ ?
11. What must be added to x — y to give 0 ?
12. When a;=3 and 2/ = 4, what is the remainder when x'^ — y^ is
subtracted from 2xy ?
13. Wliat is the difference between 2a — 6 — c and a — 6 + c ? Give
two answers.
14. To the sum of 3m — in and 2m — Sn add the sum of ?w. + 7n and
3m — 2n. — ^t ^ j, - ~ -
15. From 4a;^ + 3a;— 7 subtract the sum of 2x^ + 73;— 5 and
2a;2-8a;+7.
16. By how much is 3a;— 7 greater than 2a;+o ? For what value
of X would they be equal ?
17. Simplify a+(26-3c)-(c+a).
18. If x=a + 26 — 3c, y=:6 + 2c-.3a and 2=c + 26-3a, find the value
oi{l) x + (y + z), (■2)x-(y-z), {3) x-(y + z).
48 ALGEBRA
19. From the sum of •5x-\--4:y and -Sx—^y subtract the sum of
■2x—l-ly and •4x+l-32/.
20. By how much is 3x2 — 5a;+ll greater than Sx^ — 8x+17?
What is the meaning of the result when x = 2 ? when a;= 1 ?
21 . From |a — J6 + ^c take ^a—^b — hfi.
22. If a+6 + c = 0 when a=3x— 4^ and b = 4:y — 5z, what is the
value of c ?
23. Solve 5a;-3-(x-4)-(a;-2) = 27. (Verify.)
24. What value of x will make 3x—2 exceed x— 7 by 63 ?
25. Whena=l, 6 = 2, c = 3, the sum of x+a— 36 + 4c, 2x+6-3c + 4a
and 3x+c— a— 6 is 124, Find the value of x.
26. Using brackets, indicate that the sum of a and b is to be
diminished by the sum of c and d. If a = 2x— 3, 6 = 5 — 3x, c = 3x— ^,
d = I — 6x, what is the result ?
27. Solve 2-(x-^)-(i-3x) = 16-25.
28. Subtract 2w— 7n — 4x from zero.
29. What must be added to a — (1 — 6) — (1— c) to produce unity ?
30. Subtract the sum of 2a + 36 — 4c + d and a—'lb — c — d from the
-excess of 4o— 6 + c over a-\-b-\-c.
CHAPTER V
MULTIPLICATION AND DIVISION
35. Multiplication of Simple Positive Quajitities.
of a product may be taken in any order.
The factors
Thus,
Also
3x5x4 = 3x4x5 = 5x4x3 = etc.
axbxc=axc x6 = 6 XaXc=etc.
ab
ab
ab
ab
ab
ab
The latter product may be written abc, acb, etc.
Sax 26 = 3X0x2x6,
= 3x2xax6,
= 6ab.
Make a diagram to show that 4:XX2y=Sooy.
Similarly, 3abx5cd=3xaxbx5xcxd,
= 3x5xaxbxcxd,
= 1 Sabcd.
Thus, the coefficient of the product is obtained by multiplying
the coefficients of the factors, and the literal part of the product
by multiplying the literal parts of the factors.
36. The Index Law for Multiplication.
a^xa^ =axaxaxaxa =a^
2xx3x^=6xxxxxx =Qx^.
m^Xm^^m .m, .m, .mxm .m .m—m"^.
Similarly,
and
3y^x4y^=l2y»,
Thus, the index of the product of powers of the same quantity
is found by adding the indices of the several factors.
49 E
50 ALGEBRA
EXBRCISS 26 (Oral)
Find the product of :
1.
2x, Zij.
2.
4?w, 5n.
3.
\x, 4i/.
4.
3x, 4a;.
5.
%x, Ix.
6.
3a6, 4a;2/.
7.
a^ a.
8.
32/2, 2^/2.
9.
ab, ac.
10.
2x^, 4a;^.
11.
5j>2, 4p4.
12.
5a: 2, 3x2/.
13.
(4x2)2.
14.
Sax, 2aa;,
15.
<2, t\ t*.
16.
a6, ac, a.
17.
5a2, 3a, 2.
18.
(4a)3.
19.
fa, ih, |c.
20.
(36)2, (26)2.
21.
f m, 6w, 5mn.
37. Multiplication by a Negative Quantity.
4x3 is a short way of writing 4+4+4=12.
—4x3 is a short way of writing (— 4) + (— 4) + (— 4) = — 12.
Hence multiplication by a positive integer means that the
multiplicand is taken as an addend as many times as there
are units in the multiplier.
Also, we shall define multiplication by a negative integer
as meaning that the multiplicand is taken as a subtrahend as
many times as there are units in the multipher.
According to this definition then
4x-3 = -4-4-4=-12,
_4x-3=-(-4)-(-4)-(-4),
=+4+4+4= + 12.
We may state these results in algebraic symbols, thus :
(+rt)x(+fe)=+<*6,
{—a)x{^b)=—ah,
{■i-a)x{ — b)=—ab,
{ — a)x{ — b)=^ab.
38. Rule of Signs for Multiplication. Examine the preceding
statements and state when the product has a j)Ositive sign,
and when it has a negative sign.
The rule of signs for multiplication may be stated in the
form :
The product of two factors with like signs is positive, and of
two factors with unlike signs is negative.
MULTIPLICATION AND DIVISION 51
EXBRCISE3 27 (Oral)
State the product of ;
1.
6, -7.
2.
-4,3.
3.
-4, -5.
4.
h -5.
5.
3a;, -2.
6.
—x, y.
7.
-X, -2y.
8.
2m, —3n.
9.
—X, -~3xy.
10.
-a(-6).
11.
(-6)2.
12.
—ay{—x).
13.
— 2nit'(— t').
14.
-\x, -12a;.
15.
—2x~x—3xK
16.
a;-, —a:.
17.
-2a2, _3a.
18.
—abx—3cd.
19.
—5xy, 2x-.
20.
— a^, —a.
21.
3a%x—ab.
22.
5x^y^, —xy^.
23.
3a263c*, -ab
2c.
24.
—x^yz^, 5x^y~z.
25.
Since —4x3 = -
-12, -
-4x2=-8,
-4X1 = -
4, what would
you expect — 4 x 0 to be equal
to ? Also -
■4X-
-1 and
. -4x-2?
39. Multiplication of several Simple Factors.
Ex. 1.— Multiply 2a, —36, —4Mb, —b.
The product of 2a and — 36 is — Gab.
„ „ ,, —6ab and — 4a6 is 24a*6^.
„ 24a262 and -6 is — 24a26^
.'. the reqmred product is — 24a^6^.
Of course the factors may be multipHed in any order %ve choose.
If we mtdtiply all the negativ^e factors first, what sign will their product
have ?
What sign will the product have if we multiply four negative factors ?
Five negative factors ? Twenty negative factors ?
The product will be negative when the number of negative
factors is odd and will be positive when the number of negative
factors is even. Any number of positive factors will evidently
not aflfect the sign of the product.
Ex. 2.— Multiply 3.t, —5xy, —6x^, —y, 6y^.
(1) TTxe sign of the product is — , since there is an odd nmnber of
negative factors.
(2) The numerical coefficient = 3x 5x 6x 6 = 540.
(3) The literal part of the product = a'. a;?/, x- .3/. 2/' = a;*i/^,
.". the complete product = —o4:0x*y^.
E 2
52 ALGEBRA
Ex. 3.— Find the values of {-2f, (-2)^, {-2)K
In ( — 2)3 the number of factors is odd,
• (_2)3=— 23=-8.
Similarly, (-2)*=16 and (-2)^= -32.
SXSRCISE 28 (1-18 Oral)
Find the product of :
1. 3, 4, -5. 2. 3, -4, -5. 3. -3, -4, -5.
4. —a, —b, —c. 5. 2a, 3a, —4a. 6. —J, —J, 12.
7. 3:r, -2a:?/, -?/. 8. -3, -3, -3. 9. (-4)3.
10. -2,-3,-4,-5. 11. (-2)2x(-3)2. 12. -2a;, -2x,-2a;.
13. —a, —2a, —3a, —4a, —5a. 14. 2a:, —3x, 4(/, —y.
16. 5xy, —3xij, —2x, —2y. 16. —1, —2, —3, —J, —J.
17. What is the square of —2a, of —Sxy, of — 4a26c ?
18. What is the cube of —5, of —x, of —2x^ ?
19.* If a:= — 1 and y=—2, find the values of :
x^, y^, x^-{-y^, x^ — y^, x^, y^, x^-\-y^, x^ — y^.
20. Find the value of 3.r-+2a:— 5 when x=—2; when a;=— 3;
when 0;:=— 4.
21. Write without the brackets :
{-a)\ {-a)\ {-2)\ (-l)^ (-1)«, (-3)*, (-l)3x(-2)^
22. Find the sum of the squares of —2, —3, —4. Find also the
square of their sum.
23. Find the value of (a— 6)2+(6— c)2+(c— o)^ when a=6, 6=4,
c=2.
24. When a=2, 6=1, c= — 3, show that a^-^b^-[-c'^='iabc.
25. When a:=3 and y= — 2, how much greater is (x—y)^ than
x^ — y^ 1
If a = — 1, 6 = — 2, c-= — 3, d=— 4, find the value of:
26. 3a + 26+c-4rf. 27. 02+62+02+^2.
28. ab+ac+bc+cd. 29. aH^—bW
30. abc+bcd^cda-\-dab, 31. flS+fes+c^+rfS.
MULTIPLICATION AND DIVISION
53
40. Compound Multiplication. We multiply a compound
quantitj' by a simjjle one in a manner similar to the method
in arithmetic.
In arithmetic.
3yd. Ift. 4in. 23 = 2. 10 + 3
2 2 2
6yd. 2ft. Sin. 46 = 4. 10 + 6
In algebra.
3a + 46- 5c
2
6a+86-10c
If we wish to indicate the product of x and y + z, we write it in the
form x{y + z), which we see is equal to xy + xz.
Tlie diagram shows how this product may be illus-
trated geometrically.
Make a similar diagram to show that
a(b + c + d) = ab-\-ac + ad.
Similarly, x{y — z) = xy — xz.
Can you see that the diagram is a geometrical
illustration of this ?
Ex.— SimpUfy 3(a-6)-4(6-c)-2(a-6+c).
The expression =(3a-36)-(46-4c)-(2a-26 + 2c),
= 3a-36-46 + 4c-2a+26-2c,
=a-56 + 2c.
y
::
.xy
xz
y-s s
.x(y.z)
xz
EXERCISE!
29
Copy and supply the
products :
1. 2a+6
2. 3a— 26
3.
2m— 5n.
4
7
6.
-6
4, 4x-3«/
5. 3a;— 4?/
2a+56— c
2x
-2y
8. 3(5a;-2y).
9.
—a
7. 2(3x--ll).
-2(3x-y).
10. 3a;(a;2+5a;-2).
11. hxy{1x'^—x\j).
12.
3mp{5—mp)
Simplify :
13.* 3(a+6)+4(6+c)+5(c+a).
14. 2(x— 2«/)+3(a;-
y)_(4.r-3y/).
15. 3(2m.— 3n)— 5(m-7!)+2(m+2H).
54
ALGEBRA
16.
17.
18.
19.
20,
21.
4(a-26+c)-3(6-2c+a)-2(5c-4a-56).
i(2a-36)+i(2a+56)+|(5a+6).
x{x-l)+2x{x-3)+3x{x+5).
a(a2_a4-l)+3(a2+a_2)-2(a2+2a-3).
3a;(a;2-2a;+2)-2a;(3x2+4a;-5)+a;(4x2+5-c-6).
— 2a(6— c+d)— 3a(c— c?+6)— a(ci— 6— c).
Solve and verify :
22. 3(a;-l)=2(a;+4). 23. 5(a;-2)-2(a;+2)=70.
24. 6(2a;-3)-3(a;-3)=0. 25. 2{5a;-9)+4(x-ll)=36.
26. 3(a;+2)+5(a;-3)==2(a;-4)+4(x-l) + 13.
27. Find the sum of x{x+l), 3a-(a;— 2), 2a:(a;— 5).
28. Subtract a{2a^-a+l) from 2a(a2+3a-2).
29. If a stands for x^-\-xy~\-y^ and b for x^—xy+y"^, find the values
of a—b, 2a+b, 3a— 2b.
41. Multiplication by a Compound Quantity.
The measures of the sides of the large rectangle are a-\-b
X y and x-\-y. The measure of the area is the
prodiict of a+6 and x^-y, which is seen to
be ax^ayAr^x^hy,
:. {a-^b)[Jc+y)=ax-\-ay-\-hx-\-hy.
This diagram shows how to find the product of
a; + 3 anda;+2. What does it show the product to
be ?
Make a similar figure which will show the product
of a + 6 and a-{-h, and thus find the value of {a-\-bY,
or the square of a + 6.
The method of obtaining the product without the diagram
is similar to that used in arithmetic.
ax
ny
bx
by
2
X
3.r
2x
6
In arithmetic.
12 =
23 =
1.10+2
2.10 + 3
12x 3= 36= 3.10 + 6
12x20 = 240 = 2. 102 + 4. 10
12x23 = 276 = 2. 10^ + 7. 10 + 6
In algebra.
x+2
2.-C + 3
3a; + 6 =
2a;2+4a; =
3(a;+2)
2a;(x + 2)
2a;2 + 7a;+6 = (2x+3)(.r+2)
MULTIPLICATION AND DIVISION
55
Thus, the product of any two expressions is obtained by
multiplying each term of the multiplicand by each term of the
multiplier. The proper signs are attached to these partial
products, and the sum of the partial products is then taken.
In multiplying in arithmetic we begin at the right, but in
algebra it is usual, but not necessary, to begin at the left.
Ex.— Multiply (1) 2ft-36 by 3a-26.
(2) 3a;— 5?/ by ^x-\-y.
Check
Check
o = 6=l
-1
\
(1)
2a - 3b
3a - 26
-1
6a2- 9ab
- 4a6 + 662
6a2-13a6 + 662
-2
5
10
(2)
3x — 5y
4:x + y
\2x^-2(ixy
+ ^xy-
■5y'
12a;*— nxy — 5y^
42. Checking Results. In Chapter II. we saw how to
verify the root which we obtained in solving an equation.
We might verify our work in subtraction by addition. As
in addition, the work in multiphcation is easily checked by
substituting particular numbers for the letters involved.
Thvis to check the work in the first example in tlie preceding article,
we might substitute 1 for each letter involved.
Whena=6=l, 2a-36 = 2-3= -1,
3a-26 = 3-2=l,
and 6a2-13a6 + 662 = 6-13 + 6= -1.
Since the product of — 1 and 1 is —1, the work is likely correct.
A convenient way of exhibiting the test is shown. Of course any
numbers might be used in cJiecking, but we naturally choose the
simplest ones.
EXERCISE 30
^ind the product of the following and check :
1. x+3 2. 2x+7 . 3. x+5
4.
3a;+4
x+4: x+l 2x+2
2a;+3
li 7 ^ 'i -f
56
6.
a— 3
a-4:
2x-3
2x+S
6.
10.
a— I
a— 5
a+3
ALGEBRA
7.
11.
14.
16.
6-
26-
-4
-3
8.
12.
-4(Z).
3a— 5
2a+5
2x-
bx-
-3a
- a
9.
x+y
x—y
56). ■
3a— 7c
3a+7c
13.
15.
{3a+Ab){2
{X-
-5y)(2x-
-36)(3c-
17. Find the square of x—y by multiplying it by x—y. What is
{x—yY equal to ?
18. Find the squares of 2a -6, 2a -36, 4a +5, 3a +46. Check by
putting a =3, 6=1.
19.* Simplify (a:+l)(.T+ 2) + (.T-2)(x+ 3). ^ cC-
20. Simplify 3(a+2)(a-2)+2(a-5)(a+l). xTj-CJl^J^lSr
21. When Sa;^— 2a;— 15 is divided by 2a;— 3 the quotient is ^x-\-57~
Prove that this is correct.
22. Show that (6a;-8)(2a;-3) = (4a;-6)(3a;-4).
23. m{x-\-y)^mx-\-my. Fmd the value of mx-\-my when m=2-14,
.i;=43-7, y=5Q-3.
24. If a train goes 2a— 36 miles per hour, how many miles will it
go in 2a +36 hours ?
25. Simplify (.r+7/)2+(a;-2/)2 ; {x+y)^-ix-y)^.
26. Simplify 2(a-6)(2a+6)-3(a+6)(a-26).
27. Subtract (a;+2)(a;-9) from (a;+3)(a;+4).
28. Multiply 3(a;+3)-2(a;+4) by 2(a;-5)-(a;-3).
29. Subtract (a;+3)(a;+7) from (a;+l)(a;+ll). For what value
of X are these quantities equal ? (Verify.)
30. Show that there is no value of x which will make (a;— 10)(.'C— 1)
equal to (a;— 3)(a;— 8).
31. Subtract the sum of (3a;+2)(2.r+3) and (3a;-2)(2.T— 3) from
the sum of (4a;+3)(3a;+4) and (4a;-3)(3a,--4).
32. Simplify (a;-3)2+(a;-2)(,r+2) + (.r+l)(a;+5).
33. Subtract the product of 2a— 5 and 3a +2 from the product
of 3a +5 and 2a— 2.
( ))
MULTIPLICATION AND DIVISION 57
34. Find the sum of the squares of x+l, x-\-2, x+3. Check by
putting, x—2.
Solve and verify :
35. {x+5){x-l)={x-5){x+2). 36. {x-l)^={x-5){x-\-2).
37. (2x--l)(3x-l) = (a;-2)(6a;+4).
38. (a;+ll)(a;-2)=(a;-7)(a;-l) + 107.
39. xix+l)-ir{x+l){x+2)=2{x+l)(x+3).
40. (x+iy-+{x+2)^+(x+3y-=3{x-2)^+U.
43. Division by a Simple Positive Quantity.
To divide 24 by 6 is the same as to find the number by
which 6 must be multiplied to produce 24.
Thus division is the inverse of multiphcation as subtraction
is the inverse of addition.
Smce axb=ah,
.'. ab'^a=b and ab^b=a.
If we wish to divide 6xy by 2x, we must find what 2x must
be multiplied by to produce Qxy.
(1) 2 must be multiplied by 3 to produce 6,
(2) X „ „ „ „ y „ „ xy.
.'. 6xy -^ 2x = 3y.
Similarly, I5a6c-:-36c = 5a.
A problem in division may be viritten in the fractional
form, the dividend being the niimerator of the fraction and
the divisor the denominator.
m, 24 . ab , Qxy „
Thus, -5- = 4, —=b, -^ = 3y.
o a 2x
As in arithmetic, we may remove or cancel from the dividend
and divisor any factor which is common to both.
Thus, — X5— = 4a, on removing; the factors 3 and b.
36 *
Similarly, —-^^ = 3ay and ~^'^^ =2p.
68
ALGEBRA
44. Index Law for Division.
Since a^xa^^a^ by the index law for multiplication,
.*, a^-^a'^=^a^ or a^~a^=a",
or
^ .^ .a .a .a
= a^ and
a^ ^ .^ .i^ .a
= a'
Thus, the index of the quotient of powers of the same quantity
is found hy subtracting the index of the divisor from the index
of the dividend.
Thus,
Similarly,
I5a363
3a26
:5a3-263-i = 5a62.
The work in division may be verified by multiplication.
Thus the preceding division is seen to be correct, since
5a&2x3a26=15a363.
EXERCISE 31
Copy and supply the quotients, verifying the results by mental
nultiplication :
5. ^rl
9.
13.
X
2a
2.
abc
Qxi/^2x.
16. 10a;5^2a;^
19. 22a*b^^lla~b'
10.
5abc
ab
Ix
18^3^2
11.
24w«
3n
2a2
4a3
4.
— . 8.
12.
25xyz
~5z"
65m%
13mn
\2x^y^
14.
17.
10a3-^5a.
16a36— 4a6.
Ixy
15. \mv'^-^\v.
\bx^yH'^^2x'^yz^.
18
45. Rule of Signs for Division.
Since (+a)x (+6) = +«('->, (- a) x (+6)=— a&, (+a)x(— 6)
-—ah, {—a)x{—b) = -\-ab, it follows that
+a6 , , — a6
■■+b,
-b, +^^=-6.
—a
When is the sign of the quotient + and when is it — ?
What then is the rule of signs for division?
Compare it with the rule of signs for multiplication (art. 38).
MULTIPLICATION AND DIVISION 59
Ex.— Divide —lOxhj^ by —2xij".
(1) Wliat is the sign of the quotient ?
(2) What is the numerical coefficient ?
(3) What is the Hteral part ?
(4) What is the complete quotient ?
EXERCISE 32 (Oral)
Perform the indicated divisions :
1.
12^-3.
4.
-7^7.
7.
0-^5.
10.
ab-. — a.
13.
I0a^^-2a
16.
-I2m^n^-
-6>nn.
2,
-12^-4.
3.
-10^2.
5.
—2a^—a.
0.
-12^2-^-3.
8.
0^-5.
9.
Ga-^-2a.
11.
axy-. — X.
12.
45^-5^-3
14.
-6a^^3a.
15.
27a;*^-3.T2.
17.
x^y^z-. — xyz.
18.
-4a5^-2a3.
20.
—4:mn^
21.
6-4.r3
-2x'
in the
following :
(2)
(3)
(4)
(5)
-10x2
— lOabc
357n^n
-5c
6a
— 5m
- 2.r
2ac
19. -pir
-pq
(1)
Dividend : 6«^
Divisor : 2a
Quotient :
46. Division of a Compound Quantity by a Simple One.
If we divide 6 ft. 4 in. by 2 we get 3 ft. 2 in., or 12 lb. 6 oz.
by 6 we get 2 lb. 1 oz.
Similarly,
3)9 ft. 6 in. 4) 16 lb. 8 oz. 2)6 tens + 8 units ^
3 ftTfmT Tib. 2 oz. " 3 tens + 4 units*
3)9/+6i _ 4)16a-|-^86 _ 2)6^ + 8
3/4- 2^' 4a+26 ■ 3<4-4 '
a)ab + ac ^ 3x)C)x^-3x^ - ab):ia^b^ — 2ab
T+ c * 2x^-^x' ' - Sab + 2~ '
Thus it is seen, that we divide a compound expression by a
simple one by dividing each term of (he dividend by the divisor,
attaching the proper sign to each term of the quotient.
60 ~ ALGEBRA
EXERCISE 33 (1-15, Oral)
Divide the first quantity by the second :
1. 9a2-f6a, 3. 2. 6x^+4:X^-2x, x. 3. 15a;2-10a;, 5a:.
4. 16m^— 4m, 4m. 5. a;22/+^2/^ ^y- 6. 12a^— 4a6, —2a.
7. — aa;+ay, — «. 8. ct^+a^— a, a. 9. 6x^—4:xy, 2x.
10. — 6a6— 6a, —3a. 11. 6a3_8a2^4a, —2a.
12. a^b^-a%'^,ab''-. 13. -5a*-10a^ -Sa^.
14. — 4a;+10x2-6x-3, — 2.r. 15. 3y^—2}j\ \y.
Simplify :
.„ 3.r-|-6 , 10a;— 15 ._ ab^ac , bc4-ab , ac-\-bc
16. — +• 17. [- ! + '
3^5 a ^ b ^ c
^g^ a^+3a 3a^-\-6a ^^ (a;+2)(.T-2) + (a;-2)(a;-4)
a 3a ' ' 2
2^^ x^+xy ^ y^-xy ^i. («+2)(a+3)-(a-3)(a-2)
X y ' 2a
_„ ab—ac , be— ah „„ a'^—a'^ . a^—a ,
^Ji. h -— ' Jd. H ]-l.
— a — 0 a a
24. Subtract (a;+3)(a;— 8) from (2x— 4)(a;-(-6) and divide the
remainder by x.
■ ^ „ 1 1 * a;2-10a; , Sx^+JSx , 10a;-15 .„
25. Solve and verify + — \- =40.
a; 3a; 5
EXERCISE 34 (Review of Chapter V)
1. State the rule of signs for multiplication and for division.
2.* If a=3 and b= —4, find the values of :
a2, 62, ab, a'^ + b% a^-b^, a^, b^, a^-b^.
3. What are the values of (-1)2, (-l)^ (-1)^", (-2)^ (-3)3 ?
4. Simplify Sa^x -^b^X — 2a64-6a6*.
5. Simplify 2a(a-[-3) + 3a(2a-6).
6. What is the area in square feet of a rectangle which is (a +6) feet
long and (a — b) yards wide ?
7. Make a diagram to show that 3a;x4x=12a;^
MULTIPLICATION AND DIVISION 61
8. A merchant bought a pieces of silk at GO cents a yard and b
pieces at 80 cents a yard. If each piece contained 50 yards, find the
total cost in dollars.
9. To the product of 3a;— 2 and 2x — 3 add the product of 3x-4-2
and 2x+3.
10. From the product of ox—Sy and 2.f + y subtract the product
of 3x—2y and 2x-3y.
11. Make a diagram to show that the product of a+3 and a+1 is
a2+4a+3.
12. Divide 4o^ — 6a 2 — 8a by —2a and verify.
13. To the square of 2)n — 3n add the square of 3m— 2n.
14. Prove that when lox^—8xy—l2y^ is divided by 5x—6y the
quotient is 3x+2y.
15. Find the product of a — b, a-\-b and a--\-b^. Check by sub-
stituting 3 for a and 2 for b.
16. SunpMy ^ +
17. Solve (2x+3)(3a;+2) = (6a;-l)(a;+3). (Verify.)
18. Simplify {2a-36)(a+6) + (a-6)(3a+6).
19. What value of x will make (a;+3)(a;+9) equal to {a;+5)(x+6) ?
Could (a; + 3)(x+9) be equal to (x + 4)(x+8) ?
20. Find the sum of (a-l)^, (a-2)2 and (a-3y.
21. Subtract the product of 2x — 3y and 3x-\-2y from the product
Zx—iy and 4a; + 3«/.
22. Simplify— 2p"+(3 + 2a)(l-a).
23. Find the value of 2x^ + 30;- 1, when x= —3; when a;= —4.
24. Find the product of x — 2, x+2 and x^ + 'i.
25. If x = a^ — 3a-Ti and y = 2a^ — a—l, find the values of 2x + 3^,
4:X — 2y, —^-^ .
26. If a; = 2a + 6 and y = a~r-2b, find in terms of a and b the values of
ax — by 4x—2y x^ — y^
^ ' Sa^' 3
27. If a;=36 — 2c and y = 2b — 3c, find the value of {2x—y){3x—2y).
28. If a;=2, y=2, z= —4, find the value of x^+y^+z^ — 3xyz.
CHAPTER VI
J i SIMPLE EQUATIONS {continued from Chapter II.)
47. Definition. An equation is the statement of the
equahty of two algebraic expressions.
Thus, 2a;+3=13 is an equation, and the solving of it
consists in finding a value of x which will make the statement
true.
The beginner should clearly see the difference between the
value of X in an expression like 2a;+3 and the value of a; in
an equation like 2.e4-3=13.
In the expression 2a:+3, x may represent any number,
and for different values of x the expression has different
values. But in the equation 2;c+3=13, x can not represent
any number we please, but some particular number, in this
case 5, which when substituted for x will make 2x-\-3 have
the value 13.
48. Identity. The statement 4(a;— 2)=4a;— 8 is an equation
according to the definition we have given.
If the first side of this equation be simplified by multi-
plication, we obtain 4a; — 8, which is identically the same as
the second side. It is at once seen that this equation is true
for aU values of x.
An equation which is true for all values of the letters
involved is called an identical equation or briefly an identity,
while an equation which is true only for certain values of the
letters involved is called a conditional equation. The usual
method, however, is to call all conditional equations simply
" equations," and all identical equations, " identities."
C2
SIMPLE EQUATIONS 63
Thus, 5a;— 2 = 3x+ 10, is an equation,
and {x+3){x — 3) = x^ — 9, is an identity.
We cannot always see mentally whether a given statement is an
equation or an identity.
Thus, (a;+2)(x+3) = (a;— l)(a;— 3) + 3(3a;+l) might appear to be an
equation, but if we simplify each side, we find that each becomes
x^-\-5x+6, and this statement is therefore an identity.
EXERCISES 35
Which of the following statements are equations and which are
identities ?
1. 8(a;+3)=4.'c+4(.r+6). ^-^-^v^^
2. 3x{x+l)=x{x+l)-\-2x{x+5)-{-10. '^
3. (a;-3)2-5=.r(a;-6)+4.
4. (2x-4)(a;-5)-H->C--2)(a;-3) = (3x--2)(a;-7)+40. ^ ^'"
5. {x-\-a)(x^-{-a^)=x^-]-ax(x+a)-\-a^.
6. {x+2){x-B)=x{x+5)-\-3{x-l).
49. Transposition of Terms. In Chapter II. the method
of solving easy equations was dealt with.
The method depended almost entirely on the proper use
of the four axioms of art. 15.
The following examples will show how the methods of
Chapter II. may be abbreviated.
Ex. 1.— Solve 7x-6=4a:+12.
Add 6 to each side, 7x==^x+12 + 6.
Subtract 4x from each, 7a;— 4a; = 12 + 6.
Collect terms on each side, 3a; =18.
Divide each side by 3, x = 6.
Here we added 6 to each side with the object of causing
the —6 to disappear from the first side of the equation, so
that we might have only unknown quantities on that side.
But the addition of 6 to the second side caused +6 to appear
on that side.
We might say then, that the —6 was transposed from
the first side and written on the other side with its sign
64 ALGEBRA
changed, and similarly, that the 4a* was transposed from the
second side to the first, with its sign changed.
We therefore have the following rule :
IV Any quantity may he transposed from one side of an eqvMion
^mthe other if the sign of the quantity he changed.
Using the rule, the solution of Ex. 1 might appear thus :
7a;-6 = 4a;+12.
Transposing terms,
7a;-4a;=12 + 6,
.-. 3x=18,
.". x = 6.
Verify this result.
Ex. 2.— Solve
2{^x-5) + ^x-b)=l{x-l).
Removing brackets.
6a;- 10 + 3a;- 15 = 7a;- 7,
Transposing terms,
6a;+3a;-7a;=10+15-7,
.-. 2a; =18,
.-. a; =9.
Verification, when a; =9
first side
= 2x22+3x4 = 56,
second side
= 7x8 =56.
Ex. 3.— Solve
3(2/-2)-5(2/-3) = 17.
Removing brackets,
3y_6-52/+15=17.
Transposing terms,
3y-5v = 6-15+17,
.-. -22/ = 8,
8
.. 2/ = — = -4.
Verification : first side
= 3(-6)-5(-7)
= -18+35=17.
Ex. 4.— Solve (2a:-l)2-(a;-3)(a:-2)=3{.'K-2)2-4.
Here the indicated multiplications are first performed.
(2a;-l)2 = 4a;2-4a;+l,
(x-3)(a;-2) = a;2 — 5a;+6,
(a;-2)2 = a;2 — 4a; + 4,
4x2-4a;+l-(x2-5a;+6) = 3(a;2-4a; + 4)-4,
4x2-4a;+l-a;2 + 5a;-6 = 3a;2-12a;+12-4,
.-. 4a;2-a;2-3x2-4a;+5a;+12x= 12-4- 1 + 6,
.-. 13a;=13,
.-. a;=l.
SIMPLE EQUATIONS 65
Here the product of a; — 3 and a; — 2 is first found and enclosed in
brackets with the minus sign preceding. In the next line the brackets
are removed and the signs changed.
In ^{x—'2)-, the x—2 must first be squared and the product multiplied
by 3.
Note. — The beginner should not attempt to perform these double
operations together.
EXSRCISB 36
Solve and verify :
1. 4a;— 4=2a;+8. 2. 3a;— 7=8-2a;.
3. 3-3a;=9— 5a;. 4. 2(a;— 5)=a;+20.
5. 5(?/-2)=3(?/+4). 6. 10(a;-3)=8(a;-2).
7. ll(4x--5) = 7(6a;-5). 8. 7a;-ll+4a;-7=3x-8.
9. 14+5a;=9a;-ll+3. 10. 3(5a;-6)-9a;=30.
11. 7(x-3)==9(a;-fl)-38. 12. 5(a;-7)+63=9.r.
13. 72(a;-5)=63(5-a;). 14. 28(a;+9)=27(46-a-).
15. 7(4.r-5)=8(3a;-5)+9. 10. 4(a;+2)=3-3(2a;-5).
17. (.-c+7)(a;-3) = (a;-l)(a;+l). 18, (a;-8)(a;+12)=(a;+l)(a;-6).
19. 20(a;-4)-12(a;-5)=a;-6. 20. 5(2a;-l)-3(4a;-6)=7.
21. (2m-5)(4m-7)=8TO^+52. 22. 5(3/i + ])-7A-3(A-7)=6.
23. (a;+5)2-(a;-f3)2=40. 24. (a;+5)2-(4-a;)2=21.a;.
25. 4(22/-7)-3(42/-8)=2^-7.
26. (a;+4)(a;-3)-(a;+2)(a;+l)=42.
27. (2a;-7)(a;-f5)=(2a;-9)(a;-4)+229.
28. (a;+l)2+(a;+2)2+{a;+3)2=3(a;+l)(.r+4)-7.
~--29. 2(a;-l)2-3(a;-2)(a;+3)=32-(a:-3)(.r-4).
30. What value of .r will make 10a;+ll equal to 5a;— 9 ?
31. Prove that 3(a;— 2)+4(3a;— 5)=5(3a;— 6)+4 is true for all
values of x.
32. What value of a will make 5(a— 3) exceed 3(a— 7) by 28 ?
33. For what value of x will the sum of \2-\-lx, 4a;+3 and 9— 5a;
be zero ?
66 ALOEBRA
34. If a; =2 is a solution of the equation
find the value of k.
35. Prove that 10 is a root of the equation
(x+3)(a;+4) + (a;+5)(a:+6)=422.
36. When (3:c+2)(4a;— 5) is subtracted from (2x+7)(6a;+3) the
remainder is 141. Find x.
37. What value of y will make («/— 3)(?/+3) exceed {y-[-^){y—l)
by 40?
38. What value of k will make (5 — 3^)(7- 2h) equal to
(11-6A')(3-A-) ?
39. What is peculiar about the equation
(a;-5)2-(a;-3)(a;-7)=0 ?
40. Under what condition is the square of x+3 equal to the product
of X — 1 and a; +6 ?
41. If 3(2a:— 1) is greater than 12(a;— 3) by the same amount that
6x is greater fhan 22, find x.
42. If i:dx—4^n'^=ax-\-2a^, what is the value oi x 1
43. The lever in the diagram is balanced by the weights P and Q,
when Pa=Qb. The point of support F is called
a F ^ the fulcrum. If P=10 lb., ^=15 lb. and a=12 in.,
[ I what is the length of 6 ?
P Q 44. Two boys balance on a teeter 16 feet in
length. The heavier boy weighs 85 lb. and the point
of support is 6 feet from his end of the teeter. Find the weight of the
other boy.
45. How far from the larger weight must the fulcrum be placed,
if weights of 8 lb. and 16 lb. balance at opposite ends of a lever 12 feet
long?
46. The formula C=f,{F— 32) is used to change Fahrenheit readings
of a thermometer to Centigrade readings. If F^lT, find the value of
C.
47. Change the following readings to Fahrenheit readings :
0°C., 40°C., 100°C., -10°C., -50°C.
48. What is the temperature when the two scales indicate equal
numbers ?
SIMPLE EQUATIONS 67
50. Equations with Fractional CoefBcients.
Ex. 1.— Solve ^x-}-^x=20.
Since i+i=4, .-. ia; = 20,
.-. a; = 20^f = 24.
Instead of adding the fractions, we might get rid of them by multiply-
ing each term of the equation by 6.
Then ^xx 6 + ^a;x 6 = 20 X 6,
.-. 3a;+2a;=120,
5a; =120,
a; = 24.
Verify by substituting in the original equation.
Ex. 2.— Solve |(a;+l) + i(x-+2)=i(a;+14).
Multiply each quantity by 12 (the L.C.M. of 2, 3, 4),
.-. h(x+ 1) X 12 + i(^+2) X 12 = i(a;+ 14) x 12,
6(a;+l) + 4(x + 2) = 3(a;+14).
Complete the solution and verify.
Ex. 3.— Solve
x—2 x—3 x—l
5 6 10
Multiply by 30, .'. ^y^ x 30 - ^^-^ x 30 = ^^^ x 30,
.-. 6(.-r-2)-5(x-3) = 3(a;-7),
.-. 6x-12-5a;+15 = 3a;-21,
6a;-5a;-3a:= 12- 15-21,
.-. -2a; =-24,
a;=12.
Verification : first side = i^ — » = 2 - 1 ^ = ^.
second side =tV =1-
Note. — In this solution the beginner is advised not to attempt to
omit the line with the brackets. He may, however, omit the preceding
line when he feels that he can safely do so.
51. Steps in the Solution of an Equation. In solving an
equation the steps in the work are :
(1) Clear the equation of fractions by multiplying each term
by the L.C.M. of the denominators of the fractions.
(2) Remove any brackets which appear.
F 2
68 ALGEBRA
(3) Transpose all the unknown quantities to one side and the
known quantities to the other.
(4) Simplify each side by collecting like terms.
(5) Divide each side by the coefficient of the unknown.
(6) Verify the result by substituting the root obtained in the
original equation.
EXERCISE
37
Solve and verify :
1. %x=x-\-5.
2.
ix=^x+2.
3. ^x—^x=10.
4.
Ja;+ix+i^=26.
5. fa;+|-a;=a;+5.
6.
1 2x
7. iy=hy+h
8.
XXX
8 + 4 + 2 = ^-4.
XXX
«• 2-5 = 4 + 1-
10.
3m Im
X X 4:X , , „
13. | + 2 = li + ^-|- 14. ix-i+7x^3x+li.
15. |-j = 2|. 16. i{.T-3)=20.
7a;+2 4a;— 1 x+l
17. ^- = -^-- 18. ^-3=^0.
x x—8 x—1 a:+3
19. 5 + --J— = 5. 20. —— + -^ = 8.
3 4 4 o
21. l{x-3)+^{x-5)=0. 22. L(x-6)=U^+5)i-l(x-lS).
Sx—l 5 a: 2a;+l „^ a;+2 , ^ .r+4 , x+6
"3" + 12 = 4 + "T"
x—3 2a;-4 3rc-5
23- -^+12 = 4 + -^- 2*- -3- + 2 = ^+ 7
^25. ^- = -5- + -8-- 26. i(2/_3)-i(2/-5)=l.
SIMPLE EQUATIONS 69
27. -±--— = 1. 28. ^--rr=o.
z—2 x+2 x—3 x+1 1_ 2a:— 1
29. —2---^= 3 • 3^-2 ~"i~'^~~"3~"
x 5.T+9 _ 2x— 9 X— 1 _ x—2 _ 3— a;
31. --— g— --— g— . 32. 6 --2- -3 ^•
33. 5(a;-2)=3-65. 34. 2-34=4(a;+l-5).
35. "Sx— 3=-25x+-2a;. 36. -2(0;— l)+-5(x-— 9)=3.
3a;— 9 x+1 3a,— 14 , x+6 3a;— 16 a;+3
Y
2—x 3—x 4— X 5—x 3
a;— 1 2— a; 2x— 1 2— 3x "
-f^' -9--^ 14- + ^0-^^-
52. Problems leading to Simple Equations. In Chapter II.
we saw how certain arithmetical problems might be solved
by means of equations. The steps in the solution of such
problems are stated in art. 19, to which the pupil should
now refer.
The beginner will find his chief difficulty with step 4,
in which he is required to translate the statements given in
ordinary language into algebraic language.
Some examples are now given to illustrate how this trans-
lation is effected.
Ex. 1. — Find three consecutive numbers whose sura is 63.
If we let x represent the smallest one, what would represent the
others ? How wo\ild you now express that the siun is 63 1
We thus obtain the equation :
a; + (a;+l) + (a;+2) = 63.
Write out the full solution of this example and verify the result.
70 ALGEBRA
Ex. 2. — A is 3 times as old as 5 ; 2 years ago A was 5
times as old as B was 4 years ago. Find their ages.
Let X years represent B's age.
What will now represent -4's age ?
What will represent ^'s age, 2 years ago ?
What will represent B's age, 4 years ago ?
Now express that 3rc — 2 is 5 times a;— 4.
The complete solution might appear thus :
Let X years = B's age,
3a; ,, =^'s age,
.'. (Sx — 2) „ = 4 's age, 2 years ago,
.". (a;— 4) ,, =B's age, 4 years ago,
.-. 3a;-2 = 5(a;-4),
.-. 3x-2 = 5a;-20,
18 = 2x,
a; = 9.
.*. jB's age is 9 years and ^'s is 27 years.
Ex. 3.— How do you represent 3% of 130 ? 4% of $27 ?
5% of %x ? 21% of $(a;+50) ?
Solve the problem : " Divide $620 into two parts so that
5% of the first part together with 6% of the other part will
make $34."
Let $a;=the first part,
$(620 — a;) = the other part,
r^TT of $a;=5% of the first part,
,-. -xU of $(620-x) = 6% of the other part,
••• Tl«^+TfT(620-.T) = 34,
5.T + 6(620 -a;) = 3400.
Complete the solution and verify the result.
Ex. 4.— What is the excess of 73 over 50 ? What is the
defect of 30 from 50 ? What is the excess of x over 50 ?
The defect of x from 89 ?
Solve the problem : " The excess of a number over 50
is 11 greater than its defect from 89. Find the number."
SIMPLE EQUATIONS 71
Let x = the number,
then a; — 50 = its excess over 50, •
and 89 — x = its defect from 89
.-. a;-50 = 89-a;+ll.
Complete the solution and verify.
Ex. 5. — The value of 73 coins consisting of 10c. pieces
and 5c. pieces is $5. How many are there of each ?
Let .r = the number of 10c. pieces,
.". 73 — x= ,, „ ,, 5c. ,,
The value of the 10c. pieces = lOx cents.
The value of the oc. pieces = 5( 73 — x) cents,
.-. 10x+5(73-a;) = 500.
Complete the solution and verify.
The pupil should be careful to express each term of the
equation in the same denomination.
Why would it be incorrect to say that
10a:+5(73-x)-5 ?
SXBRCISE 38
All results should be verified.
1. A number is multiplied by 23 and 117 is then added. The
result is 232. Find the number.
2. From the double of a number 7 is taken. The remainder is
95. Find the number.
3. Three times a number is subtracted from 235 and the result
is 217. Find the number.
4. Five times a number with 33 added is equal to 7 times the
number with 18 added. Find the number.
5. Find a number such that the sum of its third and fourth parts
may be 35.
6. A has SIO more than 3 times as much as B, and they together
have §250. How much has each ?
7. The sum of two numbers is 81. The greater exceeds 6 times
the less by 4. Find the numbers.
8. Find a number whose seventh part exceeds its eighth part
by 2.
72 ALGEBRA
9. The excess of a number over 42 is the same as its defect from
59. Find the number.
10. Find 3 consecutive numbers whose sum is 129.
11. Divide 114 into three parts so that the first exceeds the second
by 15 and the third exceeds the first by 21.
12. Divide $176 among A, B and C so that B may have $16 less
than A and $8 more than C.
13. A man sold a lot for $2280 and gamed 14% of the cost. What
did the lot cost ?
14. Divide 420 into 3 parts so that the second is double the first
and the third is the sum of the other two.
15. A man buys 8 horses at $x each, 5 at ${x-\-5) each and 3 at
${x+25) each. The total cost is $2020. Fmd x.
16. Find a number which exceeds 31 by the same amount that
J of the number exceeds 1.
17. Fmd a number which when multiplied by 6 exceeds 35 by as
much as 35 exceeds the number.
18. A farmer sells 7 cows and 17 pigs for $754. Each cow sells
for $70 more than each pig. What is the price of each cow ?
19. If 10 be subtracted from a number, 40 more than J the remainder
is 30 less than the number. Find the number.
20. Find two consecutive numbers such that the sum of J of the
less and J of the greater is 44.
21. Divide 46 into two parts so that if the greater part is divided
by 7 and the other by 3, the sum of the quotients is 10.
22. Divide 237 into two parts so that one part may be contained
in the other IJ times.
23. A horse was sold for $116-25 at a loss of 7%. What did he
cost?
24. The difference between the squares of two consecutive numbers
is 17. Find the numbers.
\^ 25i* A box contains two equal sums of money, one in half-dollars
and the other in quarters. If the number of coins is 30, how much
money is in the box ?
r 26. A is 35 years old ; B is 1 years old. In how many years wiU
A he twice as old as £ ?
-t
SIMPLE EQUATIONS 73
'Ti^ 27. My age in 20 years will be double what it was 10 years ago.
What is my age ?
S<^ 28. J. is 35, £ is 7 and C is 5 years old. How long v.'ill it be before
A^s age is the sum of the ages of B and C ?
29. Find three consecutive even numbers such that the sum of
a fourth of the first, a half of the second and a fifth of the third is 17.
L. 30. A's share of S705 is i of iS's and B's is f of C's. What is the
share of each ?
31. The simple interest on a sum at 2% together with the interest
^^ on a sum twice as large at 3|% is $135 per annum. What are the
sums ?
32. Three % of a certain sum together with 4% of a sum which
is 850 greater is $12'50. Find the sums.
^33. The value of 52 coins made up of quarters and ten-cent pieces
is SIO. How many are there of each ?
34. A square floor has a margin 2 feet wide aU around a square
carpet. The area of the margin is 160 sq. ft. Find the dimensions
of the room.
35. In an}' triangle the sum of the angles is 180°. The greatest
angle is 35° larger than the smallest angle and 10° larger than the other
angle. Find the angles.
36. The length of a room exceeds the width by 4 feet. If each
dimension be increased by 2 feet the area will be increased by 52 sq. ft.
Find the length.
37. If I walk m mUes at 4 miles per hom* and 7W+2 miles at 3 miles
per hour, the whole journe}- will take 15 minutes longer than if I walked
at the imiform rate of 3| miles per hour. Fmd the length of the journey.
38. A and B together have $65, B and C have $100, C and .1 have
$95. How much has each ?
39. State problems which will give rise to the following equations :
(1) 5a;-10=60. (2) 4a;-a;=24.
(3) ? + |=a;-10. (4) 23-5a;=4a:-4
40. A fruit dealer buys apples at the rate of 5 for 3 cents and sells
them at the rate of 3 for 5 cents. How many must he sell to gain
SI -28 ?
74 ALGEBRA
41. The sum of two numbers is 147 and J of the less is 9 greater
than \ of the other. Find the numbers.
42. John has i as much money as his brother, but when each has
spent 25 cents, John has only f as much as his brother. How much
has each ?
53. Algebraic Statements of Arithmetical Theorems. If we
take any two numbers, say 23 and 13, and add together
their sum and their difference, we will find the result is twice
the larger number.
Thus, 23+13=36 and 23-13=10,
and 36+10=46, which is twice 23.
We see that it is true for the numbers 23 and 13, and we
would find it true for other pairs of numbers, but we are not
sure it is true for all pairs of numbers.
By the use of algebraic symbols and methods, we may show
that the statement is true for every two numbers.
Let the larger number be a and the smaller b.
Their sum is a + & and their difference is a — b.
But (rt+6) + (a-6) = a+6 + a-6 = 2a,
and 2a is twice the larger number.
Thus the statement (a+6)+(a— 6)=2a represents in a
brief form the theorem stated at the beginning of this article.
Besides stating it in a concise form it shows that it is true
generally.
SXBRCISS 89
Show that the following statements are true for all numbers :
1. The sum of two numbers is equal to their difference increased
by twice the smaller number.
2. The difference between the sum of two numbers and the
difference of the same two numbers is twice the smaller number.
3. Half of the sum of two numbers increased by half of their
difference is equal to the larger number.
4. The sum of two numbers multiplied by one of them is equal
to the square of that one, plus their product.
SIMPLE EQUATIONS 75
5. The square of the sum of two numbers is equal to the square
of their difference increased by four times their product.
6. The sum of three consecutive numbers is equal to three times
the middle one.
7. If two integers differ by 2, twice the square of the integer
between them is less by 2 than the sum of the squares of the two
integers.
8. Read the statement {a+b)^+{a—b)^=2{a^+b^) without using
symbols and prove that it is true.
EXERCISE 40 (Review of Chapter VI)
1. What is an equation ? An identity ?
2. Wliat rule is followed in transposing terms ?
3. Solve and verify: 6x(2x+3) = (3x+2)(4:X+^).
4. Is — ^ — = — -. — an equation or an identity ?
5. What value of x will make 5{x — 3) — 4(x ~ 2) equal to zero?
_, , x—7 , a;— 10 a;— 11 , ^
6. Solve — g- H ^ = — g h 2.
7. The sum of two numbers is 50. If 5 times the less exceeds 3
times the greater by 10, what are the numbers ?
-J. _L 3 5^; -1-6 X 2
8. Show that x — I -\ ^ = — ^ 1- —»- is true for all values
of X.
9. What value of x will make the product of 5 — 3x and 7 — 2a;
equal to the product of 11 — 6a; and 3 — a; ?
2x 3 3a; 4
10. If = 1- -262, find x correct to two decimal places.
2-5 12-5
11. A and B invested equal siims. A gained S200 and B gained
$2600. If B then had 3 times as much as A, how much did each
invest ?
12. From a cask which is |ths full, 36 gallons are drawn and it is
then half full. How much will the cask hold ?
13. Show that a; =6 is a root of
(a;-l)(a;-2)(a;-3) = 2a;(a;-5)(2a;-7).
14. A man has $115 in $2 bills and $5 bills. If he has 35 bills
altogether, how many of each has he ?
76 ALGEBRA
^K Tf 3a;— 20a 5a;— 6a , c j
15. If ^ H r — = 31 and a = ^, find x.
16. In a stairway there are 45 steps of equal heights. If they had
been one inch higher, there would have been only 40 steps. How high
is each step ?
in c 1 ^—4 a; — 5 x — 2
17. Solve— --g- = — .
18. Divide 150 into two parts such that if the smaller be divided
by 23 and the other by 27 the sum of the quotients will be 6.
19. The difference between the squares of two consecutive numbers
is 51. Find the numbers.
20. A father is 30 years older than his son ; five years ago he was
four times as old. Find the son's present age.
2a; 4-3 a; 4-5
21. If the sum of the fractions — — and — — is 9, what is the
numerical value of each fraction ?
22. Show that the difference between the squares of any two
consecutive nmnbers is equal to the sum of the numbers. Show also
that the sum of their squares is one more than twice their product.
23. Solve 2-(a;-4 + 3a;-5) = 10-a;.
24. If the product of a; + 2 and 2a; +5 is greater than the product
of 2a;-f 1 and a: + 3 by 127, find x.
25. Solve |(2-3a;)-|(a;-4)==|-(a;-5).
26. Divide -75 into two parts so that three times the greater
exceeds six times the less by -75.
„„ o , a;-3 , 2 + a; l-2a; ^
27. Solve —;, h ~ r?- = <>•
o 3 15
28. A man walked a certain distance at 3 miles per hour and returned
by train at 33 miles per hour. His whole time was 4 hours. How far
did he walk ?
29. Prove the accuracy of the following statement : " Take any
number, double it, add 12, halve the result, subtract the original
number, and 6 will remain."
30. Solve I' + ^^ - ^^ = a; - 8.
31. How many minutes is it to 10 o'clock if three-quarters of an
hour ago it was twice as many minutes past 8 ?
32. What value of a will make 2(6a;-fa)-3(2a;+a) = 4(Hx-6)
an identity ?
SIMPLE EQUATIONS ' 77
33. Solve (6a;-2)(2a;-l)-(4x-2)(3x-2) = 4.
34. A rectangular grass-plot has its length 5 yards longer than its
width. A second plot, of equal area, is 5 yards longer and 3 yards
narrower than the first. Find the dimensions of the first.
35. Solve (a;+l)(a;+2) + (a;+3)(a;+4) = 2a;(a;+ 12).
36. A man leaves his property amounting to $7500 to be divided
among his wife, two sons and three daughters. A son is to have twice
as much as a daughter, and the wife §500 more than all the children
together. Find the share of each.
37. Solve— +-g ^ = 0.
38. Find an integer whose square is less than the square of the next
higher integer by 37.
39. if — - — exceeds — : — bv — -— , find x.
3 4 -^ 6
40. How far can I walk at 3 miles per hour and return on a bicycle
at 10 mUes per hour and be absent 6 hours 4 minutes ?
41. A man invested J of his money at 3%, j at 4%, i at 5% and the
remainder at 6%. If he receives an annual income of §516, how much
did he invest ?
42. Prove that the product obtained by multiplying the sum of
any two numbers by their difference is equal to the difference of their
squares.
CHAPTER VTI
SIMULTANEOUS EQUATIONS
54. Equations with two Unknowns.
The sum of two numbers is 10. What are the numbers ?
It is evident that there are many different answers to this
problem. The numbers might be 1 and 9, 2 and 8, 3 and 7,
etc., or J and 9|, —3 and 13, etc.
If we are also given that the difference of the numbers
is 4, then only one of these answers will satisfy this new
condition. The numbers would evidently be 7 and 3.
If we follow the method previously adopted and represent
the required numbers by x and y, where x is the greater, the
first condition would be expressed by the equation
x-\-y=10.
As stated, any number of pairs of values of x and y will
satisfy this equation.
If the second condition be expressed in terms of the same
unknowns, we have another equation
x—y=4t.
It is now required to find a pair of values of x and y which
will satisfy
x-\-y=lO,
and x—y= 4.
If we add the corresponding sides of the equations we get :
2a'=14, .•. a;=7 and /. y—3,
.'. 7 and 3 are the required numbers.
SIMULTANEOUS EQUATIONS 79
55. Simultaneous Equations. Any equations which are satis-
fied by the same vakies of the unknowns are called
simultaneous equations.
Thus, x=^l, y=3 satisf}- both of the equations
x-\-y^lO and x—y=4.
To find a definite pair of values of x and y it is seen that
we must have two equations contaming these letters. To solve
any problem \Ahere two numbers are to be found we must
have two conditions given, from which the required equations
may be obtamed.
Ex. 1. — If 5 men and 4 boys earn $43 in a day, and 3 men
and 4 boys earn $29 in a day, what sum does each earn in
a day ?
Why do the first set of workers earn more than the second ? How
much more do they earn ? How much then does one man earn ?
How can we now find how much a boy earns ?
We might solve this problem algebraically, thus :
Let $a;=the wages of a man for a day,
and Sy = the wages of a boy for a day.
The conditions of the problem would now be expressed algebraically
by the equations :
$5x+ Uy= $43,
$3a;+ Uy= $29.
Or, omitting the $ sign and using only the numbers,
5a;+4i/ = 43,
3a; +42/ = 29.
Subtract the terms of the second equation from the corresponding
terms of the first,
.-. 2a;=14,
.-. x= 7.
Substitute a; =7 in the first equation and
35 + 41/ = 43,
.-. 41/= 8,
.-. y= 2.
.". the roots of the equations are x = l, y = 2,
.'. a man earns $7 and a boy $2 per day.
Verify by showing that these results satisfy the conditions of the
given problem.
80 ALGEBRA
Ex. 2.— For 3 lb. of tea and 2 lb. of sugar I pay $1-30,
and for 5 lb. of tea and 4 lb. of sugar I pay $2-20. What is
the price of one pound of each ?
How does this problem differ from the preceding ?
What change might we make in the first statement so that the
number of pomids of sugar would be the same as in the second
statement ?
Let X cents = the price of a lb. of tea,
and y cents = the price of a lb. of sugar.
Then Zx + 2y=\Z0, (1)
and 5a;+42/ = 220. (2)
Multiply the first equation by 2 and we get
6a;+42/=260, (3)
5a;+4i/ = 220. (2)
Now solve (2) and (3) as in the preceding example and verify the
results you get.
Ex. 3.— Solve 3a;+42/=39, (1)
4a;+3?/=38. (2)
Multiply (1) by 4 and (2) by 3 and we get
12a;+162/=156,
12a;+ 9?/= 114.
Complete the solution and verify.
Ex. 4.— Solve 5:r-22/=44, (1)
3a-+42/=42. (2)
Multiply (1) by 2, 10x-42/ = 88. (3)
To get rid of the term containing y, we must now add instead of
subtract. When we do so
13x=130
.-. x= 10,
Substitute a;=10 in (1) and
2/= 3.
56. Elimination. In all of the preceding examples the
object has been to get rid of one of the unknowns, so
that we might have an equation with only one unknown.
The process by which this is done is called elimination.
SIMULTAI^EOUS EQUATIONS
81
Thus in Ex. 4 we eliminated the y. We might have eUminated the
X equally well.
Solve Ex. 4 by first eliminating the x.
After performing the necessary multiplications, when do we add
and when do we subtract to eliminate the unknown ?
EXERCISE 41
Solve for x and y and verify 1-21 :
1. x+2y=8,
x+ y=.5.
4. 2x+3y=25,
2x—3y= 7.
7. 3x+5y=lS,
2x+3y=l2.
10. x-{-y=4:,
x—y=3.
13. 3x-4y=16,
7a; +3?/ =62.
16. 3x=2y+ 7,
2x=3y-12.
19. 2x+13y= 275,
14x-17?/-1385.
2. 3x+5y=l3,
3x+2y= 7.
5. 5x—2ij=18,
2x— y=^ 7.
8. 5x—&y^3\,
Qx-3y=33.
11. 3.T+ 42/= 5,
6a:+12?/=13.
14. 2x+5?/= 0,
3x-^y=23.
17. a;=3?/+20,
i/=2.r-20.
3. 6.r+5?/=23,
3x+2?/=lL
6. 5a;+2?/=24,
2x+3y=U.
9. 3x— 22/=24,
2.r-3«/=ll.
12. 3x+2?/=24,
-2.T+3i/=10.
15. 2y-3a:=-22,
2x+3i/=32.
18. 3x=2y,
2x-5?/=-33.
20. 2a;+3j/= 5a:-?/=17.
21. 4a;— 52/=10y— 14a;= — 10.
22.* If bx—y=S and 5^/— a;=20, find the values of x^-y and a;— i/
23. If 2a;-52/-31=62/-9a;+57=0, find the value of \^x-\-\3y.
24. Solve a;+3=4-2j/, 7(a;-l) + ll?/=6.
25. If ax+by equals 39 when a is 3 and b is 4, and equals 13 when
a is 5 and 6 is —2, find x and y.
26. What values of x and y will make 16x— // and 4:X-\-2y each
equal to 6 ?
27. Solve 2(x-y)+3(x+y)=3l, 3(2x-y)+5{x-2y)=53.
82 ALGEBRA
57. Fractional Equations in two Unknowns. If the equations
contain fractional coefficients of x or y. the fractions may be
removed by multiphcation.
Ex.— Solve h^-{-iy= 8, (1)
ia;+|y=32. (2)
Multiply (1) by 6, 3x+ 2y= 48.
Multiply (2) by 4, a;+ 10?/= 128.
Complete the solution and verify.
BXERCISS 42
Solve and verify 1-20 :
1.
ix+ly=3,
2,
, h^ + y= 6,
3.
i{x+y)=9.
x+ 2/=7.
. + 1=14.
l(a;-2/)=4.
4.
^ + ^ = 14,
5.
ia;+3y=2,
6.
- + ^=15,
8^3
? + ^ = 24.
9^2
a;+4y==0.
^ 2/_ 4
4 5
7.
^^-iy= i»
8.
^ + ^ = 41,
3^8
9.
^4-9v= 91.
9 "^
fx+f2/=26.
3x—4:y= 0,
Qx + ^^ 167.
10.
l6 + 24 ^ ''
11.
.=ii/.
12.
ia;+-li/=6,
^ 2/ _i
4 12
9y-nx=S0.
2/-i(.-2/)=7.
13. •3x+-5^=-23, 14. •lx+ 3?/= 2-6, 15. ■05x+-03y=29.
6x-+ 5^=2-6. a;-l-6y=10-2. -OSx-'Oiy^ 0.
16. a: = ^:il = ^=^. 17. ?_^ = ^-^ = 3. 18. ^=.^^6-y
3 2 5376 25 ^
19. a; + | = 2/ + | = 7. 20. ^ _ | = 3x + 7^ + 26 - 6.
o o o 4
SIMULTANEOVJ EQUATIONS 83
21.* ^ + ^=2^, 22. x+y-^=y+^{x+y),
^ + ^= 4^. 2y-x+l=i{2x+y+3).
23. x+y=y-2, 24. 5{x+y)-l{x-y)=26,
y+lx=x+6. (3x+7^)-^4-(6a:-2/)H-3.
^„ cc+1 3y— 5 a;— «
25. 8a;-72/= 12, 26. ^ = -^ - -^^ •
x—2y 2x—y _ ^
4 "^ 3
27. ly-\x+24.=fy+\x+ll=0.
SXERCISE 48
Solve, by using two unknowns, and verify :
1. The sum of two numbers is 40 and their difference is 12. Find
the numbers.
' 2. The sum of two numbers is 19. The sum of 3 times the first
and 4 times the second is 64, Find the numbers.
3. If 4 lb. of tea and 7 lb. of sugar cost $2-42, and 5 lb. of tea
and 3 lb. of sugar cost $2*68, find the cost of each per lb.
4. Find two numbers such that 7 times the first is greater than
twice the second by 23, and 5 times the first and 3 times the second
make 136.
^5. If 5 horses and 6 cows cost $840, and 3 horses and 2 cows cost
$440, find the cost of a horse.
6. If either 9 tables and 7 chairs, or 10 tables and 2 chairs, can be
bought for 8156, what is the cost of each ?
7. If 3 men and 4 women earn $164 in 4 days and 5 men and 2
women earn $135 in 3 days, find the daUy wages of a man and of a
woman.
8. Find two numbers such that \ of the first and J of the second
is 26, and ^ of the first and | of the second is 8.
^9. Three bushels of wheat cost 20 cents more than 5 bushels of
corn, and 2 bushels of wheat and 1 bushel of corn cost $2*30. What is
the price of each per bushel ?
G 2
84 ALGEBRA
10. In 10 years a man will be twice as old as his son, but 8 years
ago the man was 8 times as old as his son. Find their present ages.
" 11. If the sum of two numbers be added to 3 times their difference
the result is 18 ; if twice the sum be added to their difference the result
is 26. Find them.
12. A merchant seUs 33 suits, some at $35 each and the others
at $25, and receives $945. How many did he sell at each price 2
13. Find two numbers such that 5% of the first is greater than 6%
of the second by 3, and 7% of the second is greater than 4% of the first
by 7-5.
14. If 3 algebras and 4 arithmetics cost $2*95, and 2 algebras and
3 arithmetics cost $2*10, find the cost of 6 algebras and 2 arithmetics.
■~ 15. A bull's eye counts 5 and an inner 4. In 10 shots a marks-
man scores 46 points, each shot being either a bull's eye or an inner.
How many of each kind did he make i
--.16. A classroom has 25 seats, some double and some single. If
there is seating accommodation for 42 pupils, how many double Seats
are there ?
. 17. A man bought 8 cows and 50 sheep for $900. He sold the cows
at a gain of 20% and the sheep at a gain of 10%, and received in all
Find the cost of a cow ?
18. If 10 men and 8 boys receive $37, and 4 men receive $1 more
than 6 boys, how much does each boy receive ?
19. A man bought 20 bushels of wheat and 15 bushels of corn for
$36 and 15 bushels of wheat and 25 bushels of corn, at the same rat«,
for $32-50. How much did he pay per bushel for each ?
20. Find two numbers such that, if the first be increased by 8 it
wiU be twice the second, and if the second be increased by 31 it will
be three times the first-
21. A farmer bought 100 acres of land for $4220, part at $37 and
the rest at $45 per acre. How many acres were there of each kind ?
22. Find two numbers such that 7 times the greater and 5 times
the less together make 332, and 51 times their difference is 408.
23. The quotient is 20 when the sum of two numbers is divided
by 3, and the quotient is 7 when their difference is divided by 2. Find
the numbers.
SIMULTANEOUS EQUATIONS 85
/^ 24. A grocer bought tea at 60c. a lb. and coffee at 30c., the total
cost being $96. He sold the tea at 75c. a lb. and the coffee at 35c.,
and gained $21. How many lb. of each did he buy ?
25. Three times the greater of two numbers exceeds twice the less
by 90, and twice the greater together with three times the less is 255.
Find the numbers.
26. The sum of two fractions whose denominators are 2 and 5
respectively is 2-9. If the numerators be interchanged the sum would
be 4"1. Find the fractions? " "^'-
27. Divide 142 into two parts so that when the larger part is divided
by 17 and the other by 19 the sum of the quotients will be 8.
28. A farm was rented for $650, part of it at $6 and the rest at $8
per acre. If the rates had been interchanged the rental would have
been $750. How many acres were in the farm ?
29. A's age 3 years ago was half of 5's present age. In 7 years
the sum of their ages will be 77 years. Find their present ages.
30. A man travelled 240 miles in 4 days, diminishing his rate each
day by the same distance. The first two days he went 136 miles.
How far did he go each day ?
EXERCISE 44 (Review of Chapter VII)
1. Solve 2x+3y = 38, 3x+2.y = 37.
2. I fire 20 shots at a target. If a hit counts 5 and a miss counts
— 2, how many hits did I make if my net score is 51 ?
3. Solve 7a: -2?/ =13, 2a; +3y = 43.
4. The average marks of those who passed an examination was
65, and of those who failed was 25. If there were 1000 candidates
in all and their average was 53, how many passed ?
5. Solve 2(a;-2/) = 3(a;-4?/), U{x+y) = ll(x+S).
6. At an election ^'s majority was 384, which was f", of the whole
number of votes. How many votes did A receive ?
7. Solve i(x+5)-5 = J(2/ + 2), ^{y + 8)-S=l{x-S).
8. Divide S5600 into two parts, so that the income from one part
at 3% may be equal to the income on the other part at 4%.
9. Solve I + I = 3x - 72/ - 37 = 0.
86 ALGEBRA
10. Two numbers differ by 11, and J of the larger is 1 more than
I of the smaller. Find the numbers.
11. If px + qy is 74 when p = 5 and q = 3, and is 76 when p = 6 and
q = 2, find x and 2/.
12. If 3% of ^'s salary plus 4% of B's salary is $93, and 5% of ^'s
plus 3% of i?'s is $111, find their salaries.
13. Solve 2l2/ + 20x=165, 77?/ -30a; = 295.
14. Divide 100 into two parts so that | of the greater part exceeds
^ of the less by 2.
15. Solve 5x — 22/=7cc+22/=a;+2/+ 11.
16. If 3 men and 4 boys earn $26, and 5 men and 2 boys earn $34,
what would 7 men and 3 boys earn ?
17. Solve i(x+l)-i(2/+2) = 3, ^x + 2) + i{y + 3) = 4.
18. If 3x~4:=ax-\-b when x=2 and when x=5, show that a=3 and
6=-4.
19. I bought a horse and carriage for $400. I sold the horse at
a profit of 20% and the carriage at a loss of 4%, and on the whole
transaction I gained 5%. What did each cost ?
, 20. Solve ~~2y = 2x-^-^ = 7.
21. A man pays a debt of $52 in $5 bills and $1 bills. If the
'^'number of bills is 24, how many are there of each ?
22. Solve 19a;- 21?/ = 100, 21a;- 19?/= 140.
23. ^'s wages are half as high again as B's, but A spends twice as
much as B. If A saves $5 and B $10 per week, what are the wages
of each per week ?
24. If 23a;+l% = 91, and y is 50% more than x, find x and y.
' 25. When a man was married his age was A more than his wife's
age. His age 8 years afterwards was } more than his wife's age. How
old was he when he was married ?
26. If 3(5a;-22/) = 2(3a;+62/), find a; in terms of y.
27. A man has two farms rented at $5 per acre and the total rent
is $1100. When the rent of the first is reduced 20% and the second
is increased 20%, the total rent is $1120. How many acres are there
in each ?
28,
SIMULTANEOUS EQUATIONS . 87
If ^ + ^ = ? + i^ = 9, find the value of t + '{ ■
3 4 6 Id I z
29, Seven years ago B was three times as old as A, but in 5 years
he will be only twice as old. What are their present ages ?
31. Solve ?^ = 5 + ^-±i, €±? ^ 3 + 2±-^
CHAPTER VIII
TYPE PRODUCTS AND SIMPLE FACTORING
58. Factor. When a quantity is the product of two or
more quantities, each of the latter is called a factor of the
given quantity.
Thus, the factors of 36c are 3, b and c.
The product oi b-\-c and a is ah + ac,
.'. the factors of ab-\-ac are a and fe + c,
or ab-\-ac = a{b-\-c).
Similarly, ab — ac = a{b — c).
When a:-(-?/+2 is multiplied by a, the product ax-f-ay-\-az
contains the factor a in each term.
If we wish to factor ax-\-ay^az, we recog-
vm,Q that since a is a factor of each term, it ")"^+"y+°^
must be a factor of the whole expression.
The remaining factor is the quotient found by dividing the
expression by a.
Then ax+ay+az=^a{dc-\-y-\-z).
This is seen to be similar to the method in arithmetic.
If we wish to factor 485, we see that 5 is a factor. How do
we obtain the other factor ?
Ex.— Factor 4a2_6a6.
Here we see that 2 and a are factors of each term and therefore 2a
is a factor. On division the other factor is 2a — 36.
.-. 4o2-6a6 = 2a(2a-36).
Similarly, 36a;4-6ca;=3a;( ).
ab—a^—a^— a{ ).
The result of the factoring may be verified by multiplication
and this may usually be done mentally.
TYPE PRODUCTS AND SIMPLE FACTORING 89
EXERCISE 45
Fill in the blanks in the following :
1. 4x+6=2{ ). 2. 3a-9=3( ).
4. ax+^x=^x{ ). 5. bx—bi/=b( ).
7. Tp^-6p=p{ ). 8. 6//2+3«/=3i/(
Factor the following and verify :
10. 2y+4. 11. 6«t-12.
13. ab-i-ac. 14. am—bm.
16. mx-^-my—mz. 17. x'^—lx.
19. 4a;3+6x-2+2a;. 20. a^^+a^y-aS.
22. 2aa:— 4a</+6az. 23. a:2— 3a:2;/+x?/2.
25. {x+y)a+{x+y)b. 26. a-(a-6)+2/(a-6).
27. 2a;(6— c)— 2(6— c).
59. Definition. An algebraic expression containing only
one term is called a monomial, one of two terms is called a
binomial, one of three terms a trinomial, and one of more
than three terms a multinomial or polynomial.
Thus, 2x — 5 is a binomial and o.^+3ct + 7 is a trinomial.
60. Product of two Binomials. The pupil should be able
to write down mentally the product of two simple
binomials hke x+2 and a;4-3. x +3
3.
5.r-10?/-5( ).
6.
a;2-j-.r=a;( ).
9.
8a;3-2.r2=2.r2( )
12.
3.r--15.
15.
ab-\-ac-{-a.
18.
5a2+10a6.
2J.
\5x^~\0xy.
24.
4a6+6a262_8ff6c
What is the source of the first term {x^) in the product ?
What is the source of the last term (6) ? What two ^^ + 2a;
quantities were added to give the middle term (5a;) ? ' "*"
How were these two quantities obtaiiaed ? x^ 4- 5x4-6
In the product of a;+ 1 and x+1, what would be the
first term, the last term, the middle term ? What is the complete
product ?
In the product of x—2 and x— 3, what is the first term, last term,
middle term ? How does the product differ from the product of x+2
and a;+3 ?
Ex. — Multiply x—5 by .r+3.
Why is the last term negative ? The middle term is the sum of
+ 3x and — 5a; or — 2a;. What is the complete product ?
What is the middle term in the product of a; +5 and x— 3 ?
90
ALGEBRA
The middle term in every case is seen to be the
sum of the two cross products, each taken wdth
the proper sign.
EXERCISE] 46 (1-22, Oral)
State the products of :
1. x+1 2. x+ 5 3. x—3 4.
x+2 x+n jc— 4
x + 5
13.
y—6
y+5
x+4:y
x—3y
ab-1
ab—3
a;+ll
6.
m—2
m-\-4i
10,
y-5x
y-\-5x
14.
xy—1
xy+7
a+ b
a+2b
11.
P~ 6g
p+Uq
15.
pq-r
pq+r
12.
X— 5
a;-12
x—Zy
x-2y
a— 2
a—h
16. ax—2by
ax—3bu
17. (a+2)(a+l). 18. {x-y)(x-4.y).
20. {x—Zy){x+2y). 21. (?H + 4?j.)(m— 5/t)
Remove the brackets, simplify and check :
23. 3(x+2)+2(3.r-l)-(x-3).
{x+\)(x+2) + {x+2){x+Z).
(2/+3)(2/-2) + (.v-5)(i/+4).
(x+\)^+{x-\){x+\)+{x+\)(x-2).
2(TO+l)('m+2) + 3(TO— l)(m-2).
4(a;+3)(a;+l)-(x-+l)(a:+12).
19. ip+q){p-q).
22. (b-3){b-^).
24.
25.
26.
27.
28.
61. Factors of Trinomials. The product of two binomials,
like those in the preceding exercise, is seen to be a trinomial.
To find the factors of a trinomial we must reverse the
process of muItipHcation.
Ex. 1.— Factor a;2+6.r+8.
Since the last term is positive, the last terms in the factors must have
like signs, and since the middle term is positive, the signs must both
be plus.
.•. the factors are of the form {x-\- )(x+ ).
TYPE PRODUCTS AND SIMPLE FACTORING 91
The last terms in the factors must be factors of 8, so they must be
1 and 8 or 2 and 4.
x+l x+2
X+8 X+4:
Which of these when multiplied will give the proper middle term ?
What are the factors of a;^ — 6a; + 8 ?
The factors of a;^ — 9x+14 must be of the form {x— ){x~ ).
What are the factors ?
Ex. 2.— Factor a;2_2a;— 15.
Here the factors must be of the form (x— )(x-\- ), since —15
must be the product of two numbers differing in sign. The possible
combinations are :
a;— 15 a;+15 a;— 5 a; + 5
a;+ 1 a;— 1 a; + 3 a; 3
Which of these sets of factors is tlie correct one ?
In factoring a trinomial like a;^ — 8a;+15, we require two factors
of 15 whose algebraic smn is —8. They are evidently —5 and —3.
.-. a;2-8x+15 = (a;-5)(.r-3).
In factoring a;^ — 4.r— 21, we require two factors of —21 whose
algebraic sum is — 4, and they are evidently — 7 and 3.
.-. a;2-4a;-21 = (a;-7){a;+3).
The pupil is advised to write the factors under each other,
below the expression he is attempting to factor.
Thus,
a;* — 6a;— 16
X +2
.-. a;*-6a;-16 = (a;-
8)(x+2)
]
Factor :
iJXERCI
1. a;2+8a;+7.
2. i
4. a^+22a+2l.
5. .
7. a^+3ab+2b^.
8. 1
10. x2-5.r+6.
11, c
13. x^-4xy+3y^.
14.
a;^+ \\xy — ^2y^
X -\- 14i/
X — 3?/
.-. a;2+llxz/-422/«=(a;+14i/)(x-3y).
6.r+5. 3. 2/2+8?/+ 15.
8a;+12. 6. h^+\Qh-^24t.
w2+77nn+10?i2. 9. i/2 + 40.r?/+39a;2.
a;2-7a;+6. 12. a;2-12.c+ll.
o2_lla6+2862. 15. m^-lmn+\2n\
92 ALGEBRA
16. x^-x-2Q. 17. ir-y-ZQ. 18. a^+a-SO.
19. a;2— 5a;— 14. 20. m^—Gm— 40. 21. x^— lOx— 24.
22. a262+8a6 + 15. 23. .tV-11^»/+30. 24. a;«-10a;2+9.
25. a2_f_6a+9. 26. a;2-14;i-+49. 27. ?/«- 122/2+36.
Use factoring to simplify the following :
^ a2+5a+4 a24,4a_5 ^^ m^— 5m+6 _ w^— 7m+12
a+4 a+5 * m— 3 w— 4
(a;2+3a:+2)(a:-5) Sx^-Ba; 2a:3-4x2 a:2-5x+4
^2_3^_10 • ^ • 3a; ^ 2x2 ^ a._l •
32. What factor is common to
(1) a;2— a;-30 and .r2— 2a;— 35 ?
(2) a^+ab and a2+3a6+262 ?
Find three factors of :
33. 2.T2-10.r+12. 34. 3a2+3a_36. 36. a;3-8a;2+7a;.
36. If the expression a;2+TOa;— 6 has two binomial factors with
integral coefficients, what are all the possible values of m ?
37. Is the expression ;<;2— 3a;— 10 factored when it is written in the
forma;(a;-3)-10?
62. Square Root of a Monomial.. When a number is the
product of two equal factors, each factor is called a square
root of the number.
Thus, 16 = 4x4, therefore a square root of 16 is 4.
But 16=— 4x —4, therefore a square root of 16 is also —4.
Similarly, the square root of 25 is +5 or —5,
and the square root of 9a ^ is +3a or —3a.
Thus it is seen that every number has two square roots
differing only in sign.
It is customary to call the positive square root of a number
the principal square root.
63. Radical Sign. The symbol y/~ , called the radical
sign or root sign, is used to indicate the principal square
root of a number.
Thus, V25 = 5, A/a^ = a, V'9¥V = 3xy.
TYPE PRODUCTS AND SIMPLE FACTORING 93
When both the positive and negative square roots are
considered, both signs must precede the radical sign.
Thus, ^9 = 3 not -3; —V'9= —3 not +3, but ±V'9=±3, and
is read " plus or minus the square root of 9 equals plus or minus 3."
Thus, V4+V9=2+3 = 5,
but ±\/4±V'9=±2±3=±5or ±1.
If we represent the square root of 16 by x, then x^=16.
To solve this equation, take the square root of each side,
.'. x~±4.
We might have said ± x~ ± 4, which includes the four
statements :
+a;=+4, -{-X——4, — x=+4, — a;=— 4.
If both terms of the last two be multiplied by — 1, the
statements become the same as the first two. which are
represented by a;— ±4.
We see then, that it is necessary to attach the double sign
to the square root of only one side of the equation.
Ex.— Solve (.r-f 1)2=25.
Take the square root of each side,
.-. a;+l = ±5,
a;=±5-l = 5-l or -5-1,
= 4 or -6.
Show by substitution tliat each root satisfies the given equation.
E3XERCISE
48 (1-16, Oral)
T
State the two square roots of :
1. 36. 2, 81.
3.
121.
4.
2i.
5. y^. 6. b^c^.
7.
25a2.
8.
UxhjK
9. ia2. 10. Imhi^.
11.
25?*-
12.
6ix\
Solve the following equations :
13. x2=9. 14. 3x2^75.
15.
x2=^4a2.
16.
x^=a%\
17.* (a;+2)2=81. 18. (
[x-Sf.
=49.
19.
(a;-5)2=0
94 ALGEBRA
20. If the area of a square is 100 square inches, find the length of
its side.
21. If r is the radius of a circle the area is -rrr^, where 7r=31
approximately. If the area of a circle is 154 square inches, what is the
radius, or what is the value of r, if 3}r^=154: ?
22. Find the radius of a circle whose area is 616 sq. in.
23. If y is the radius of a sphere the area of its surface is given by
the formula, area =47rr^. If the area of the surface of a sphere is
154 sq. in., what is the radius ?
64. Squares of Binomials. If we multiply x-^-y by x-}-y,
the result, which will be the square of x-\-y, is x'^-\-2xy-\-y^.
The diagram shows a geometrical illus-
tration of this identity.
The first and last terms in x^-\-2xy-\-y^
are the squares of the terms of x-\-y, and
the middle term is twice the product of x
and y.
Therefore, the square of the sum of two numbers is equal to
the sum of the squares of the numbers, increased by twice their
product.
Also {jc—y)^=^oc^—2xy-{-y^.
Therefore, the square of the difference of two numbers is equal
to the sum of the squares of the numbers decreased by twice their
product.
In the square of a sum all the terms are positive, and in
the square of a difference the middle term is negative.
Thus, (3a + 26)2 = (.3a)2 + 2(3a)(26) + (26)2,
= 9o2 + 12ab +462.
(5x-37/)2 = (5x)2-2(5x){32/) + (3y)^
= 25a;='- 30xy +9y^.
(^x-4y"~) = {^xr^-2(hx){4y) + (4y)\
.r=
.VJ
XJ'
y'
TYPE PRODUCTS AND SIMPLE FACTORING 95
EXERCISE 49 (1-16, Oral)
What are the squares of :
1. a+l. 2. y+2. 3. to— 1. 4. x— 4.
5, 2a+l. 6. l-3x. 7. p— 9. 8. 2a;-l-3.
9. 2a— 3. 10, m— 2tt. 11. 3x— 2^. 12. 4x— 3a.
13. |— X-. 14. 2^—^. 15. 3x— |. 16. —a;— 2.
Simplify :
17.* {x+l)^+ix-l)K 18. (a-i)2-f(a+6)2.
19. {2x+iy-+{x~2)\ 20. (a+6)2-(a-6)2.
21. (3to— n)2-(2TO+/?)2. 22. i3x+2y)^-{2x-3y)^
23. (a;+l)2+(a;+2)2+(.r^3)2. 24. {x-iy^+{x-2)^-{x-3)".
25. 2(a+l)2+3(a-l)2-5(a-2)2.
26. Find the value of a--{-b^-\-c^ when a=x—y, b=^x-\-y, c=x—2y
27. SimpUfy {x+iy-+(x-2)^+{x-3)^-3{x-4:y-.
28. From the sum of the squares of x-\-2, x+3, a; 4-4, subtract the
sum of the squares of a;— 2, x—3, x— 4.
29. SimpUfy (2a-36)2-f (3a+26)2-(2a4-26)2.
30. If two numbers differ by 2, show that the difference of their
squares is equal to twice their sum.
2
31. Bv how much does the square of a; -f - exceed the square of
2,
X •
x
32. Show that the sum of the squares of three consecutive numbers
is greater by two than three times the square of the middle number.
33. The square of 1234 Is 1,522,756. Find the square of 1235.
34. The square of 2^=2x3+^^6}; the square of 5^=5x6+}=
30}, etc. In the same way find the squares of 6|, 8|, 20^. Prove
that this method may be used to find the square of any number ending
in |. (Let the number be %+i.)
65. Square Roots of Trinomials. Any trinomial which is
of the form a'^-\-2ab-{-b'- or a'"—2ab'\-b^ is a perfect square.
In order that a trinomial may be a square, the first and
96 ALGEBRA
last terms must each be a square and the middle terra must
be twice the product of the quantities which were squared
to produce the first and last terms.
Thus, Qx- + 2'ixy-\- \Qy^ is a square, because
(1) 9a;- is the square of 3rc,
(2) 162/^ is the square of 4^,
(3) 24:xy is twice the product of 3a; and ^y.
:. 9x2 + 24x2/+ 162/2 = (3a; + 4?/)2.
.'. the square root of 9a;2 + 24a;2/+162/2 is 3x-{-4y.
Is 4m2— 12mn+9n2 a perfect square ? What is it the square of ?
What is its square root ?
Similarly, 25a;2- 10a;+ l = (5a;- 1)2,
36a;2 + 24a; + 4 = ( )2,
a262_6a6 + 9 = ( )2.
Why is CT2 + 5afe + 25&2 not a square ? Is it the square of a + 56 ?
How would you change it so that it would be a square ?
The square root of a^-^2ab-\-b^ is a+6, but —{a-\-b) or
—a—h is also a squai'e root, since
(-a-&)2=:a2^2a6+&2.
It is customary, however, in stating the square root of a
trinomial to give only that one which has its first term
positive.
BXSRCISE! 50 (1-24, Oral)
Express as squares :
1. x^-^2xy+y\ 2. y^—2y+l.
4, 4a2+20a+25. 5. 9a2_24a+16.
7. a%'^—2ah+\. 8. \—&y+%if.
10. a%'C'—2abc+\. 11. x^+x+J.
What is the square root of :
13. 9a2+12a+4. 14. a;^— 4a;?/+42/2.
16. 4a262_20a6+25. 17. 4m2+2m+i.
19. 4— 4a+a2. 20. 9-12a:+4a;2.
Supply the missing terms, so that the following will be perfect
squares :
11. a2+. . .+6'^. 23. .^2— . . .+42/2. 24. x^+^x . . .
25. 4m2— . . .+9. 26. 9a2+18a. ... 27. . . .—^xy+'^yK
3.
4a;2+4a;+l.
6.
16a;2-8a;+l.
9.
9a;2— 18a;2/+92/2.
12.
y^—xy+lx\
15.
1— 6a;+9.r2.
18.
a2_i4a6+4962.
21.
9a;2-30a;y+252/'
TYPE PRODUCTS A AW SIMPLE FACTORING
97
a +b
a —b
a^ -\- ab
-ab-b^
28. If 16a2— ma+4 is a perfect square, what is the vakie of m ?
Give two answers and verify each.
29. What is the square root of 9a;^-(-6x'+l ? Check by putting
a;=10.
30. Solve the equations and verify :
(1) Vx'^+2x+l + Vx-+10x+25=^U.
(2) SVx^—4:X+4:—2Vx^+6x+9= —2.
(3) V'9x2+6x+l + \/4x2+4x+i + v'a;2— 2x+l = 13.
31. Show that
2Va^—6a+9—Va^-4a+4:^SVa^—2a+l — Via^+'ia+l.
66. Product of the Sum and Difference. The product of
a-\-b and a — h is a^—h",
:. {a-Vh){a-h)=a^-b^.
Here the two factors multipHed are the sum
and difference of the same two quantities a
and h, and the product is the difference of the
squares of a and h.
Therefore, the product of the sum and dijference of the same
two quantities is equal to the difference of their squares.
Thus, {x-\-y){x—y) = x'^ — y^.
(2a+36)(2a-36) = (2a)2-(36)2 = 4a2-962.
(3a2-6)(3a2+6) = (3a2)2-62 = 9a4_{,2,
(i + 3a;)(J-3x) = (i)2-(3x)2=i-9x2.
67. Factors of the Difference of Two Squares. ^.^
Since a'^—h'^=^{a^h){a—h), the factors of
the difference of two squares are the sum and
the difference of the quantities squared.
The diagram shows how this identity may
be ilhistrated geometrically.
Thus, 9x2- 251/2 = (3a;)2-(5i/)2, which shows that
it is the difference of the squares of 3x and 5y.
Therefore one factor of 9x' — 25y^ is the sum of 3x
and 5y, and the other is the difference of 3x and 5y.
That is, 9x'^-25y^ = {dx + 5y){3x-5y).
Similarly, 16?H2-9 = (4m)2-32 = (4TO+3)(4m-3).
(-- - 1
I -v ]
X 1
h I
98 ALGEBRA
If we wish to factor 8x^—2?/^, we should recognize that 2
is a factor of each term.
.-. 8x^-2y^ = 2(4:x'^-y^) = 2(2x + y)i2x-y).
EXERCISE 51 (1-24, Oral)
State the products of :
1. m-{-n, m—n. 2. p—q, p-^q- 3. ct~\-2, a— 2.
4. a:— 5, a;+5. 5. 2a-f-l, 2a— 1. 6. 3x—2,3x+2.
7. {2a-3x){2a-\-3x). 8. {4:X+5y){ix-5y).
9. {x+i){x-i). 10. {x'—2y){x'~+2y).
11. {5x+ab){5x-ab). 12. ( 2.r - | j ( 2x + 1).
State the factors of :
13. x^-1. 14. ?/2— 4. 15. a2_4j2 j^^ 4w2— w2,
17. 4j92_9^2_ -i8_ x2— |. 19, 9-xK 20. 1 — 16a262^
21. 25-49a;2. 22. a*-25. 23. a262-49. 24. 992-982.
Simplify :
25.* (a-2)(a+2)+(2a-l)(2a+l).
26. {2a-3b)(2a+3b)-{a+b)(a-b).
27. 2{x-3y){x+3yn2(3y~x){Sy+x).
28. 2(p-qr^ + 3(p+q)ip-q)-5{p + 2q)ip-2q).
29. Find the product of x—a, x^a and a:2-|-a2_
30. From the product oi x—l, x-\-\ and x^-\-\, subtract the product
of x—2, x-^2 and a;2-|-4.
Find three factors of :
31. 3x2— 3?/2. 32. 5x2—20. 33. a^—a.
34. mx^—ma^. 35. 5— 45^^. 36. x*—y*.
37. 7r7?2_7rr2. 38. a{x-—l)-Jrb{x^—l).
39. Why is the difference between the squares of any two consecutive
numbers always equal to their sum ?
TYPE PRODUCTS AND SIMPLE FACTORING 99
40. Simplify (a"-b^){a--5ab^%b-)^{a^~Zab+2b'^).
41. SimpUfv ""-^^ + t^y^ and ""^^^ - ^!ll?-
x—y x-\-y a-— 4 x+S
42. Solve ^i!rd + •:^!l^ ^ 10 ; 2(x--5)(a;+5)=15+(.r-I)(a;+l).
x+l a: — o
68. Numerical Applications of Products and Factors.
In this Chapter we have developed certain formulaj
concerning products and factors.
(1) (a-6)2 =a2-2a6+6?.
(2) (a+6)2 =a2^2a6+62.
(3) {a+h){a-h) =a^-b^.
These formulse are true for all values of the letters involved.
By substituting particular numbers for the letters we will
see how some arithmetical operations might be simplified.
(1) Since {a-by = a^-2ab + b\
992= (100-1)2=10000-200+1 = 10001-200 = 9801.
372= (40-3)2= 1600-240+9= 1609-240=1369.
9982 = (1000-2)2= = =
892= (90-1)2= = =
(2) Since (a + 6)2 = a2 + 2a6 + 62,
922= (9o_^2)2= 8100+360+4= 8464.
1212 = (120+ 1)2= 14400+240+ 1 = 14641.
752= (70+5)2= =
(3) Since {a+b){a-b) = a"--b\
92x88 = (90+2)(90-2) = 902-22 = 8100- 4 = 8096.
65x75 = (70-5)(70+5) = 702-52 = 4900-25 = 4875.
27x23 = (25 + 2)(25-2)= = =
87x93 = ( )( )= = =
(4) Since a^-b' = {a-Jrb){a-b),
532- 522 = (53 + 52)(53-52) = 105x 1 = 105.
412- 3i2 = (4i-|-3i)(4i_3i)= 72x10=720.
.272-6272 = ( )( )= =
672- 332 = ( )( )= ^
H 2
a
100 ALGEBRA
69. Some Geometrical Applications.
(1) If a is the length of the side of the large square and
h the side of the small square, the area of the shaded portion
is evidently a^—b^.
If we wish to find the area of the shaded part when
a = ll and 6 = 23, we have
a2- 62 = 772- 232 = (77 + 23)(77- 23) =100x54 = 5400.
If a = 225 and 6=125, find the difference in the areas
of the two squares.
(2) The radius of the large circle is R and of the small
circle is r. The area of the large circle is -y-iJ^ and of the
small one is -y-r^,
.'. the area of the shaded part is '^f-{R^—r'^).
If i? = 39 and r = 31, find the area of the ring.
The area =-V-(i?2_r2) = ^^(392-312)
= -V{39 + 31)(39-31) = -V-X 70x8=1760.
If i? = 89 and r=82, show that the area of the ring is 3762.
(3) In the right-angled triangle in the figure it
^ is shown in geometry that
62-f c^^a^ or b^=a^—c^ or c^=a^—b".
If a = 41 and c = 40, find the length of 6.
62 = a2_c2 = 4i2_402 = 81xl = 81,
.-. b=Vsi = 9.
If a = 61 and 6=11, show that c = 60.
BXERCISE 52
Use short methods in the following :
1. Find the squares of 98, 999, 119, 58, 799.
2. Find the products of 91 x 89, 61 X 59, 47 X 53, 203 x 197.
3. Find the values of 522-48^, 79^-782, 215^-2052, 7252-2752,
6732-5732.
4. If x^=b^—c\ find x when 6=13, c=12 ; when 6=25, 0=24.
5. If 7a;2=642-572, find the value of x.
TYPE PRODUCTS AND SIMPLE FACTORING 101
6. Find the difference of the areas of squares whose sides are
a and b for the following values :
a—
41
13 29
83
15m
2-85
b=
40
12
21
17
14to
2-15
a^~b^^
7. Find the difference in the areas of circles whose radii are R and
/• for the following values :
R=
4
14
25
51
19a
3-25
r—
3
7
24
44
5a
2-35
3l{R'--r'~)=
Factor
1.
4.
7.
10.
13.
16.
EXERCISE 53 (Review of Chapter VIII)
18.
19.
3x+6y.
6ac — 36c.
4a2_9:c2.
2a^-lS.
a^-b*. 17.
Write down the squares of :
2a;— 3, Sx— 6, 4x-:iij, a-\, bc — h
What are the square roots of : a^ + Ga + d, p^—8p-]-l6.
2. 4m-12n.
3. ax — bx.
5. a;2 + 4x+4.
6. a2-2o+l.
8. x^-3x+2.
9. 2/2-2/-110.
11, 100/32- 81r/2.
12. a2-19a-20.
14. 300-3x2.
15. (x + 2/)2-l.
Solve x2= 100; x^- =
\; 9x^ = 4; 5x2=1-
m2n2-10mn + 25, a'^-a-{-\, \Qx--^Qxy + -25y^ ?
20. How much must be added to the middle term of 4:a--\-Za-{-Q
to make it the square of 2a 4- 3 ?
21. What middle term must be inserted in Ox" . . . +25?/2 to
make it a complete square ? Give two answers.
102 ALGEBRA
22.* Find three factors of x^ — x, 3a;*— 12, a^ — 3a^-\-2a.
23. Solve (x-3)2 = 25; 4(x-^)2 = 9.
24. If a = irr^, find r when a= 12-56, 7r = 314.
25. Find the values of 9972, 8752-75^, 97x103, 81x81, 86x94,
using algebraic methods.
26. Find four factors of 2a;^-32, a*-13a2 + 36, 2m3-18TO and
o2(a;2-2/2)-62(a;2-2/2).
27. Simplify {b+l)^ + {b-iy + (c+iy + {c-l)^.
28. Simplify {x + y)(x~-y) + (x + 2y)(x-y) + (x + y)(x-2y).
29. If a = 92 and 6 = 88, find the values of ab, a^-b^, a^ + b^, using
algebraic methods.
30. Simplify (a-26)(a + 26) + (a-46)(a + 4fe)-2(o-36)2.
31. What are all the possible values of 6, if x^-\-bx-\-4:2 is the product
of two factors with positive integral coefficients ?
32. Simplify ^^^^' + ^^' + "'-^^' -
x—y x~2y x—3y
33. If the square of 426 is 181476, find the squares of 427 and 425.
CHAPTER TX
SIMPLE APPLICATIONS OF FACTORING
70. Highest Common Factor. \^nien a factor divides two
or more expressions it is called a common factor of those
expressions.
Thus, 4 is a common factor of 8, 12 and 20,
and a is a common factor of a^, 2a and Sab.
As in arithmetic, the highest common factor (H.C.F.) is
the product of all the simple common factors.
Thus, the simple common factors of 3a -6, 606^ and 9abc are 3, a and b,
and therefore the H.C.F. is Sab.
In the case of monomials the H.C.F. may be written down
by inspection.
Ex. 1.— Find the H.C.F. of Qm^n, Urnhi^ and dmhi^.
(1) The H.C.F. of 6, 12 and 9 is 3.
(2) The highest power of m which is common is m^.
(3) The highest power of n which is common is n.
:. the H.C.F. is SxOT^xn or Sm^n.
If the expressions are not monomials they must be factored
when possible, after which the H.C.F. may be written down
by inspection.
Ex. 2.— Find the H.C.F. of a^+ab, ab+b^, a^-\-3abi-2b2.
a--\-ab=^a{a+b),
ab + b^ = b{a + b),
a^ + 3ab+2b^ = {a + b){a + 2b),
:. the H.C.F. = a+6.
103
104 ALGEBRA
BXEROISE 54 (1-12, Oral)
Find the H.C.F. of :
1. 3,9,12. 2. 16,24,40. 3. 2«, 46, 8c.
4. 3x, 6x, \2x. 5. 4.ax, Qax, 2x^. 6. a%, ab\ a%\
7. 3x2, 4^.3^ 5^4 8^ 5^2^ loa^, 15a. 9. llx^y-, Mxhj, blx^.
10. 2a, a2+a6. 11. Gx^, 4x2+2x. 12. (a+6)2, a^— 62.
13.* 2a+46, 3a+66. 14. a^-h"^, ah-h\
15. m2— n2, 7n^—2nin-\-n^. 16. x^-\-xy, xi/^i/^, {x-{-y)^.
17. mn+2n, m2+3m+2. 18. a2— 3a+2, a2— 5a+6.
19. a;2-9, :c2-7x+12,x2-4x+3. 20. y'^-ir2ij-^,rj^+y-2.
21. a2-{-2a6+62, 2a2-262. 22. x2-10.r+25, 3x2-75.
23. 6a6+462, 6a2+4a6. 24. a3-2a2+a, a^-\.a^-2a.
25. If a-|-6 is a common factor of a"-\-mab-\rb- and a2+?ia6-|-262,
what are the values of m and n ?
26. The H.C.F of a26 and ab'^ is ab. Find the greatest common
measure of the numbers to which a^b and ab" are equal when a=2 and
6=4, and compare the result with the value of a6 when a=2 and 6=4.
71. Algebraic Fractions. A fraction has the same meaning
in algebra as it has in arithmetic.
Thus, I means 3 of the 4 equals parts of a unit, or the
quotient of 3-^4.
Similarly, 7- means a of the h equal parts of a unit, or the
quotient of a-^h.
The fraction y is read " a divided by h " or " a over 6."
b -^
72. Changes in the Terms of a Fraction. As in arithmetic,
both terms of a fraction may be muItipUed or divided by the
same quantity (zero being excepted) without altering the
value of the fraction.
SIMPLE APPLICATIONS OF FACTORING 105
Thus, f|=| = J.o,= 3o_etc.
Similarly,
a
b~
ac
be
OCX
bcx
=
etc..
a^b^
ab^
62
6
a^bc
abc
be
c
and
73. Lowest Terms. A fraction is said to be in its lowest
terms when its numerator and denominator have no common
factor. If it is not in its lowest terms, it may be reduced by
dividing both terms by all the common factors.
Examples.
18 _ 18^6 _ 3
42 "42-4-6^ T
Iba^b _ I5a^b^5ab _ 3a*
25a62 ~ 25ab^^5ab ~ 56 *
g x'-y' ^ {x+y){x-y) ^ x—y _
x^ + 2xy+y' {x+y)ix+y) x+y'
^ x^ + 5xy + 4y^ ^ {x + y){x + 4:y) ^ x + y _
x^ + Sxy-'iy^ (x-y)(x + 4:y) x — y'
The attention of the pupil is drawn to the fact that it is
factors and not terms which are cancelled from the numerator and
denominator.
7 + 2
ThvLS, in the fraction ~ — ^ we cannot cancel the twos and say that the
9 + 2 ^
fraction is equal to I, for the value of the fraction is j\, which does not
7x2
equal J. But if the fraction is - — ^ we can now cancel the twos and
the resulting fraction is l-
Similarly, — = - after cancelhng, or div-iding by the common factor a.
_ a+6 . b
But IS not equal to -.
a-\-c c
It is thus seen, that no cancelling can be done until both
terms of the fraction are expressed as products.
106
ALGEBRA
EXERCISE 55 (6-21, Oral)
Fill in the blanks in the followins; :
2.
4.
5.
15
20
ax
bx
30
Gxy
12 4a
hm
Sam
h b{m-{-n)
6a^x a^x
662
12a 2^:2
(
a^-2ab+b'^ (
Reduce to lowest terms
2x2
6.
14
21
3x
6
10. —
14.
18.
25.
28.
2a+4
6
x{x—l)
a;2-l
11.
15.
19.
L5a
"25
lOm^n
9. ^
35a26
74. Multiplication and Division of Fractions. The methods
by which fractions are multipHed and divided in algebra
are the same as in arithmetic.
SIMPLE APPLICATIONS OF FACTORING 107
Examples.
3 5_ 3xo_ 15
4^7~4x7~28*
10 7^^_10 7 15_25^
^- 21 ^ 3 • 15~ 21 ^ 3 ^ 4 ~ 6 ^'
a c ay.c ac
b d ~ bxd bd
ab x^y . x^ a6 x'^y a __b
4 . X IT T" ^ X 7C X 5 ^
xy a^ a xy a^ x^ x
a^ + ab cd + d^ _ a{a + b) djc + d) _ ad
^' c'^+cd ^ a6 + 62 - c{c + d) ^ b(a + b} ~ be'
EXBRCISE 56
Simplify :
,453 ^ 2^5.3 2a 96
'• 5^6^4- ^' 15^7-^14 '• 36^4^
4. ^X^-X^ 5. ^xf. 6. 1-^1^
6 c a xy be ob So
7 «^^. 8. ?^^^1^\ 9. ^Vl2«.
' b - d 14?/- ■ 7y 56
in* ^•^' ^~^ 11 4a +66 lOx- x-+a:y , xy+y^
3^^^l0~' 5a; 2a+36' "' a2+a6a6+62*
x^-l a;2-5a;+6 . . a^-3a+2 a^-'7a+l2 a2-4a+3
^'- ^^iHi ^ a;2_4a;+3" " a2-5a+6 a^-Ga+S a2-5a+4
a2_62 a2^2a6-862 a2+3a6-462
-IK Y -^ •
a2-3a6+262 a2_2a6-362 ' a2-4a6+362
75. Lowest Common Multiple. A product is a multiple of
any of its factors.
Thus, 3xy is a multiple of 3, of x, of y, of 3.T, of 'Sy, of xy, and a* is a
multiple of a, of a*, of a'.
When an expression is a multiple of two or more
expressions it is a common multiple of those expressions.
Thus, I2a^b' is a common multiple of 2a'- and 3a62.
108 ALGEBRA
The lowest common multiple (L.C.M.) of two or more
expressions is the expression containing the smallest number
of factors which is a multiple of each of the given expressions.
Ex. 1.— Find the L.C.M. of Qx-y, 9xy^ and I2xy^.
The numerical coefficient of the L.C.M. is evidently the L.C.M.
of 6, 9 and 12 or 36.
The highest power of x in any of the given expressions is cc' and of
y is 2/', so that the L.C.M. must contain the factors x^ and y^.
:. tlie L.C.M. = 36 xa;2xj/3 = 36x22/3.
Ex. 2.— Find the L.C.M. of a^-b^ and a^-2ab-\-b^.
o2-62 = (a-6)(a + fe).
a2_2a6 + &2 = (a_6)2.
.-. the L.C.M. =(a-b)^a + b).
Why is (a-b)(a + b) or (a-6)2(o + 6)2 not the L.C.M. ?
EXERCISE 57 (1-9, Oral)
Find the L.C.M. of :
1. 3, 4, 5. 2. 10, 15, 20. 3. 2a, 4a, 6a.
4. a, ah, a^. 5. x\ xij, y^. 6. 2ah, Sac, 66c.
7. 10a2, 15a2, 5a. 8. Sa^, 2a-, 4a. 9. M^-h, 4:ab\
10.* a\ a^+a. 11. 3a;, 3x2+6^. 12. ab+ac, b^+bc.
13. 2a;+2, a;2-l. 14. x^+xij, {x^y)K 15. x^—l,x^-—3x+2.
16. a^—ab,ab—b^. 17. a"—b\a^—2ab-\-b^. 18. x^—x,x^—x.
19. 2x, 4x+4:, 2x2-2. 20. ?/2-3?/+2, y^--y-2, y^-1.
21. Show that the product of x'^+x—2 and x^—x—6 is equal to
the product of their H.C.F. and L.C.M.
76. Addition and Subtraction of Fractions. If we wish
to add or subtract fractions we must reduce them to a
common denominator. As in arithmetic, the lowest common
denominator is the L.C.M. of the denominators.
SIMPLE APPLICATIONS OF FACTORING 109
Examples.
3 5 2 9 10 8 9+10-8 11
■ 4 "^6 3 "12"^ 12 12" 12 "12*
o a a _ac ab ac + ab
b c be be ~ be '
3^_ £ _5 _ ^_ 4a2 5a6 Sb^-4a^ + 5ab
**• „2 U2 'T' „U „2;,2 „2»,2 "T
4.
ab a^b- a'^b^ a^b^ a%^
b a b ay — bx
x^+xy xy+y^ x{x+y) y(x+y) xy{x+y)
2x 2 2x 2 2a:-2(x-2)
a;2_4 x+2 ~ (x+2)(cc+2) ~ a;+2 ~ (a;+2)(a;-2)
2a;-2a;+4
(^.+2)(a;-2) (x + 2)(.x--2)
EXERCISE 58 (1-8, Oral)
Reduce to fractions with the lowest common denominator :
1.
2 5
3' 9'
2
3 a
4' 46'
3.
1
a'
1
a-'
4. 1,A
3a; 2x
5
3a 4a
(3
in n
7
1
b
8. 2, i^
a a-
4 ' 3
n m
a
c
9.
2 5
3
x». ?.
b
c
11.
2x2
3bx 4
3a' 4a2'
2a ^
c
a
6 ''
' 2 ' 36c*
12 — ^, A. 13 ^, X, ^. 14. ^Ltl, "~^ «+2
Sij-' 3xy' 2x ' 3ab' 26c' 4ac ' a ' 2a ' 3a
Perform the operations indicated :
a a a+b a~b a+4 5— a
lo. 3+5. 16. -2-+nr- '"• ^-+^-
._ u.—^ u, iQ 3 2 a + 6 , 6+c a+c
18. -,; ,. 19. -_^ + ^— . 20. __^-^+_-.
21. . + T— • 22. -^ g-. 23. ^- + ^ g--
24. :::^_ii:^^^±^. 25. ^ ^.
3 ' 4 X— 2/ a;+2/
a
— X
a
3
5
1
-,',.
1
— X
x
-y
x—y
110 ALGEBRA
26 4 4 8a ^7 a+x a—x a^—x^
a+4 a— 4 a2_i6 a x ax
28. — ^ ^— • 29.
3a;+6 2a;+4 ' a^— a& a6— 6-
30. = \ 1 (Check when a= 1.)
a2+3a+2 ^ a2^5a+6 ^ a2^4a+3
a2_i ' a2^3Q^^2 a2_|_a_2
77. Mixed Expressions. An expression which is partly
integral and partly fractional is called a mixed expression. A
mixed expression in algebra corresponds to a mixed number
in arithmetic.
Thus, 3? is a mixed number and 0+ is a mixed exj^ression.
Note that in a mixed number the sign of addition is
omitted and 3| means 3+|. But in algebra the sign must
be inserted, as a- would mean ax- and not a + -.
c c c
78. Reduction of a Mixed Expression to a Fraction. Since
every integral quantity may be written as a fraction whose
denominator is unity, it follows that the reduction of a
mixed expression to a complete fraction is a problem in
addition or subtraction.
Exj\.mples.
1. 3S„3 + |="'+? = y-
000
b ac , b ac + b
2. a + -= \- - =
c c c c
X _ 5y X _ 5y—x
4.
y y y y
ac a{b-^c) ac _ ab+ac—ac _ ab
b+c ~ 6+c 6+c ^ b\.c ~ 6+c
SIMPLE APPLICATWNS OF FACTORING 111
79. Reduction of a Fraction to a Mixed Expression. To
, ab-\-bc . .
separate into two ira'ctions we merely reverse the
operation of addition.
n,, ab-\-bc ab be , be
Thtis, — = = b A ,
a a a a
, ab — bd — cabbdc c
EXERCISES 59 (1-9, Oral)
Reduce to complete fractions :
1. 2+t. 2. 1 + ^. 3. 3 + ?.
2 y
4. a + ~. 5. x — ±. 6. a — -.
•* o C
7. x-"^- 8. 2.r + ^. 9. ah-^1.
n X a
10. a-^. 11. x + ^^. 12. 2a + M.
o+c x—y a — b
^o 11,2 ., , a-\-h x^-\-v^
13. a;+l+ — . 14. a-h 3^. 15. a; - y + ^^
Separate into fractions in their lowest terms :
6a+26 aa:+6x 5a;— 8?/
Id, . X ^ , — • Xo, •
4 a/> 10a
19. 6«'-3^' 20. "+'^^-^g . 21. ?i^i?.
3a6 * 21a6 ' Qxy
.^„ Smn—4n „., 6a6c— Qfec+c^ „, (a— 6)2+a;
JJ. . Jo. . J'i. •
2n 36c a— 6
EXERCISE 60 (Review of Chapter IX)
1. Define highest common factor and lowest common multiple.
2.* Find the H.C.F. and L.C.M. of 3a;- 6, 4a;- 8, 5a;- 10.
112 ALGEBRA
3. Find the H.C.F. and L.C.M. of x^+xy, xy+y^ and x^y + xy^.
4. Find the H.C.F. and L.C.M. of x^-Tx+lO and x2+2a;-8.
Show that the product of these expressions is equal to the product
of their H.C.F. and L.C.M.
5. Reduce to lowest terms :
a^-{-ab x^ 6a* — 9a6 abx—bx^
a^ ' x^ — xy 8ab—l2b^ acx—cx^
T.T li- 1 Sax 4cy 5bd
6. Multiply^, si' 2^x'
7. Simplify _x— -.g-,-
8. Reduce to lowest terms :
x'^—2x x^+4:X+4: 20;"- 18 a*— 6*
x^-5x+~6' x^ + 5x + 6' 3ic2 + 3:c-18' a^-2ab + b^
x—y x-'ry 2ax xy—y^
x^-\-2xy-\-y^ x^—2xy-{-y^ y x^-\-xy
10. Divide -^^- by ^,-^^-, and — ^ by ^^-^ •
_ „. ... a;-3 , a;+4 2x-l 8-4a;
11. Simphfy g H ^ and — \ g— •
12. Find the sum of
x—y y—z z—x
xy ' yz zx
^o T. .u , 3& + 40 , &-6c ,^ ^ a+6c
13. From the sum of — jp^— and —^ — subtract —; •
2ab 26c 4ac
14. Simphfy -3- - ^^- and — -^--- •
15. Express „,„ as the difference of two fractions in their
^ a^b^
J,2 g2 c* a*
lowest terms. Do the same with — , s-i and — s— s— and find the sum
of the three results.
y — 4 y — 5
16. By how much does - exceed ^-- — ?
42/ 5y
17. Find the sum of ,-^, -, and -^ — , ,•
a + b a — b a^ — b^
SIMPLE APPLICATIONS OF FACTORING 113
18. By what must be multiplied to give -5 — ij~~_^^•) ^^ ^^®
product ?
19. Find the quotient when - is divided by ^ — ^*
20. Solve {a-b)x = {a^-b-){a + b).
21. Find the difference between
a , 6 , c , a; , X , a;
\- T 1 and h r \- --
a— a; 0 — x c~x a — x o — x c—x
by first subtracting from , etc.
° a—x a~x
22. Find the missing term in the following identity :
x^ — 5x-{-Q x^-\-5x . . . x--\-2x—8
x^-Zx-\ ^ a;2-9 ~ ^ x^-x-l2'
CHAPTER X
REVIEV/ OF THE SIMPLE RULES
80. In this Chapter will be found such exercises as will
furnish a review of the elementary rules. In it is also
included matter which it was not thought advisable to
present to beginners in the subject of algebra.
81. Use of Brackets. We have abeady seen that
(1) a+(6 + c)=a+6+c,
(2) a-\-{b—c)=a-\-h-c,
(3) a—{b+c)=a—b—c,
(4) a—{b—c) = a—b-\-c.
That is, when brackets are preceded by the negative sign,
as in (3) and (4), the brackets may be removed if the signs
of all terms within the brackets be changed ; but when they are
preceded by the positive sign, as in (1) and (2), no change
is made in the signs ivhen the bracJcets are removed.
In (3) the sign of b in a—{b-\-c) is positive as the
expression might be written a—{-\-b-\-c). When the brackets
are removed we follo\y the rule and change -\-b into —b.
Sometimes we find more than one pair of brackets in the
same expression.
Ex. 1.— Simphfy a-(3a— 26) + (5a-46).
Following the rule, this expression becomes
a-3a + 2b + 5a — ib or 3a-26.
When one pair of brackets appears within another, it is
better to remove the brackets singly, and the pupil is
advised to remove the inner brackets first.
BE VIEW OF THE SIMPLE RULES 115
Ex. 2.— Simplify 4x-{2x-{3^x)\.
Removing the inner brackets, we get
4a;— [2a; — 3 — x}.
Removing the remaining brackets, we get
4x-2x+3+x or 3a;+3.
Ex. 3.— Simplify 3a—[a+b — {a—h—c—{a+b-c)\].
The expression = 3a — [a + 6— {a — 6 — c — a — 6 + c|],
= 3a — [ct-1-6 — a + fo + c + a + fe — c],
= 3a— a — 6 + a — 6 — c — a — 6 + c,
= 2a-3b.
After removing the first pair of brackets, the quantity
a — b — c — a — b-'rc
might have been changed into the simple form —26. Work the
problem again, simplifying at each step.
When brackets are used to indicate multiplication, the
multipHcation must be performed if the brackets are
removed.
Ex. 4.— Simplify 4a;-.3(x— 2^)+2a;— 4i/.
The expression =4.r — (3a;— 6y) + 2a;— 8?/,
= 4x~3x+6y + 2x-8y,
= 3x-2y.
Note. — When the pupil has had some practice he should be able
to remove the brackets and perform the multiplication in a single step.
EXBRCI3B
61 (1-9, Oral)
Remove the brackets from :
1. {a—b) + {c-d).
2.
{a-b)-{c-d).
3. -ia-b) + {c-d).
4.
-{a-b)-{c-d).
5. a-{b-c)-{d-e).
6.
a-{-b)-{-c).
7. a-}-',b+{c-d)}.
8.
a+',b—{c—d)\.
9. a-{b+{c-dy,.
10.
a->b-{c-d):.
11. a-{-b-{c-d)\.
12.
-[a-ib-{c-dn
T 2
116 ALGEBRA
Simplify :
13. 4a-26-(2a-26). 14. 2(lx-3y)-3{2x-3y).
16. B{a—b+c)-2{a+b—c). 16. 2a- l3a+2(a— 26)}.
17. 3(a+b—c)—2{a—b+c)+5{b—c+a).
18.* l5x-{4.-[3-5x-{3x-l)]\.
19. Add 3(a+b)-5(p+q), -2{a+b)+{p+q) and Mp+q).
20. Add l+rc— ?/, 1—x—y and 1— a;+?/.
21. Add 3x-2{y-z), 3y-2{z—x), 3z-2(x—y).
Remove the brackets and express in descending powers of x :
22. 3{5x~3^2x^)-2{x^-5+3x)-3{4.-5x-dx-^).
23. 2a:(3a;-2)-5(a:-3)+6a;(a;-l)-2(a;2-5a;).
24. ^(4a;-3)-J(6-a;2)+J(a;2+8a;-12).
Solve for x and verify :
26. 4(a;-3)-7(a;— 4)=6— x.
26. 5a;-[8.f-3;l6-6a;-(4-5x)|]=6.
27. 3(2a;-7)-(*'-14)-2(5a;+17) = 6(5-8a;) + 21a;+149.
28. ^(27-2x)=|-/5(7a;-54).
29. Simplify a-[5b-{a-(3c-3b)+2c-{a-2b-c)\].
30 SimBlifv 3(a-6+c)+2(6-c+a)-(c-a+&)
^ -^ 5(a-26+c)-2(6-3c+2a)-(llc-2a-116 "
31. Solve (7ix-2|)-[4i-|(3i-5x)] = 18i.
82. Insertion of Quantities in Brackets. The trinomial
a—h-'rc may be changed into a binomial by combining two
of its terms into a single term. This may be done in a number
of ways.
Thus, a—b + c = (a -b) + c = {a+c) — b
= a — {b — c) = a+{c—b).
Remove the brackets mentally and see that each of these is equal
to a—b-\-c.
REVIEW OF THE SIMPLE RULES 117
Ex. 1. — Express a—h-\-c—d as a binomial.
As we have seen, this may be done in many ways as a — (6 — c + d),
(a-6) + (c-rf), {a-{-c)-(b+d), (a-d)-ib-c), c-{b+d-a).
Note. — The pupil must exercise particular care when dealing with
brackets which are preceded by the negative sign. The signs of all
terms inserted in such brackets must, of course, be changed. He
should, in every case, remove the brackets mentally to test the accuracy
of the work.
Ex. 2. — Express a-\-b — c as a binomial by combining the
last two terms within brackets, preceded by the negative
sign.
a-\-b — c = a — {—b-\-c''
= a~{c — b).
Either form is correct, but it is usual to make the first term within
the brackets positive, so that the second form is preferable.
83. Collecting Coefficients. Brackets are frequently used
for the purpose of collecting the coefficients of particular letters
in an expression.
Thus, ax-{-by—cx—dy=x{a—c)-\-y{b—d),
and ?nx — ny+px+qy = z(m-irp) — y(n — q),
= x{m+p) + y{q-n).
Verify these by removing the brackets.
EXERCISE 62
1. Express 3a — 26+ 4c as a binomial in three different waj's and
verify in each case.
2. Express p—q—r-^s as a trinomial in four different ways and
verify.
3. Express x—y—z—k as a binomial in four different ways and
verify.
Collect the coefficients of x and y :
4. ax— by— ex— dy.
5. mx—ny—px-\-qy—ax-{-by.
6. a{x-y) + b{2y-3x)+c{5x+2y).
7. x{a—b)+y{b—c)—d(x+y).
8. 2ax—3by—l0x—5y+6bx-4:ay.
9. {a-3)y-{2-b)x+4y+2ax-{3x+by).
118 ALGEBRA
10. Enclose a—h—c—cl—e\-f in alphabetical order in brackets,
with two terms in each ; with three terms in each.
Arrange in descending powers of x :
11. a(a;2+4— 3a;)-6(3a;-5a;2)_c(l-4a;).
12. ax''-—hx-\-c—{2px^—Zqx+r) — {l(lx^+Zcx+f).
84. Multiplication with Detached Coefficients. The method
of multiptying two binomials has already been shown in
Chapter V. The same method is followed when the factors
are not binomials.
Ex. 1.— Multiply :i;2— 3x-+4 by x-2.
(1)
a;2-3a;+4
X -2
-8
(2)
1-3 + 4
1-2
-8
(3)
Check
a;=l
.t3-3x-2+ 4x
-2x--\- 6a;-
1-3+ 4
-2+ 6-
+ 2
-1
x^ — 5x'^-\- 10a;-
-8
1-5+10-
-8
-T
The second method is called multipHcation with detached
coefficients. The processes in the two methods are the
same, with the exception that the letters are omitted in the
second method and the coefficients only are used.
When the second method is used the first coefficient in
the product must be the coefficient of the product of .r^ and
X, that is, of x^. The next must be the coefficient of x^ and
the next of x, as the product will evidently be in descending
powers of x, as both factors multipHed are so written.
In (3) the check is shown as explained in art. 42.
Ex. 2.— Multiply 3.r3-7:r+2 by x^-2x+^.
Here the term containing x^ in the first 3 + 0 — 7+ 2
expression is missing and a zero is sup- 1 — 2 + 3
plied in order to bring coefficients of like ■
S-l-0— 74- ^
powers of x under each other in the partial <J-rv < -r -
products. -6-0+14- 4
The first term in the product is 3a; «. +9+ 0-21 + 6
Write down the complete product and 3_6 + 2 + 16 — 25 + 6.
check the work.
REVIEW OF THE SIMPLE RULES 119
Ex. 3. — Find the coefficient of x^ in the product of
dx-'—Qx^+3x-2 and x^—2x^—3x-^4.
Here the complete product is not required, but only the term which
contains x^.
The partial jDroducts which will contain x^ are evidently those whicli
we obtain by multiplying —2 by x^, 3x by —2x'", —6x- by — 3.r and
5x^ by 4.
.-. the coefficient of x^ = - 2 - 6 + 1 8 + 20 = 30.
Ex. 4. — Multiply ax--{-hx-\-c by mx—n.
ax- -\-bx +c
mx —n
Here the multiplication is
performed in the usual way.
In adding the partial pro- amx^-{-bmx'' -\-cmx
ducts, the coefficients of the —anx^ —bux —en
powers of x are collected.
amx^ + (bm — an )x^ + (cm — bn)x — en.
EXERCISE 63
Multiply and check :
1. .T2-3.C+2, x-2. 2. 2x2-5.c-3, 3.f-2.
3. x~—x-\-\, a;+l. 4. a'^-\-ab-\-b-, a—b.
5. 'X'—x-\-\, X'^x-\-\. 6. a^—5a'^—2, a^+a— 1.
7. 3x2-2a;-5, .r2+a;-3. S. 2a'—oab+Zb^, 2a2+5a6-362.
9. a+6— c, a— 6+c. 10. .T3+2a;2+4T+8,a;2— 4a;+4.
11. 6--6 + 1, 62+6+1, fe^-^Hl-
1*2. a;2— a;i/+«/2+a;+?/+l, x+2/— 1. . •
Use detached coefficients to multiply ; check the results :
13. 3a;3— 4a;2+7x— 3 by x''-—2x—\.
14. 5a*— 6a3_2a2-a+2 by 2a2_3a+2.
15. 4x^— ox— 2 by 4a;2— 3.^-1.
16. (x2-x-2)(2x---x-l)(3.r-2).
Simplify :
17.* (.T-l)(x-2)+(.r-2){.T-3)+(rr-3)(.T-I).
120 ALGEBRA
18. {a+x){b-c)+{b+x){c-a) + (c+x)ia-b).
19. (a^b){c+d)-{a-b){c-d).
20. {a+b-c){a-b) + {b+c—a){b—c) + {c+a—b){c—a).
21. (x+l){x+2){x+3)-(x-l){x-2){x~3)
Find the product of :
22. (l-x){l+x)(l+x^){l+x'^). 23. (.r-l)(.r-2)(.r-3)(a;-4).
24. (a;-l)(a;-3)(a;+l)(a;+3). 25. («-l)(a2+a+l)(a3-|-l).
Find the coefficient of x^ in the product of :
26. 3x2— 5x+ll and dx^+Qx-i.
27. a;3+4.r2— 5x+2 and x--2x-3.
28. 3x2— 12a;+15 and 2x2-7x-38.
29. Multiply l+x+x^+x^ by l+2x-|-3x2-f-4.r^ retaining no powers
higher than the third.
30. Add together (x-l)(x+2), (x+2)(x-3), (x+3)(x+4)(.r-l),
(a;+4)(a;2-2x+3) and 7— x^+Sx.
Check by putting x— 2.
31. Multiply lx^—5xhj—xy"+Qij^ by 4x2+3xy— 2y2_
32. Show that the expression x(x+l)(x+2)(x+3) + l is equal to
(x2+3x+l)2.
33. Find the first four terras only in the product of :
2 + 3x+4x2+5x3 and l-2x+3x2-4x3
34. Find the coefficient of x* in the product of :
l+4a;+7a;2+l0x3+13x« and l+5x+9x2+ 13x3+17x4.
35. Solve and verify :
(x-2)(x-4)(x-6)(x-10) = (x-l)(x-5)(x-7)(x-9).
36. Multiply ax^+bx+c by bx'^—cx+d. Collect the coefficients
of X and write in descending powers.
37. Multiply px^—qx->rr by px+q, and (a— l)x2+ax— 1 by ax+1.
38. Simplify {ay^-by+c)imj+b)^{ay^-^bij-c){ay-b).
39. Subtract the product of x~+x{p+l) — l and x— 2/3 from the
product of x^- x(p— 1)+2 and x+p.
REVIEW OF THE SIMPLE RULES 121
40. Point out two obvious errors in each of the following statements :
(i) ab{a+b){a-+b^)^a*b+a%+a^b^-ab\
(ii) {2x+Syf=6x^+S6xy-54xy^+21y^.
(iii) x^-6x'-y-3xy^~+2y^^ix-2y)ix^^~4:x+y^).
41. Use the formula {a-{-l){b-\-l)=ab+{a+b) + l to find the
product of 2146 and 3526, being given that the product of 2145 and
3525 is 7,561,125.
85. Division by a Compound Quantity. The method of
dividing by a monomial has akeady been shown in Chapter V.
The method of dividing by a quantity containing two or
more terms is in many ways similar to long division in
arithmetic.
Divide 672 by 32.
(1) (2)
.32)672(21 3 . 10 + 2)6 . 10^ + 7 . 10 + 2(2 . lO + l
64 6 . 102 + 4. 10
32 3.10 + 2
32 3 . 10 + 2
In (2) the divisor is expressed in the equivalent form
3 . 10+2 and the dividend 6 . 10^+7 . 10+2.
If we substitute a; for 10 the problem would be :
Divide Qx^-\-lx-\-2 by 3.r+2
The method here is so similar to the 3a; + 2)6x2+7a; + 2(2.T+ 1
method in arithmetic, that little explana- dx^ + ix
tion is necessary. The first term in the
quotient is obtained by dividing 3x into ox-^^
6x2. The product of 3a; + 2 by 2x is then 3x+2
subtracted from the dividend and the
remainder is 3x+2. The last term of the quotient is obtained by
dividing the first term of the remainder (3x) by the first term of tlie
divisor (3x).
In more complicated examples the method is precisely
the same as here. The division is continued until there is
122 ALGEBRA
no remainder, or until a remainder is found which is of lower
degree than the divisor.
86. Verifying Division., The work may be verified as in
arithmetic, by multipUcation. It is simpler, however, to
test by substituting a particular number for each letter
involved.
Thus, in the preceding problem if we let x=l, the divisor is 5, the
dividend is 15 and the quotient is 3, which shows that the result is
veiy likely correct.
If on substituting particular values for the letters involved, the
divisor becomes zero, other values should be selected.
87. Division with Detached Coefficients. As in multipUcation
the method of detached coefficients may be used.
Ex.— Divide Ux^-x^- 29x^-^12 by 7;r2+3a;-6.
7 + 3-6)14-1-29 + 0+12(2-1-2
14 + 6-12
Check
x= 1
4)-4(-l
-7-
-7-
-17 + 0
- 3 + 6
-
-14-6+12
-14-6+12
Here the first term in the quotient is 2x^, since l4:X*-^lx^ = 2x^
The complete quotient is 2a;^ — a;— 2.
Divide also by the visual method.
EXERCISE 64 (1-6, Oral)
State the quotients in the following divisions:
G.
^ a;2+3x+2 ^
a2
-3a -1-2
a;+l
a-I ■
^ a;2-4
4 5
a2
+2ab+b^
* x+2
a+b
Divide and verify :
7, ijx-+x-l5hy2x-3.
8. e
9. 5a;2— 31a7/+6.y•-by.T-
-6//.
10. S
8. 6x^+xy—12y^hy3x—4y.
9a2+6«6-3562 by 3a+7b.
REVIEW OF THE SIMPLE RULES 123
11. 7a;3+96a;2-28x-by7a;-2. 12. lOOx-^-lSz^-S^by 25x-+3.
13. 3 + 7a:— 6.c2by3— 2a;. 14. 6a-+3o— 31a by 2a— 7.
15. a;3+13a;24-54a;+72bya-+6. 16. 2a3+7a2+5a+100by a+5.
17. x^-\-^x-y+oxij'^-\-y^hyx+y. 18. —x^-\-Zx-y—^xy^-]-y^hyx—y.
19. IGm^'— 46??«2-|-39„i_-9 by S.vi— 3.
20. 6.r3— 29a:22/+18a;?/2+35?/3 by 2a;-7?/.
21. o4+a3_^4ct2_j.3cj^9 by a2-a-i-3.
22. a;«-a;3-6a;2+15a;-9 by a:2+2a;-3.
23. 5x4-4.r3+3a;2+22x+55 by 5a;2+llx+Il.
24. 2.t-3— 8.r+x-*+12-7a;2 by a;2+2— 3a;.
25. 30— 12.r2+x*-.r by a;-5+a;2.
Use detached coefficients to divide :
26. a;3-3.rH3a;-l by a;2-2a;+l.
27. 6a;'*-a;3-lla;2-10a;-2 by 2a;2-3.r;-l.
28. a5-5a3+7a2+6a+l by a2+3a + l.
29. 4.r2+9+a;4+3x+a;3 by 2a:+3+a;2.
30.* (a;2_a--2)(2.r2+a;-l) by 2x-2— 5a;+2.
31. 10.i-3+17a-*— 2.1-5- lLr2_x-+l by 2a;2+a;— 1.
32. Divide a^- 1 by a— 1 and a^+1 by a+1.
x^-\-y^ .r^ — y^
33. Simplify
34. Simplify
.T+y x-y
ffl2+a"-fl"^a2-^+l'
.,. ., , 6x24-a;— 2 3a;2+8x-3 „
3o. Solve — ^— ^ — =11.
2a;— 1 3a.- 1
36. If .r(3a2-a+l) + 2 = 3a3-7a2-L3a, find x.
37. The dividend is a*-i-6a^-r6a2— 9a-f 2, the quotient is a--\-2a—\.
Find the divisor.
38. Divide 3fi-{-x'^y'^-ry^ by x'^—x-y'^-\-y'^ and divide the quotient
by x^—xy-\-y\
124 ALGEBRA
39 Simplify "^^^— ^(Q^+^^)+^<^ i acx^+x{ad+bc)+bd
bx—c cx-\-d
Without removing the brackets divide :
40. ax--\-{b-{-ac)x-\-bc by ax+b.
41. x^-\-{2p—l)x-'rp{p—l) by x-\-p.
42. a^x'^—2abx-{-b^—c^ by ax—{b—c).
43. ahf+{2a^+a)ij^+{a'+2a)y+{a+l) by ay^+ay+1.
88. Inexact Division. As in arithmetic, the divisor may
not divide evenly into the dividend, and so there may be a
remainder.
Thus, 34-^5 gives a quotient 6 and a remainder 4,
.-. :u = 6+4 or 34 = 6 . 5 + 4.
Similarly, when a--\-3a+5 is divided by a+1, the quotient is a + 2
and the remainder is 3.
. a^ + 3a + 5^ 3 *
a+1 ^a+1'
or a2+3o + 5 = (a+l)(a + 2) + 3.
That is, dividend = divisor xquotient+ remainder.
Ex. — Express -^j — — as a mixed expression.
Here the quotient is 1— a;)l +a;*(l + a; + x^ + a;'
\+x + x^+x^ ^-^
and the remainder is 2x*. +a;
. l+x* , , , , , , , 2a;« +a;-x2
1 a; 1 a; ^^t
Divide 1— x^ by l + .r and show tliat -{-x^ — x^
1 3-5 9~5 ■
\-^ = l-x + x^-x^ + x* -,-'''. +a;3 + x4
l+x l+X J^x^ — X*
In such cases the division mav. of
course, be contmued to any number of
terms.
REVIEW OF THE SIMPLE RULES 125
EXERCISE 65
Find the remainder on dividing :
1.* X--— lO.r+25 by a;-7. 2, a2+20a + 70 by a+5.
3. x-3-4x2+5.r+20by a;-l. 4. if -ly-+?,y-l hy y^-y-^rl.
5. x^-\-y^ by x—y. 6. x'^—y^ by x-'f-y-
Express as mixed quantities :
7 ^±? 8 " + ^^ 9 2a -36 ^^ 5x2+7.1-3
x+1 ' * a—b ' ' a+b ' ' x+2
Find four terms in the quotient of :
11. l + (l_x). 12. l+(l+.r). 13. i±^'. 14. l+«^+2«'
1—x 1— a+a-
15. When the dividend is a-— 3a+7, the quotient is a and the
remainder is 7. Find the divisor.
16. Divide x-~5x+a by x—2 and determine for what value of a
the division will be exact.
17. If x^—mx-\-12 is divisible by x—3, what must the quotient be
and what is the value of m ?
18. By division show that
a2+62 26'- a2+62 26^
a—b a—b a+b a+b
EXERCISE 66 (Review of Chapter X)
1. Add z^-2ax^-\-a^x + a^, 3x^ + 3ax^, 2a^-ax^-x\
2. Add ^a-J6, |a-J6, |a+J6.
3. Subtract 8a+2b~~5c from lla-26 + 5c-3rf.
4. Subtract — 3a + 46 — c from zero.
5. Subtract ^a — ^6-f Jc from fa + f6 — fc.
6. How much must be added to ix — ^y-\-^z to produce x — y-\-zl
7. Subtract the sum of 3a + 26, 26 — 3c and 3c — a from the sum of
-6, 6 — c and c — a.
8. Simplify a-(36-4c)-(6 + c-a)-2(a-c).
126 ALOE BRA
9, Subtract 4a;^ — 3a;^— 2;+2 from lx^—6x^-\-2z—l and check
by substituting 2 for x.
10. Find the value of a^ + fo^ + c^ — 3a6c when a = 2, 6 = 3, c= — 5.
11. SimpHfy (a+6 — c) — (6 |-c — a) — (c + a— 6) — (a + & + c).
12. Multiply 1-40;- lOx^ by 1 - 6x + 3x^.
13. Find the product of x+l, x-f 2 and re— 3.
14. Divide the product of x+2, 2a;— 3, 3a;— 2 by 3a;2 + 4a; — 4 and
check when a;= 1.
15.* Multiply a^ + b'^ + c'^ — ab — bc — ca by a + b + c.
16. Find the product of a-2,a + 2,a2 + 4,a*+ 16.
17. Divide a;* + 64 by a;2 + 4a;+8.
18. Divide a;* + a;* — 24a;2 — 35a; + 57 by a;2 + 2x— 3, using detached
coefficients and verify by multiplication using the same method.
19. Find the coefficient of x^ in the product of 3x^ — 2x^ + lx — 2
and 2a;3 + 5a;2+lla;+4.
20. The expression 44a;*— 83a;3— 74a;24-89a;+56 is the product of
two expressions of which 4a;2 — 5a;— 7 is one. Find the other.
21. Divide x* + 4:X^ + 6x^ + ^x+l by a;2 + 2a;+l and check by
substituting a; =10.
22. Subtract ax^ + bx + c from cx^ + clx+f, collecting the coefficients
of powers of x in the result.
23. Find the remainder on dividing a;^+6 by x'—l.
24. Show that
(a-6)(a;-a-6) + (6-c)(a;-6-c) + (c-a)(a;-c-a) = 0.
25. Show that {a-b){b-c){c-a) = a{b^-c^) + b{c~-a-) + c{a^-b-).
26. If a = x^ + 2xy + 2y\ b=x'-2xij + 2y^, c=x*-?/S find the
value of ab — c.
27. Subtract 2x-3(y + 2z) from 3»/-(82-3a;).
28. If s = a + b + c, find in terms of a, b, c the value of
a{s — a) + b{s — b) + c{s-c).
29. Arrange in descending powers of x,
c{ax — b) — x(a — b) + bx{x" — cx).
30. When a = 5, find the value of
2a- |3a-(46 + 2a)} + 5a-(46-a).
REVIEW OF THE SIMPLE RULES 127
31. What quantity when divided by a;'— 2a;+3 gives x'^-\-2x—'i as
quotient and 9 as remainder ?
32. If a=a;2-3a;+2, 6 = 3a;-- 10x-+8, c = 4a;2-9a; + 2, find the
value of (a+26 — c)-h(a; — 2).
33. Arrange in descending order of magnitude and find the average
of: 30, -15, 27, 0, 3, -10, -2, 6, -8.
34. What number must be added to 5a;'— 13a;2 + 2a;— 1 so that
the sum may be divisible bj' x—2 ?
35. Find the coefficient of x^ when \-\-x-\-x'^ is divided by \~x — x-.
36. Divide Zp''-lp{\-p'^)-{2-^p^) by (3p+l)(p+l).
37. If a;2— Ox+c is divisible by a; +4, find c.
38. Find the sum of the coefficients in the sqviare of 2a;- — x — 3.
39. Find the product of x-\-a, x-\-b, x-\-c. Collect the coefficients
of the powers of x in the product. From the result, write down the
product of a;+l, x+2, a;+3 and of a;— 1, a;— 3, a;+4.
40. Divide a;«-2a;3+l by a;2-2a;+l.
41. When o = 3, b = 2\, c=2, find the vakie of
~ + Vlab{2c^-ab) - (2a -36)^.
42. Prove that {l + xf{\ + y-)-{l + x-){\+y)- = 2{x-rj){\-xy).
43. If p = X and q = x^ ^, show that ^^(^^ + 4) = q^.
44. Divide a' + 6' + c' — 3a6c by a-\-h-^c. What are the factors
of a^-\-b^-\-c^ — Zabc ? Compare witli Ex. 15.
45. Multiply x'^-\-bx-\-c by x'^+px-[-q, arranging the product in
descending powers of x.
46. Divide 9a2-462-c2 + 46c by 3a-2^ + c.
47. Multiply .i;^ — a;(a- 1)- 1 by a;2 + aa;+l.
48. Divide a*-166«c* by a-26c.
49. Arrange the product of x — a, x — 6, x~c in descending powers
of X.
50. Divide a^b-\-b'^c-\-c^a — ab^ — bc^ — ca''' by a — b, and divide the
quotient by a — c.
51. What expression will give a quotient of x^-\-\ and a remainder
of 2a-2 — 7a;-f 6 when divided by 3a;*— 10a;- — 6 ?
52. Divide x^ — \j^+Q>y^-\2y~\-^ by x-~y+2.
CHAPTER XI
FACTORING (continued)
89. In Chapter VIII. we have already dealt with the
subject of factoring m simple cases. This Chapter will
furnish a review of the methods already used, and an
extension of those methods to more difficult examples.
90. Type I. Factors common to every Term. When every
term of an expression contains the same factor, that factor
can be found by inspection (art. 58).
Thus, 2xy is a factor of 4:X^y — 6xy^-\-2axy,
.'. 4x^y — (ixy^-^2axy = 2xy(2x~3y-\-a).
Also, x-\-y is a factor of a{x-'t-y)-'t-b{x-\-y). When this expression
is divided hy x-\-y, the quotient is a + 6,
.-. a{x + y) + b(x + y) = (x + ^j){a + b).
91. Type II. Factors by Grouping. When every term has
not a common factor, if the number of terms be changed by
grouping, we may sometimes obtain a common factor.
Ex. 1. — Factor mx-\-nx-\-my-\-ny
(2)
(1)
mx -\- tix + 7ny + ny,
= x{in-\-n)-\-y{'m-{-n),
= {m+n){x+y).
mx -\- nx + my + ny,
=m{x+y)-\-n{x-{-y),
= {x+y){m+7i).
Here we changed from four terms to two, and we found a common
factor in the two terms. The other factor of the expression was then
found by division.
The two solutions show that different methods of grouping may be
employed. If the first method tried is not successful, try others.
128
FACTORING 129
Usually those terms are grouped which contain a simple common
factor. In the example we should not expect to be successful by
grouping mx with ny, as these terms have not a common factor.
Ex. 2.— Factor x^-\-x^-\-2x+2.
Use two different methods of grouping and obtain the factors x-\-\
anda;2 + 2.
Ex. 3. — Factor {a—h)^—ax-\-hx.
{a — h)^ — ax-\-hx={a—b)''- — {ax — hx),
= (a~b)^-x{a-b),
= {a — b)(a — b — x).
Note. — When quantities are enclosed in brackets, the pupil must
not forget to verify by mentally removing the brackets.
EXERCISE 67 (1-9. Oral)
Fac
1.
tor:
3.r-27.
4.
&2_56.
7.
a{x+y)+b{x-iry).
2.
2ffl-6.
5.
3a2_i5o6.
8.
p{m—n)-\-{m—n).
3.
a 2- 3a.
6.
6x'-i/—l2xi/'~.
9.
x{a—b)—2y{a—b).
Factor, using two different methods of grouping and verify by
multiplication :
10.
ax-\-bx-\-ay-\-by.
11.
am—bm \-nn—bn.
12.
x^—ax-\-bx—ab.
13.
bx —ax-\-ab— x ''.
14.
2ac + 3ad-2bc-3b(l.
15.
x^-j-x^-JrX+1.
16.
a^-a^-3a+3.
17.
x-y-xy+1.
18.
x^+^x^-3x-12.
19.
a3-7a2-4a+28.
20. Factor x^-\-r*—x^—x--{-x-\-l by making three groups each
containing two terms, also by making two groups each containing three
terms.
21.* Find three factors of 3x^—6x--{-3x—Q and of axy—ay—ax-\-a.
22. Find a common factor of
am-\-bm-{-av.+bn and ax-\-ay-{-bx-\-bi/,
and of x^—x^+x—l and x^—x^-i-2x~2.
23. Factor l0x^—5xy—Qxz+3yz and a^b+a^c—3a^^—3ab"C.
K
130 ALGEBRA
24. Find a common factor of 2x^ —6x^ —3x -\-9 and x^—3x^-{-2x—6,
and show that it is a factor of their difference.
25. Factor ■iSax—56ay-35bij+30bx.
26. Factor (x+y)^+4:X+4:y and 2(a-by—a+b.
92. Type III. Complete Squares. We have already seen
in art. 64 how the square of a binomial may be written
down. We have also seen in art. 65 how the square root of
a trinomial may be found, when the trinomial is a perfect
square.
93. Square of a Trinomial. A trinomial may be squared
by expressing it in the form of a binomial or
hy multiplication.
Thus,
(a + 6 + c)2=[a + (6 + c)|2,
= a2_|_2a(6 + c) + (6 + c)2,
= a^ + 2ab + 2ac + b^ + 2bc + c'^,
:. {ft + b + cy = a^ + b^+c^ + 2ab + 2ac+2bc.
Multiply a + 6 + c by a-^-b + c in the ordinary way and compare the
results. Examine the diagram and see that the same result is obtained.
Similarly, (^a + b~c)^={a + {b-c)}'^,
and (a_6_|_c)2=[a-(6-c);2.
Complete these two in a manner similar to the one worked in full.
If we examine these products we see that they consist of
two kinds of terms, squares (a^, 6-, c^) and double products
{2ah, 2ac, 26c).
We might express the result thus :
The sq7iare of any expression is equal to the svm of the squares
of each of its terms, together ivith twice the svm. of the products
of each pair of terms.
In writing down the square, care must be taken to attach
the proper sign to each double product.
Ex.1. (2.T-32/ + 42)2
= 4x^ + 9y"+\6z^-\2xy+\ Gxz - 24yz.
2
a
ab
ac
ab
b"
be
ac
be
c^
FACTORING
131
Ex.2. (a-26+c-d)2
= a2 + 462 + c2 + d2-4a6 + 2ac-2ad-4fec + 46rf- 2cd.
Ex. 3, Factor a;^+4y- + 22—4a:^ — 2.r2 4-41/2.
This is evidently the square of an expression of the form x+2j±z
Which of these when squared will give the proper arrangement of signs ?
Verify by writing down the square.
9.
x+h
13.
a — h — c.
10.
2a -\.
14.
a + 6+c+cZ.
11.
x+y—z.
15.
p—q—r-\-s.
12.
x~y+z.
16.
x-y^-z-\.
^xy
+ 9;/2. 19
. 4a;2+4a;+].
EXERCISE 68 (1-33, Oral)
What are the squares of :
1. m-\-n. 5. x—2y.
2. m—n. 6. 4x—y.
3. 3a;+2. 7. 2a-3b.
4. .3a— 5. 8. 3?n—5n.
Express as squares :
17. a;«+4a-?/+4?/2 18. a
20, 4a2_20a6+2562. 21. 9a^-l2ab+ib^. 22. mhi^-8mn + ie.
23. 4rt2+2a+i. 24. l-10a + 25a2. 25. a*+2a^b^+b*.
What are the square roots of :
26. a;2?/2-Ul0.ry3+2oz2. 27.
28. 4-20a2+25a*. 29,
30. m^+n^+p^+2mn+2mp+2np,
31. a^+b^+c^—2ab+2ac—2bc.
32. a;2+42/2+22+4a:2/+2xz+42/z. ;
33. 4a2+624-9c2— 4a6— 12ac+66c.
Simplify :
34.* (3a;-2/)2+(a;-32/)2+(2x+3i/)2.
35. (a-6)2+(6-c)2+(c-a)2+(a+6-fc)*.
36. (x2+a;+l)2+(x2-x+l)2.
37. (a-6+c)2-l-(6-c+a)2+(c-a-f6)2.
38. (3a;-2?/+z)2-(a;-22/+32)2
K 2
16x2-24x!/-f 9j/2.
(a+6)2-2c(rt+6)+c2.
132 ALGEBRA
Complete the squares by supplying the missing terms :
■^ 39. a;2— +25. 40. 4cX^+ +25y^
41. a^+4:ab 42 — Uimi+dn^
43. a^+%"+ —6ab—2ac
44. 9a;2+ + — Qxy—I2xz
^'45. Find three factors of 3a;2+6x+3 and of a^+ia^b+Aab^.
46. Factor
(a+6)2+4c(a+6)+4c2 and {a+b)^-2{ai-b){c+d)+{c+d)\
47. Show that the square of the sum of any two consecutive integers
is less than twice the sum of their squares by unity.
48. Divide the sum of the squares of a—2b-\-c, 6— 2c+a,
by the sum of the squares of a—b, b—c, c—a.
49. If .r + - = 4, find the value of x^ -\ — -.
X x^
50. Factor {ax-\-hy)"-\-{bx—ayY-^c~{x'^-\-y'^).
51. Express a^x-'+fe'^'+a^^/^+ft^x^ as the sum of two squares.
52. Find the value of x^-\-y^-\-z~-\-2xy^2xZ'^2yz, when
x=a^2b—^c, y=b+2c-3a, z=c+2a—3b.
94. Type IV. The Difference of Squares. The product of
the sum and difference of the same two quantities is equal to
the difference of their squares (art. 66).
Conversely, the difference of the squares of two quantities
is equal to the product of their sum and difference (art. 67).
Or, in symbols, {a-{-b){a—b)^a^—b^,
and a^-b^={a-\-b){a-b).
95. The formula for the product of the sum and difference
may sometimes be used to find the product of expressions of
more than two terms.
Ex. 1.— Multiply 2a—b-\-c by 2a— b—c.
Here 2a — 6 + c is the sum oi2a — b and c,
and 2a — b — c is the difference of 2a — b and c.
•♦ $
FACTORING 133
They might be written (2a — b)-\-c and (2a — 6) — c. The product
is therefore the difference of the squares of 2a— 6 and c.
.-. 42a-6 + c)(2a-6-c) = (2a-6)2-c2,
9 =4a- — 4a6 + ^" — C-.
Ex. 2. — Find tl^ product of 2x-\-y—z and 2x-— y-j-z.
Here the first expression = 2a; +(2/ — 3),
and the second =2x—{y~z),
:. the product ={2xY — {y — z)-,
= 4a;^ — (2/^ — 2?/2 -\-z^),
= 4:X^ — y^-\-2yz—z^.
Verify by ordinary multiplication.
Ex. 3. — Multiply a—b-\-c—d by a-\-b—c—d.
Note that the terms with the same signs in the two expressions are
a and —d. These should be grouped to form the first term in each
factor.
a — b-\-c — d — (a — d) — (b — c),
a + 6 — c — d = (a — c/) + (6 — c),
.". the product = (a— rf) 2 — (6 — c)^.
Simplify this result and verify by multiplying in the ordinary way.
Ex. 4. — Factor ;:>-— 4^jq'+4g'2— a;2.
Here the first three terms -form a square and the expression may
be written :
(p2 _ ^pq _j. 4^2) _ ^2 _ ^p _ 2q)^ — x^,
= {p — 2q + x){p—2q-x).
What two quantities were here added and subtracted to obtain
the factors ?
Ex. 5.— Factor a^-h'^+2hc-c'^.
Here the last three terms should be grouped to form the second
square.
.-. a2-62 + 26c-c2 = a2-(62-26c + c2),
= a2-(6-c)2,
= {a+{b-c)\\a-{b-c)\,
= {a-{-b — c){a~b-\-c).
Verify by multiplication.
134 ALGEBRA
Ex. 6.— Factor x^-^ij'^—a-—h^^2xy-^2al).
Evidently three of these terms form one square and the remaining
three the other square.
The expression ={x'^-\-2xy-{-y^)—{a^ — 2ab-{-b'^),
= {x + yY-{a-b)\
= {x-\-y^a-h){x+y-a + b).
EXERCISES 69 (1-10, 17-32, Oral)
Use the formula to obtain the following products :
^ 1. (2a4-3){2a-3). 2. (4x-l)(4x4-l).
3. (a;y/+5)(x7/— 5). 4. {ah~-c){ab^-c).
5. {2m^-\'^n)(2m'^—'in). 6. {abc-\-x!/){abc—xy).
7. (x + iKx-l). 8. (.x2-y2)(.,;2^_y2).
9. {x-'ry-\-z){x-\-y—z). ). 10. {a—b—c){a — b~[-c).
11.* {a+b-c){a-b^c). 12. {2x+3y-5){2x + :\y+5).
13. {p-2q+3r){p+2q-3r). 14. {l-X']-x-){l+x+x-).
15. {a+b—c+d){a—b—c—d).
16. (ri-26-fc-2rf)(n— 26-c+2(Z).
Factor and verify :
17. ^2-9.
20. a^b^—x^.
23. l-a~b^
26. (x+?/)2-25.
29. (a-f6)2-(c-d)2.
32. a2_2aft_^62_c2_
35. (4.r-f-3)2-16x-2.
38. a2-l-62-f2a&— c2-
40. a2_2a_^i_62^26c-c2.
42. 4x2— 4x-i/2 4-4a?/-4a2-(-l. 43^ i_^a2_62_4c2^4jc_2a,
44. Find three factors of 2x-2— 8, a'^—a, a*— x"*.
18.
4x2—25.
19.
a2-462.
21.
16x2-0?/2.
22.
932-42.
24.
25 -x*.
25.
(a-6)2-c2.
27.
c2-(a+6)2.
28.
x^—{y—z)K
30.
(a+26)2-4c2.
31.
x^~Ar2xy+y^—a\
33.
a2_^,2_c2-26c.
34.
a^-b^-4:c^+ibc.
.36.
l-x2+2x//-2/2.
37.
a^-x^^2ay+y^.
-d2-
-2crf. 39. a2-,
&2 + c2
-d^-2ac-2bd.
\bc-
■c2. 41. X*-
x2-4
-2xhf--4x+y*:
FACTORING 135
45. Find three factors of 5a^— 10a6+56-— 20c- ami of
(.r-36)3-462.i; + 1263.
46. Find four factors of a~b'^—a-c~—b-d'--{-cH- and of
47. Factor a-x'— 6-(/-+2ac.c+c^ and m^—9m^n'^-\-n^—2mn.
48. Find the simplest factors of 'ix^—2x'^~^x-\-2 and of
49. Simplify (a— 6-[-c)(a— 6— g)-(-(«+6— c)(a— 6+c).
50. AiTange x%x^—a-)—y'^{y'-—a-)-\-2xy{x'^—y-) so as to show
that x^—y'^ is a factor of it, and thus find the simplest factors.
51. Use factoring to simplify :
(1) {a^~3ai-l)^-(a--'Sa)'.
(2) {x-2y+Szy'-{3z-x+2yf.
(3) (a2_3rt_4)2_(a2^4)2.
(4) {5x^—2xy-^y-r—(5x-'+2xy~yY".
52. Multiply a-f 6+c by a + b—c and a— ft-fc by a—b—c and use
the results to obtain the product of
(a+6+c)(a4-6-c)(«— 6+c)(a-6-c).
53. Show that x{y"—z-)^y{z^—x~)-\-z{x''^—y-) is equal to
{'V-y){i/—z-)—{x-—y''-){y—z)
and then find the factors of this expression.
54. Arrange a{b'-—c-)^b{c~—a-)-\-c{a-—b") hi the form
a(6-— c-)— 6c(6— c)— ft-(6— c)
and thus obtain the factor b—c. Find the other two factors.
96. Type V. Incomplete Squares. We have already factored
many expressions which were seen to be the difference of two
squares.
Sometimes the two squares of which an expression is the
difference are not so easily seen.
136 ALGEBRA
Ex. 1.— Factor x'^-\-x^y^+y'^.
This expression would be the square oi x^-\-y' ii the middle term ^W6re
2x^y^. We will therefore add x^y^ to com.plete the square and also
subtract x^y^ to preserve the value of the expression.
Then x'^+x^y^+y*=x'^+2x^y^+y^-x^y^,
= (x^+y^)^ — (xy)^,
= {x^ + y''-\-xy)(x^-\-y^—xy).
In order that this method may be successful, it will be
seen that the quantity we add to complete the square must
itself be a square.
Thus, to change a^-{-ab-\-b^ into a^-\-2ab-\-b^ — ab is of no value as
06 is not an algebraic square.
Ex. 2.— Factor a'^-\-4:M.
This can be made the square of a^-\-2b^ by adding 4a'b^.
Complete the factoring and verify by multii^lication.
Ex. 3. — Factor im^— IGm^n^-f 9n*.
What must be added to make it the square of 2m^ — 3n^ ? Complete
the factoring.
Try to factor it by making it the square of 2m^-\-3n~.
Ex. 4.— Factor a^-^b'^^c^-2a^b^-2b^c^-2cW'.
How does this expression differ from the square of a^ + fe^ — c' ;
ExjDress it in the form {a--\-b^ — c^)- — 4:a-b^.
Write down the two factors and see if you can factor each of them
again and tlius obtain the result
{a+b + c){a + b-c)(a-b + c)(a-b-c).
EXERCISE 70
Factor and verify :
1. a*+a^-+l. 2. .T*+.r2+25. 3. x^+lx^+lG.
4. x*+2xhj^+Qy\ 5. 4a* + l. 6. dx*+8x~y^+iey*.
7. 46^-1362+1. 8. 9a*- 15^2+ 1. 9. da'^-52a%''+Ub\
FACTORING 137
10. 25.r'*-89.r2j/2+64^«. 11. x'^+y'^-\\xhj\ 12. x^-lx^+\,
13.* Find three factors of 2a;*+8 and x^+x^-\-x.
14. Find four factors of Qa'^-lQa^b'^+b'^.
15. Find three factors t)f a:®4-x*+l.
16. Find four factors of a'^+h'^-\-c'^—2a-b'—2b-c-—2c~a- by
completing the square of a^—b^-\-c^.
17. Factor (a + l)H(«--l)-+(«-l)*.
97. Type VI. Trinomials. We have already dealt with
the factoring of expressions of the type x'^-\-j)x-\-q, where the
coefficient of the first term is unity (art. 61).
We now wish to factor expressions of the type mx^+^xx'+g,
where m is not necessarily unity.
98. First Method, by Cross Multiplication.
Ex. 1.— Factor 2x^+lxy+^^-.
The product of the first terms of the factors is 2x^, and therefore
the first terms must be 2x and x ; similarly, the last terms must be
Zy and y and the signs are evidently all positive.
.'. the factors must be
2x + Zy 2x+ y
x-{- y or x+Sy
It is seen, by cross multiplication, that the coefficient of xy in the
first product is 3 + 2 = 5, and in the second is 1 + 6 = 7.
.'. the correct factors are {2x+y){x-\-3y).
Ex. 2.— Factor S.r^-T.r-G.
Here the numerical coefficients of the first terms of the factors must
be 3 and 1, and of the last terms may be 6 and 1 or 3 and 2.
Since the third term is negative, the signs of the second terms of the
factors must be different.
The possible sets of factors, omitting the signs, are :
3x 3 3a; 2 3x 6 3a; 1
x 2 x 3 X I X G
138 ALGEBRA
Since the signs are different for the last terms, when we cross
multiply to find the coefficient of x in the product, the partial
products must be subtracted.
It is easily seen that the second arrangement is the only one from
which Ix can be obtained.
Since the middle term is negative, tlie larger of the cross products
must be negative.
.'. the factors are (3a;+2)(a;— 3).
This method is Hable to be found tedious when the coefficients
have a number of pairs of factors, but in ordinary cases the
puj^il will find little difficulty after he has had some practice
in the work.
99. Second Method, by Decomposition. In the process of
multiplying two binomials like 2,<-l-3 and 3x-{-5, we have
(2.r+3)(3^+5)-3,r(2.r+3)+5(2x+3),
=-6a;2+9.r+10x+15,
-6.r2+19:r+15.
If we wish to factor a trinomial like 6a;'^-t-19a;-|-15, we
may do so by reversing the process.
Thus, Qx^+ 19.T+ 15=6.rH9.r+ 10a:+ 15,
= 3a;(2.c+3)+5(2.r+3),
= (2a;+3)(3.c+5).
The only difficulty in this method is in finding the two
terms into which the middle term, 19.r, should be decom-
posed. This difficulty may be overcome in the following waj'^ :
{ax-\-b)icx^d)=acx^+x{ad+bc)-\~bd.
Note that the product of the two terms in the coefficient
of X, ad and he, is the same as the ]iroduct of the coefficient
of x^, ac and the absolute term, bd.
In the trinomial 6;r2-|-19.r+15 above, the product of 6 and
15 is 90 and the two factors of 90 whose sum is 19 are 9
and 10, which shows that the middle term, 19a;, should be
decomposed into 9x+10a;.
FACTORING 139
Ex. 1.— Factor 6,r2+13a:+6.
The product of the coefficient of x^ and the absokite term is 36.
The two factors of 36 whose sum is 13 are 4 and 9.
.-, 6a;2_^13.T+6 = 6:c2 + 4a; + 9x+6,
= 2a;(3a;+2) + 3(3a; + 2),
= (3a; + 2)(2a;+3).
Ex. 2.— Factor \2x^—\lx-5.
Here we require two factors of —60 vvliose sum is —17, and tliey
are evidently —20 and 3.
.-. 12x2-17a;-5=12a;2-20a; + 3a;-5,
= 4a;(3a;-5) + (3x-5), »
= (3a;-5)(4x+l).
EXERCISE 71 (1-18, Oral)
Factor and verify :
1. a;2+4a;+3. 2. rtHll«+30. 3. ?/2-f8«/-fl5.
4. a2— lla+18. 5. x'^— 14a;+48. G. l+5a;+6a;2.
7. a;2-15.c+14. 8. aW^-5ah^Q. 9. a^-\5a+m.
10. l-21a;+38x2. 11. x'^-Qxy^Sif. 12. a2_i3f^ft_^3662.
13. a;2-4x-5. 14. a2_ga_22. 15. x--2Sx-2^.
16. ?/2_4y_2i. 17. l-2a-15a2. 18. a^-ay-2y\
19. 2a;2+5a;-f-3. 20, 4x-+8xy+3yK 21. Oa^— l8a6 + 862.
22. 8x2+a;— 9. 23. 3x'—x—2. 24. Ga^— a— 2.
25. 4x2+a;-5. 26. 1562-196-8. 27. 10a;2-23a;-5.
28. 1062-896-9. 29. {}x^-3lxy+12y\ 30. 10a2-29a6-f-1062.
Find the sir. ■ lerj'^ factors of :
31.* 3x2-3a:-21G. 32. 2a2+8a+6. 33. x-Bx^+Qx"^.
34. a:«— 5x2+4. .35. a^-~\0a^+9a. 36. 9a*— 10^2+1.
37. (a;2+4x)2-2(.r2-|-4.r)-15. 38. (.r2_9x)2+4(x2-9a;)-140.
39. Without multiplying show that
(x2-a;-2)(x2+2x--15)=(x2+6x+5)(a;2-5a:+6).
40. An expression is divisible by x-—2, the quotient being x-—x—Q.
Show that it is divisible by x-\-2 and find the quotient.
140 ALGEBRA
V 41. Show that the product of 6x^~ldx-\-6 and 2x~—lx-\-5 is
divisible by 3x^—5x-\-2 and find the quotient.
42. If 3x^-\-ax—l4: is the product of two binomials with integral
coefficients, find all the different values that a may have.
43. By factoring, find the quotient when the product of
6a2+7a6-2062 and 22a^-13ab-Ub^
is divided by 4a-— 4a6— 356'.
44. Factor x''^-'r5xy-\-iy'^-^x-\-y.
45. Factor 3a^—ab—2b^+6a+4:b.
J^[^ 46. Divide the product of x^-\-3x-\-2 and x*—l by the product
of a;24-2x+l and x'^+x—2.
100. Type VII. Sum and Difference of Cubes.
Divide x'-^-}-y^ by x^y, x+y)x^ +y^(x^-xy+y^
and x^—y^ by x—y. ^ +^ ^
and x^-if^{x-y){x^+i)cy-^y^). ~^'^~^^''
Examine carefully the signs in these ^^^al^^-i
factors.
xy^^-y'
It is thus seen, that the sum of the cubes of tiuo quantities
is divisible by their sum, and the difference of the cubes is
divisible by their difference.
The quotient in each case consists of the square, product and
square of the terms of the divisor, with the proper algebraic signs.
Ex. 1.— Factor 8a^+27b^.
Here 8a^ = {2a)^ and 2763 = (36)3,
.'. the expression may be written (2a)3-|-(36)3,
.'. the first factor is 2a + 36 and the second is
(2a)2-(2a)(36) + (36)2 or 4a2-6a6 + 962.
.'. 8o3 + 2763 = (2« _,_ 36)(4a2- 6a6 + 962).
Ex. 2. — Factor a^x^—My^.
a3a;3— 64?/* = (a.r)3 — (4!/")3,
= (ax—iy'^){a-x- + 4:axy'^+lQf/*).
FACTORING 141
Ex. S.^Factor x^~y^.
This may be expressed as the difference of two squares or of two
cubes.
• .x-«-?/« = (x3)2- (1/3)2, OT {x-)^-{y-Y,
= {x^-\-y^){x^ — y^), or {x- — y^)(x*-\-x-y^'\-y^).
Complete the factoring by each method and decide which you will
use, if you have the choice, as here.
EXERCISE 72 (1-12, Oral)
State one factor of : ' ^
1. a'^+h^. 2. .r3-L8. 3. x'^-21. 4. lOOO-a^.
5. x^-Uy^. 6. 27-fe3. 7. Sa^Ar\25. 8. \25a^-%b^.
9. 1-27x3. 10. 343.1-3-8. 11. {a+bf+cK
12. (a-fe)3-c3.
Factor and verif}^ 13-21 :
13. a3_^27. 14. .r^-S^/^.
16. 27.r3-64;/3. 17. 8-27a3.
19. a^+h^. 20. x^-b^.
22.* 2rt3-16. 23. 81-t-3y3.
25. a36+6*. 26. a^+b^.
28. (.T-2)3+8. 29. (a-6)3+ff3.
31. What is one factor of {2x—y)^—{x—2y)'^ 1
32. Show that (2a-Zbf^{^a-2bf is divisible by a-b.
33. Factor (a2-26c)3+863c3 and 21xhfz-y^z*.
34. Find six factors of a^^ — 6 ^2.
35. Find two binomial factors of (2.r-— 3x'-f-3)3— (.r^— 2.r-t-5)3.
36. If X + 1 =2, find the value of x^ -}- 1 .
x x^
37. By factormg show that {a+bY-^ab{a+b)^={a-^b){a^-^h^).
101. Type VIII. The Factor Theorem. \Vliat are the values
of 0x5, axO, Ox(— 4), 1x0, -1000x0?
15.
8a.3 + l.
18.
1000.f3-2/3.
21.
a^— ^«.
24.
«*+«.
27.
{x^yf+a^
30.
(a-6)3+(a-|-6)3
142 ALGEBRA
If one of the factors of a product be zero, the product
must also be zero.
If the product of two numbers be zero, what can Ave infer ?
If a6 = 0, it follows that either a=0 or 6=0.
If (a;— 3)(a:— 4)=0, then either a;— 3=0 or a;— 4=0.
Since {x—2){x^—lx+\2)=x^-Qx^+2Qx-2i:,
.'. .r^— 9.r"^4-26.r— 24 must be equal to zero when .r=2, for
then one of its factors, x—2, is zero.
If we substitute 2 for x, we see that this is true.
x3-9a;2 + 26a;-24=23-9 . 22 + 26 . 2-24,
= 8-36 + 52 -24 = 0.
Conversely, when any expression becomes zero when a;=a,
then x—a is a factor of it.
Substitute .t = 3 in x^ — Qx'^-\-\\x—% and it becomes
33-6 . 32+11 . 3-6 = 27-54 + 33-6 = 0,
a; — 3 is a factor of a;3 — 6x2+ 11a;— 6.
Divide it by x — 3 and the other factor is aj2 — 3a; + 2.
.-. a;3-6a;2+lla;-6 = (a;-3)(a;2-3a;+2),
= {x-3)(a;-2)(a;-l).
If a;+l is a factor of an expression, the expression must be equal to
zero when x= — 1 , for then a; + 1 = 0.
Thus, a;+l is a factor of a;' — a;2— lOx— 8, since
(-l)3-(-l)2-10(-l)-8=-l-l + 10-8 = 0.
Divide by a;+ 1 and complete the factoring.
Any expression is divisible by x—a if it vanishes (becomes
zero) when a is substituted for x.
This is called the factor theorem.
Show that x — a is a factor of .-c^- 7aa;2+ lOa^a;- 4a'.
Show that x + a is a factor of 5x^~\-6x^a+\lxa^+l0a^.
Ex.— Factor .r^— 9a:+10.
If it has a binomial factor it must be of the form
x+1, a;±2, x±5ora;+10.
Testing for these factors we find that a; — 2 is a factor,
.-. x»-9x+10 = (x-2)(x2 + 2x-5).
The factoring is complete as x^-\-2x— 5 has no simple factors.
FACTORING 143
102. Special Case. It is easy to see when x— 1 is a factor
of any expression, for when 1 is substituted for x, the vahie
of the expression becomes equal to the sum of its coefficients.
Thus, ifa;=l, a;3-2a;2- 19a;+20,
= 1 -2 -19 +20 = 0,
.'. X— 1 is a factor. Complete the factoring.
Similarly, x — a is a factor of 'ix^—\Q)X^a—lxa^-[-20a^, since
3-16-7 + 20 = 0, and a-b is a factor of a^-Qa^b + Zah^ + 2b^, since
1-6 + 3 + 2 = 0.
EXERCISE 73
Each of these expressions is divisible by x—\, x—2 or .r- 3. Find
all the factors of each and verify.
3.r2-12x+14.
4a;2+a;+6.
4a-3-9.r2-10a;+3.
1. a;3— 10x2+29a;— 20.
2,
a;'
3. .r3+5x2-2a:-24.
4.
X
6. 2.i-3-7.r2+7.T-2.
6.
4:
Factor :
7.* 2a;3-ll.r2+5a;+4.
8.
X
9. .x-3— 7.r+6.
10.
x^
11. a3+a2— lOa+8.
12.
a
-2x2— .T+ 2.
19^+30.
-3a62_263.
13. Show that x-\-2 is a factor of a;^— .r-— x+lO.
14. Show that x+o is a factor of x^-\-lx^a-{-Qxa^Ar'ia^.
15. Show that a:+3, a;+4 and x—1 are the factors of x^— 37.r— 84.
16. If x^— 10a;+a is divisible by x+2, find a.
17. Show that a—b is a factor of a^-\-Aa%-\-ab'^—(Sb^, and find all
the factors.
18. Noting that x^— 2x-3 = (x+l)(a;— 3), show that x^— 2.r— 3
is a factor of x*— 4x3+2x-+4x— 3.
19. Show that a—b, b—c and c—a are factors of
a(62_c2) + 6(c2_a2)+c(a2-62).
20. If x—1 and x— 2 are factors of x^- 5x2+a.T+6, find a and b.
21. If px^—3x^+qx—\0 and qx^-\-2x^—l'Jx+p are both divisible
by X— 2, find p and q.
144 ALGEBRA
103. Equations Solved by Factoring. We have seen that if
(.r-3)(.c-4) = 0,
then a;— 3=0 or a;— 4=0.
Thus the equation (x— 3)(x-— 4)=0 is equivalent to the two
simple equations x— 3=0 and x— 4=0.
But if a;— 3=0, .r=3,
and if .r— 4=0, .t=4,
.'. the roots of the equation (.r~3)(.r— 4)=0 are 3, 4.
The truth of this may be seen by substitution.
Ifa; = 3, (x-3)(x-4) = (3-3)(3-4) = 0x-l = 0.
Ifa; = 4, (x-3)(a:-4) = (4-3)(4-4) = lx 0 = 0.
Since (a;-3)(x-4) = a;2_7a;-I-12,
the given equation may be written
.T2-7.r+12 = 0.
104. Quadratic Equation. Any equation which contains
the square of the unknown and no higher power is called a
quadratic equation or an equation of the second degree.
The preceding shows that if we wish to solve a quadratic
equation we may do so by finding, by factoring, the simple
equations of which it is composed.
The simple equations when solved will give the roots of the
given quadratic equation.
Ex. 1.— Solve x2-6.r-7 = 0
Factoring, {x—7)(x+l) = 0,
:. a;-7 = 0 or a;+l = 0,
x = l or — 1.
Verification: if a; =7, x2-6a;-7 = 49-42-7 = 0,
iix=-l,x^-6x-7= 1+ 6-7 = 0.
Ex. 2.— Solve 3x^+1 x=6.
Transpose the 6 so as to make the right-hand side zero, as io the
previous problem
3.T2 + 7.r-6 = 0,
(3.'r-2)(a;+3)=0,
.-. 3.T-2 = 0 or a;+3 = 0,
.T=5 or —3.
Verify both of these roots.
FACTORING 145
Ex. 3. — Form the equation whose roots are 2 and —5.
The required equation is at once seen to be a combination of the
two simple equations
X— 2 = 0 and a;+5 = 0,
and therefore is (a;— 2)(a;+5) = 0,
or a;2+3a;-10 = 0.
Ex. 4. — If x=2 is a root of the equation
.T3+3a;2-16.T+12=0,
€nd the other roots.
Since x=2 is a root, then x — 2 is a factor of a;^ + 3a;-— 16a;+12 and
the other factor, found by division, is x^-\-5x — Q.
:. x^ + 3x^-l6x+12={x-2){x-l){x+6) = 0,
«— 2 = 0 or a;— 1 = 0 or a;+6 = 0,
a; = 2 or 1 or —6.
.'. the other roots are 1 and — 6.
EXERCISE 74 (1-16, Oral)
To what equations of the first degree is each of the following
equivalent :
1. (a;-l)(a;-2)=0 2. (a;-3)(.-c+5)=0.
3. a;(a;-5)=0. 4. (a;-l)(a;-2)(a;-3)=0.
5. a;2— 4=0. 6. a;^— 4a;+3=0.
7. a;2+5a;+6=0. 8. a;2-a;-20=0.
9. x^+3ax+2a-=0. 10. x^—bx—12b^=0.
State the equations whose roots are :
11. 2 and 3. 12. 4 and -5. 13. -2 and -4.
14. a and 6. 15. 2, 3 and 1. 16. 4, 5 and —6
Solve and verify :
17. x'-8x+15=0. 18. x2+8.r+15=0.
19. x2+2a;-15=0. 20. a;2-2x-15=0.
21. 3x2-8x+4=0. 22. 4a;2-2a:-2=0.
23. 2x2+a;=15. • 24. a;(3a;— 1) = 10.
25. a;3-a;=0. 26. x^~ax+bx-ab=0.
146 ALGEBRA
27. If x=2 is a root of a;^— 19a;+30=0, find the other roots.
28. Solve x^-6x^+Ux-Q=0 and ix^~l2x'^ + UxS^0.
(Note that the sum of the coefficients is zero.)
29. The sum of a number and its square is 42. Find the number.
30. The sum of the squares of two consecutive numbers is 61.
Find them.
31. The sides of a right-angled triangle are x, x-\- 1 and x-\-2. Find x.
105. Notes concerning Factoring. The subject of factoring
is one of the important parts of algebra, as it enters into so
many other processes. We have already had examples of
its use in solving equations and m performing operations on
fractions.
In the preceding exercises, in this Chapter, the expressions
to be factored have been classified for the pupil. In the
practical use of factoring, however, he must determine for
himself the particular method to be used.
This is usually done by determining the type or form
to which the expression belongs. The examples in the
review exercise which follows will give the required practice.
The types which have been discussed in this Chapter are
here collected for reference :
I. ax-\-ay. (Common factor in every term.)
II. ax+ay-\-hx-\-hff. (Factored by grouping,]
III. x^±2xy-\-y'^. (Complete squares.)
IV. a^—b^. (Difference of two squares.)
V. x^+x^y^^y^. (Incomplete squares.)
VI. ax^-{-bx-{-c. (Trinomials.)
VII. x^ ± y^. (Sum or difference of cubes. )
VIII. Factored by the factor theorem.
FACTORING 147
EXERCISE 75 (Review of Chapter XI)
1. State the squares of a-\-b, a — b, x—3y, 2a;— 1, 3x— 5, 5a + 26,
3x—4y, 7a — 3, a^—l, a-{
2. State the squares of a-\-b-\-c, x-{-y~z, a — b — c.
3. Writedown the products of x{a — b), a{a — b-\-c), {x-\-\){x+l),
(a;-3)(x-5), (2a;-3)(2a;+3), (ax -(S){ax -{-!).
Use short methods to find, in the simplest form, the value of :
4.* (x+a+6)(a;+a-6) + (a;-a-6)(x-a + 6).
5. (x2 + a;+l)(x2 + a;-l)-(x2-a;+l)(a;2-x-l).
6. (o + 6 + c)2 + (a + 6-c)2 + (a-6 + c)2 + (&-6-c)2.
7. (2a+36-c)2 + (3a+6-2c)2 + (a-26 + 3c)2.
8. 99992-99982. 9. 57432-42572.
10. 503x497-502x498. 11. (a + 99)2-(a + 98)2
Find the simplest factors and verify 12-29:
12. a;2-a;-42. 13. a;3-3a;2-a;+3
14. a;3-4x. ^ 15. a2 + a-56.
16. x^-a+ax-x. 17. 27x2-127/2.
18. a;» + 5a;2-4a;-20. 19. a;=»-3x2+2a;.
20. 15a2+32a + 9. 21. 343-a;».
22. a;*-4x2. 23. 18a;2 + 48x+32.
24. (a;-3)2 + (a;-3)(x+4). 25. 15a;2- 15^/2- 16a;2/.
26. l + 2a6-a2_62. 27. x{x-2) + y{x-2)-x-\-2.
28. abc^ + a^cd->rabd^ + b~cd. 29, 25x«2/-40a;»i/2+ 16.^2^/3.
30. 4(x-2)2-x+2. 31. 24a*-3a63.
32. (a+26-3c)2-(3a+26-c)2. 33. 12x2-x-20.
34. 108o*-500. 35. x'^ + x-y^-[-y.
36. x*-7x2-18. 37. x^-y^-2x^y + 2xy'^.
38. x2-x2/- 1327/2. 39. (a + c)(a-c)-6(2a-6).
40. a;3^y3_j. 33,^(3, _^^) 41_ ax2-x(3a6 + 2) + 66.
42. a2-462_3a-66. 43, 2x{2x + a) -y{y-\- a).
44, a^ + 2ab + b^^ac-\-bc. 45. a2-2a6-f 62-a+i;.
T '>
148 ALGEBRA
46, x^+x^-\-x—y^—y^—y. 47. a^b-a^b^-a^b^-^abK
48. 4a2-2562+2a+56. 49. 8(a + 6)3-(2a-6)3.
50. a;Hy*- 18x22/2. 51. a«-o2-9-2a262_|_64^6a.
52. a;3-llx2+7a;+3. 53. 3a3-5a2-8a+ 10.
54. a;2c3— cs+a;2— 1. 55. a'-a3 + 8a*-8.
56. Show that a — 6 + c is a factor of
(2o-36 + 4c)3 + (2a[-6)3.
57. Factor 4a4— 37a262-[-964, (i) by cross multiphcation, (2) by
completing the square of 2a~ — Zb", (3) by completing the square of
2a2 + 362.
58. Without multiplication show that
(x2-4a;+3)(a;2-12a;+35) = (x2-6x+5)(x2-10x+21).
59. Make a diagram to show the square of a + b-\-c-^d.
60. Factor {a-b){h^-c'^)-{b-c){a^-b^).
61. Find the factors of 6.^3 — 7x2— 16a;+ 12, being given that it
vanishes when x=2.
62. Find four factors of (a;2-5.'c)2+ 10(a;2 — 5x) + 24 and of
(a;2 — 6)2 — 4a;(a;2 — 6) — 5^2.
63. Use the factor theorem to solve
x3-31a;+30 = 0 and a;*-43a;2 + 42a; = 0.
64. If two numbers differ by 6, show that the difference of their
squares is equal to six times their sum.
65. Find the quotient when the product of x'^ — {b — c)x — bc and
.c2 — (c — a)x — ca is divided by x^~\-{a-'b)x — ab.
66. Multiply a2- 62 _c2 + 26c by |i^i^-
67. If a;*+a;^ + aa;2 + 6a; — 3 is divisible by x— 1 and a;+3, find a and
b and the remaining factor.
68. Factor 2x^-ax-\-bx — ab — a'^.
69. Express a262-f c2cZ2 — a2c2 — 62^2 ^s the difference of two squares
in two different ways.
70. Factor a* + 6* + c*- 20*6^ 262c2 — 2c2a2 by completing the
square of a2 — 62 — c*.
71. Find four factors of (a2-62-c2 + d2)--4(ad-6c)=.
CHAPTER XII
SIMULTANEOUS EQUATIONS {continued)
106. In Chapter VII. the sohation of simple examples
of equations in two unknowns has been considered.
The method there followed was to make the coefficients
of one of the unknowns numerically equal by multiplication,
and then that unknown was eliminated by addition or
subtraction.
Other methods of eliminating one of the unknowns are
useful in certain cases.
107. Elimination by Substitution.
Ex.— Solve x-2i/= 2, (1)
5x+ly=lS. (2)
From(l), x = 2 + 2y. (3)
Substituting this value of x in (2),
5(2 + 2^/) + 72/ = 78,
.-. \0+lOy + ly = lS,
172/ = 68,
y= 4.
Substituting 2/ = 4in(3), x=10.
Here we eliminated x by finding the value of x in terms
of y from (1) and substituting that value in (2). We thus
obtained an equation which contained only the luikiiown y.
This is called the method of elimination by substitution.
We might take the value of x from (2) and substitute in (1).
Thus from (2), 5x=18-7y, :. x - ~ ^-
5
Substituting in (1), ^^~ - 2?/ = 2.
Complete the solution and verify the roots.
149
150 ALGEBRA
The value of y might have been found from either equation and
substituted in the other.
Thus from (1), 2ij = x~2, :. y = ^~^.
Substituting in (2), bx + '^~~^ = 78.
Complete the solution.
Solve also by finding y from (2) and substituting in (1).
If the four solutions be compared it will be seen that, in
this problem, the first is the simplest.
In solving equations with two unknowns, the pupil
should examine them carefully and choose the unknown
which he thinks will be the simpler to deal with.
108. Elimination by Comparison.
Ex.— Solve 2a:- 3^= 7. (1)
3x-\-5y=39. (2)
From (1), a; = ''-^. and from (2), x = ^^~^,
T + 3y ^ 39-%
2 3 '
.-. 3{l + 3y) = 2(3d-5y).
Complete the solution and verify the roots.
Here we effected the elimination of x by comparing the
values of x from the two equations.
This is called the method of elimination by comparison.
We might have compared the values of y obtained from
the two equations. Solve it that way.
109. Three Methods of Elimination. We have illustrated
three methods of elimination, by addition or subtraction,
by substitution and by comparison. When no particular
method is specified, the pupil is advised to use the first
method as no fractions appear in it.
SIMULTANEOUS EQUATIONS 151
EXERCISE 76 (1-6, Oral)
State the value of eacli unknown in terms of the other in :
1. x+y=5. 2. x—7j=3. 3. .r+2^=ll.
4. 3x—y=Q. 5. 2x+3^=12 6. 5.^—4^=19.
Solve by substitution and verif\' :
7. x+2i/=lS, 8. Sx+ y= 7, 9. 2x— ?/=19,
2a;+5«/=41. 4x+3?/=ll. 5a;-3^=46,
10. 2x-3y=\4:, 11. 3x-4?/=10, 12. 8.r+ Qy= 7,
x-5//= 0. 2x+6^=ll. 10.c+21(/=12.
Solve by comparison and verify :
13. x+2y=\Q, 14. 2x-\-y=2Q, 15. 3.r+42/=10,
.T-j-52/=14. 3a:— ?/=14. 4;«:— 3?/= 5.
Solve by any method and verify :
16. \x-\y=2, 17. Zx=2y, 18. y=lx+Q,
|a;+f2/=9 i^=ii2/-2. 3^-l2/=3|.
19. - = y, 20. ?^ + '^ = 6, 21. 3v-7.r=T,
2 3 5^3 •
' \2 ^ 4 ^ 2/
22. 3x + 2--^ = 2w + ^^ = 10.
11 -^ 7
23. x-52/+3=2x-8?/+3 = 7a;-10?/+16.
24. (x-l)(?/-2)-(y-3)(.T+l) = 17,
(x-3)(y-5)-(.i--5)(2/-3)=-22.
25. •la;+-21?/+-52=.-01x-+-01i/+3=0.
26. x+5 = 3(.y-3), ^^ + 2/ = -~^+19.
27. ^^3^-1:5 ^ ^^._ 23^ 2^^-9^,
4 ■^6 2
28. If the sum of two numbers is i of the greater number, the
difference of the numbers is how many times the less ?
152 ALOEBRA
110. Equations with three Unknowns.
Ex.— Solve 2.T+3J/— 4z=12, (1)
3:f- y+2z=\5, (2)
4a;+ 2/-32=19. (3)
This system of equations differs from the preceding by
containing three unknown quantities.
If we can obtain from these three equations, two equations
containing the same two unknowns, the solution can be
effected by preceding methods.
How can we obtain from (1) and (2) an equation containing
X and z only ? How can we obtain another equation from
(2) and (3) containing x and z only ?
Perform these two eliminations and find x and z from the
resulting equations.
Now find y by substituting in any one of the given equations
and verify by showing that the values you have found for x,
y and z will satisfy all of the given equations.
The solution might be written in the following form :
liminate y from (1) and (2),
.-. lla;+22 = 57.
(4)
„ „ (2) „ (3),
.-. 7a;-
2 = 34.
(5)
z „ (4) „ (5),
x= 5.
ibstitute x = 5 in (4),
2= 1.
,, a; = 5 and 2=1 in (1),
y= 2.
.-. x = 5, y = 2, 2=1.
Of course it will be seen that any other unknown might have been
eliminated twice from two pairs of the equations.
Thus we might have eliminated z from (1) and (2) and also from
(1) and (3), and thus obtained two equations in x and y. We might
then have completed the solution as before.
Solve the equations by this plan. Also solve them by two
eliminations of x.
Which letter do you think is easiest to eliminate twice ?
Note that the solution is completed only when the values of all
of the unknowns have been found.
SIMULTANEOUS EQUATIONS 153
BXBRCISB 77 (1-4. Oral)
1. What operation will eliminate both x and a;-fi/+2=35 (1)
y from (1) and (2) ? What is the value of z ? x+y—z=25 (2)
of x+yl x-y-\-z=\5 (3)
2. What operation will eliminate both y and z from (2) and (3) ?
What is the value of a: ? oi y~zt
3. How can you eliminate both x and z from (1) and (3) ? What
is the value oiy t oix-\-zt
4. In No. 5 below, which letter is simplest to eliminate from two
pairs of the equations ? Which in No. 6 ? Which in No. 7 ?
Solve and verify :
5. a;+2;/+ 33=16, 6. 2x-y+^z= 7,
a;+3^+ 4z=24, Zx+y-Az= 7,
a;+4?/+10z=41. %x-y+5z=2\.
7. 4x-3;/+ z==10, 8. a;+2/— ^=16,
6a;— 5(/+23=17, a;— ?/+ 2= 4,
x-\- y+ 2= 8, a;+2/+22=22.
9. x+2^+32=32, 10. x+y^25,
4:x-5y+6z=27, y+z=15,
lx+8y—9z=l4:. 2+a:=70.
11. x+2y=12, 12. 3(2-1) =2(?/-l),
3y+4:z= 2, ^y+x) =9z -4,
52_2.c=-21. 7(5a;-32)=2;/ -9.
13. 3+a;=5-42/, 14. ir+ J?/+ Jz =36,
2+x=3;/, i^+i^s2/+2V2 =10,
Ty=z+2. lx+ ky+^z -43.
2^3^4 3^4^5 4^5^6
16.* If x-\-2y—2o, 2/+3z=5o, z-|-4a;=35, find the value of x-{-y-\-z.
17. If X— ?/+z=9, 2x--|-?/=8, «/— 4z=5 and x+y-{-z+io=\2,
find ?<?.
18. If ax'^—hx-\-c is 8 when .<;=1, 8 when a;— 2, and 10 when .T=3,
find a, 6 and c.
154 ALGEBRA
19. If ax"-\-bx-\-c is 9 when x=\, —3 when x= — \, 18 when
x=2, find its value when .x=3.
20. Determine three numbers whose sum is 9, such that the sum
of the first, twice the second and three times the third is 22, and the
sum of the first, four times the second and nine times the third is 58.
21. If «+6=12, 6+c=15, c4-rf=19. find a+d.
111. Special Forms of Equations. Two equations of the
first degree iii x and y will usually determine the values of
X and y.
Consider the following sets of equations
(1) 2x-Zy=lO, (2) 2x-^y=\0, (3) 2a:-3y=10.
4a;+5i/=42. 4.T-6?y=20. 4a;— 6!/=30.
In (1), if the two equations are solved in the usual way
we find that x— 8, ?/=2 will satisfy both of the equations,
and no other values of x and y will satisfy them.
We therefore say that these equations are determinate, that
is, the}' determine the values of x and y.
In this ease the second equation can not be deduced from
the first, nor the first from the second. We therefore say
that the equations are independent.
In (2), the second equation may be deduced from the first
by multiplying by 2. These equations are dependent and
not independent as in (1).
Any number of values of x and y will satisfy both equations,
because any values which will satisfy the first will also satisfy
the second. These equations are therefore indeterminate.
In (3), if the first equation is true, the second can not be
true. They are therefore said to be inconsistent or impossible,
and no values of x and y can be found to satisfy both of them.
We thus see that two equations in two unknowns can have
a definite solution only ivhen the equations are independent
and consistent.
In this set of equations, the third may be 3x-\-'2y— z= 5,
obtained by adding the other two. They are 4a;— i/ + 32 = 20,
therefore dependent equations- and consequently lx+ y-\-2z = 25.
indeterminate.
SIMULTANEOUS EQUATIONS 155
BXBROISB 78
1. Find three pairs of values of x and y which satisfy the equation
2x-3y=l2.
2. Solve 2.r+3?/=13, Sx— ?/=24. Is it possible that 2x+3y=\3,
5x — y=24 and 4a;-|-5?/=19 can be true at the same time ?
3. What is peculiar about the equations 4.t-|-«/=17, 8x-\-2y=35 ?
Also about 8x+l2y=60, 6.r+%=45 ?
4. Shov/ that the equations
x+z+4: = 3y, 3x+z=2y+6, 2x+y = l0,
are indeterminate. If 2=5, solve the equations.
a. Find two solutions of the simultaneous equations .
xJi-y-\-z=lO, 3x—2y—z=l.
For what values of a will the following sets of equations be consistent :
6.* 3x— y= 5, 7. 3x+2y= 1, 8. 9.r— ay=Q,
x-^2y=25, 10.r-4?/= 2, 3.c- y=2,
.r+4^y= a. 3x'+rt?y=ll. 5.r— ^if;y=^-.
9. Show that these equations are inconsistent :
2x+3y— 3z=20, 3;r-L7//— 22=5, a;+2y— 2=6.
(1)
(2)
Here we coiild obtain the solution in the usual way by removing
the fractional forms, by multiplying each equation by xy. See if you
can complete the solution by this method.
It is simpler, however, to eliminate y from the equations as they
stand.
112.
Special Fractional Equations.
Ex.-
-Solve
X y
4 2
^ + - = 21.
X y
156 ALGEBRA
Thus, multiplying (2) by 4 and adding
— = 95, .-. 95a; = 19, .-.
X
1 8
Substitute a; = > in (1) and 15 = 11.
5 ' ' y
y ^
The solution, therefore, is x = i, y = 2.
Verify this result.
Solve and verify :
EXERCISE 79
1. ^ +'- =2,
X y
24 21 _ J
X y
2.
' + ' =29,
X y
5_6 _2.
X y
3.
? + '=19,
a; y
4. ^ _4 ^8,
a; 2/
1? + ' =101.
5.
- + 2^/ = 15,
a;
!-3, = 0.
6.
1+2=11,
X y
?-?=2,
!( »
*-l=I7.
Z X
7. 3y—5x=xy,
2y-\-Sx=26xy.
8. ^ + o^ =
a; ^2^
15
2a:
+^=^.
^32/ 6
9 5 3 _ 135 75 _
X y X y
30
10. 3a; H 1 = 12a; +
2/
5
y
+ 14 = i - 2a; - 14.
y
11. ? + 2_^^^1^4
X y X y
= ■
122 -- + - = 83 + 17.
X y
113. Problems leading to Simultaneous Equations. In
Chapter VII. we have had illustrations of problems which
were solved by using equations of two unknowns. We now
give some further examples on special subjects which were
not then considered.
SIMULTANEOUS EQUATIONS 157
The number 47 might be written 4 . 10 + 7. What is the sum of the
digits of this number ? What number would be formed by reversing
the digits ? What is the sum of the number and the reversed number ?
What is the sum of the digits of the reversed number ?
Ex. 1. — A number has two digits. If IS is added to it
the digits are reversed. The sum of the two nuinbers is 88.
Find the number.
Let a; = the units digit and y the tens digit,
.". the number =lOy-i-x,
and the reversed number =lOx+y.
:. I0y + x+l8=10x+y, (1)
and I0y + x+10x + y = 88. (2)
Simplifying (1), 9a; — 9?/= 18 or a; — ?/ = 2,
(2), Ux+ny = 88 or x+y = 8.
Solving x = 5, y = 3.
.". the required number is 35.
Verification : 35+18 = 53,35+53 = 88.
Ex. 2. — If 4 be added to the numerator of a fraction
and 3 to the denominator, the fraction becomes h. If 2
had been subtracted from the numerator and 5 from the
denominator the result would have been ^. Find the
fraction.
Let - = the fraction,
y
x+i I , x-2 1
~r5 = o and = ^ •
2/ + 3 2 y-5 6
2a;+8 = 2/ + 3 and 6x—l2 = y — 5.
2x — y=: —5 and dx— y = l.
Complete the solution and verify.
Sometimes the solution of a problem may be simplified
by using some function of x instead of x to represent one
of the imknowns.
Thus, if two numbers are in the ratio of 7 to 6, we might
represent the larger number by x and then the smaller ^ould
be yo:.
158 ALOEBRA
A better way, however, would be to represent the larger
by Ix, and then the smaller would be Qx. By domg so we
get rid of the use of fractions.
Ex. 3. — The incomes of A and B are in the ratio of 3 to
2, and their expenses in the ratio of 5 to 3. Each saves
$400 a year. Find their incomes and expenses.
Let $3a; = .4's income, then %2x = B's income.
Let %5y = A's expenses, then $3?/ = Z?'s expenses.
.-. 3a; - 52/ = 400 and 2x - 3^/ = 400.
Solving X = 800 and y = 400.
.-. .4's income = %Zx= $2400 and B's = $1600.
.-. ^'s expenses = $5y= $2000 and B's = $1200.
Note. — In solving the problems in the exercise following, the pupil
will find that he frequently has the choice of using one, two or more
unknowns. Except in special cases, he is advised to use as small a
number of unknowns as possible. In each case the results should be
verified.
EXERCISE 80
1. If 10 men and 4 boys, or 7 men and 10 boys, earn $96 in a
day, find a man's wages per day.
2. Two numbers are in the ratio of 5 to 7 and their difference is
10. What are the numbers ?
3. The sum of three numbers is 370. The sum of the first two
is 70 more than the third, and six times the first is equal to four times
the third. Find the numbers.
4. Find three numbers such that the results of adding them two
at a time are 29, 33, 36.
6. Divide 429 into three parts so that the quotient of the first
by 7, the second by 4 and the thu-d by 2 will all be equal.
6, A workman can save $200 a year. He goes to another town
where his wages are 10% greater and his expenses are 5% less, and he
can now save $310 a year. What are his wages now ?
7. The denominator of a fraction exceeds the numerator by 3.
If 2 is subtracted from each term, the fraction reduces to |. Find
the fraction,
SIMULTANEOUS EQUATIONS 159
8. Divide 120 into three parts, so that f of the first part is
greater than the second by 5 and J of the second part is greater than
the third by 10.
9. If 6 men and 2 boys earn S56 in 2 days and 7 men and 5 boys
earn S57 in 1| days, how long wUl it take 3 men and 4 boys to earn S60 ?
10. A number between 10 and 100 is 8 times the sum of its digits,
and if 45 be subtracted from it, the digits are reversed. Find the
number.
11. The difference of the two digits of a number is 4. The sum of
the number and the reversed number is 110. Find the number.
12. The sum of the two digits of a numbers is 14, and when 18 is
added to the number the digits are reversed. Find the number.
13. When 1 is added to both terms of a fraction the result is ^. If
9 had been subtracted from the denominator only the result would
have been J. Find the fraction.
14. A number consists of two digits whose sum is II. If the order
of the digits be reversed, the number thus obtained is greater by 7
than twice the original number. What is the number ?
15. The difference between the digits of a number less than 100 is 6.
Show that the difference between the number and the number formed
by reversing the digits is always 54.
16. The sum of the reciprocals of two numbers is ^g. Six times
the reciprocal of the first is greater than five times the reciprocal of
the second by J. Plnd the numbers. (The reciprocal of x is -•)
17. Divide 150 into two parts such that the quotient obtained by
dividing the greater by the less is 3 and the remainder is 2.
18. I wish to obtain 100 lb. of tea worth 34c. per lb. by mixhig
tea worth 30c. per lb. with tea worth 40c. per lb. How much of each
must I take ?
19. Three pounds of tea and 10 of sugar cost $2-40. If tea is
increased 10% in price and sugar decreased 10%, they would cost
S2-52. Find the price of each per lb.
20. Two numbers are in the ratio of 7 to 5. What quotient
is obtained when three times their sum is divided by six times their
difference ?
160 ALGEBRA
21. Show that the sum of any number of two digits and the number
formed by reversing the digits is always divisible by 11 and that the
difference is divisible by 9.
22. A number has three digits, the middle one being 0. If 396
be added the digits are reversed. The difference between the number
and five times the sum of the digits is 257. What is the number ?
23. Divide 126 into four parts, so that if 2 be added to the first,
2 be subtracted from the second, the third be multiplied by 2, and the
fourth be divided by 2, the results wiU aU be equal. (Let the result=a;.)
24. There are three numbers such that when each is added to twice
the sum of the remaining two the results are 44, 42, 39. Find the
numbers.
25. The sum of the three digits of a number is 12. If the units
and tens digits be interchanged the number is increased by 36, and if
the hundreds and units be interchanged it is increased by 198. Find
the number.
26. Find three numbers such that the first with ^ of the sum of
the other two, the second with J of the others, and the third with
i of the others, shall each be 25.
27. A piece of work can be done by A working 6 days and 5 21
days, or by A working 8 days and 5 18 days. In what time could
each of them complete it alone ?
28. Divide 84 into four parts, so that the first is to the second as
2 to 3, the second to the third as 3 to 4, and the third to the fourth as
4 to 5.
29. Of what three numbers is it true that the sum of the reciprocals
of the first and second is J, of the first and third is J and of the second
and third is J ?
30. Two numbers consist of the same three digits but in inverted
order. The sum of the numbers is 1029. The sum of the digits of
each is 15 and the difference of the units digits is 5. Find the numbers.
31. A stream flows at 2 miles per hour. A man rows a certain
distance up stream in 5 hours and returns in If hours. How many
miles per hour could he row in still water ?
32. A rancher sold 50 head of horses, part at §125 a head and the
balance at $150 a head. After spending $50 he was able to make the
first payment of ^ of the purchase price of 1200 acres of land at $18
per acre. How many horses did he sell at $125 a head ?
SIMULTANEOUS EQUATIONS 161
33. A number consists of a units digit and a tenths digit, the units
digit being the greater by 1. The sum of the digits is less than twice
the number by 2. Find the number.
34. A grocer spent $120 in buying tea at 60c. a lb., and 100 lb.
of coffee. He sold the tea at an advance of 25% on cost and the coffee
at an advance of 20%. The total selling price was $148. Find the
number of lb. of tea purchased.
35. When 2 is subtracted from each term of a fraction the result
is equal to .^. Show that the result would have
been the same if 1 had been subtracted from
the numerator only.
36. It is shown in geometrj^ that the two
tangents drawn to a circle from a point are
equal. Thus, in the figure AD=AE, etc. If
AB^\b, BC=14:, CA = 13, find x, y and z.
37. If the sides of a triangle are 10, 15 and
19, where will the inscribed circle touch the
sides ? (See figure of preceding example.)
38. li A can do a piece of work in m daj's and B can do it in n days,
in what time can they do it working together ?
If X is, the number of days required,
1 1,1 *"«
then - = 1
m n m-\-n
39. Use the preceding result to find in what time A and B working
together can do a piece of work which could be done by A and B
separately in the following number of days :
(1) yl in 10, £ in 15. (2) A in 20, S in 5. (3) A in f, B in 1^-.
EXERCISE 81 (Review of Chapter XII)
Solve and verify :
7a;-
-8y=
10,
3x-
-2y =
= 10.
10
X
12
= 14,
7
X
4.
= 16,
4:x+7y=~l, 3. 73x+y= 75,
3a;- 2/ = 3. x+ldy^Ul.
^ ^ 3, 6. 12y~-8x = 2xy,
X y
2 25
Sy-]-4:X = 2xy.
M
162
ALGEBRA
7.
"+'-7, ,-^ =6.
y 1+2/
8.
X+l 03+3 ^
2/ + 2 2y+l
9.
x-y 2x + 3y
3^5
10.
x~2y Zx-y
3 7 '
^x+y) = 8.
11.
3. '^-^ = 6,
5
12.
x— 1 2/ — 3 z— 5
2 "~ 4 ~ 6 '
4y + ^-' - 12.
a;+2/+2=33.
13.
x+ y+ 2=-3,
14.
2x+3i/- 2 = 5,
x + 2y+ 2=-0,
3x — 42/ + 22=l,
303+ 2/ + 62 = 0.
4a;-6y + 5z = 7.
15.
3 = 8 + ^-2-^-
16.
a;-2 10-a; y—\0
5 3^4'
a; + y + z = 24.
22/ + 4 2a;+2/ x-\-\Z
3 8 ~ 4
a;— 1 y + 5 ^ a;+2
3 12 "' 60 '
{x^\l){y-n) = xy-5.
18. ia;-f2/ + 2+l = 3(a3-2/) + 52 + 4=a;+62/-22-9 = 0.
_^ 5x~3y . _ 102 + Ja;
19. — ^^ = 42/ - 22 = 2 = ^•
20. x+y = 5, y + z = Z, z+w=l, x—iv=3.
21. 31a; + 282/=146, 28x+31?/=149. (Add and subtract the
equations and remove common factors.)
22. 97a;-592/ = 329, 59a;-97y=139.
23. What values of x and y will make ^ and — ^ each equal
toa3-10 ?
24. Show that x+y + z=l2, 3a; + 4?/ - 5z = - 22, 10x+ 122/-6z = 4,
are indeterminate.
25. Divicfe unity into two parts so that 18 times the first part may
exceed 12 times the second by 13.
26. A number of two digits is four times the sum of its digits,
and if 18 be added to the number the digits are reversed. What is
the numlxT ?
27. The tens digit of a number is twice the units digit. When the
number is divided by the sum of the digits what must the quotient be ?
SIMULTANEOUS EQUATIONS 163
28. Find a fraction equal to | such that ^ of the denominator
exceeds f of the nvunerator by 8.
29. Two persons who are 30 miles apart are together after 5 hoiu-s
if they walk in opposite directions, but are not together for 15 hours
if they walk in the same direction. What are their rates ?
30. ^'s age is equal to the combined ages of B and C. Ten years
ago A was twice as old as B. Show that 10 years hence ^ will be twice
as old as C.
31. A biU of S19-50 was paid in half-dollars and quarters and four
times the number of quarters exceeded twice the number of half-dollars
by 12. How many of each were used ?
32. If 5 lb. of tea and 8 lb. of coffee cost $5*80, and coffee advances
10% in price and tea 15% and they now cost $6'53, find the prices
per lb. of each before the advance.
33. I invest a certain sum at 4% and another sum at 6% and
receive $42 interest. If the sums had been interchanged I would
have received $8-50 more. What were the sums ?
34. If each side of a rectangle is increased by 5 feet the area is
increased by 275 square feet. If each side is decreased by 5 feet the
area is decreased by 225 square feet. Show that the sides can not
be determined from these conditions.
35. Solve ^-^t|:=-^=^ = 4x-.= l. ■
36. A grocer wishes to mix tea worth 30c. a lb. with tea worth
40c. to make a mixture weighing 601b. worth 36c. a lb. How many lb.
of each must he use ?
37. If 3x^~2x-\-5=ax^-\-bx-\-c, when x=l or x=2 or x=3, show
thata=3, 6= — 2, c = 5.
38. The tens digit of a number exceeds the units digit by 3. By
how much is the number decreased by inverting the digits ?
39. A train is 27 minutes late when it makes its usual trip at 28
miles per hour and is 42 minutes late when it runs at 27 miles pei
hour. What is the distance ?
40. A piece of work can be done by A working 6 days and B 10 days
or by A working 9 days and B 14 days. How long would it take each
alone to do it ?
41. A number has three digits, the units being J of the tens and ^
of the hundreds. If 396 be subtracted the digits are reversed. Find
the number.
M 2
164 ALGEBRA
42. When the greater of two numbers is divided by the less the
quotient is 5 and the remainder is 2. When 12 times the less is divided
by the greater the quotient is 2 and the remainder is 12. Find the
nvunbers.
43. Find four numbers such that when each is added to twice
the sum of the remaining three, the results are 46, 43, 41 and 38
respectively.
44. If the sum of two numbers is a times the greater and the
difference is b times the smaller, show that a—b + ab = 2.
CHAPTER Xm
GEOMETRICAL REPRESENTATION OF NUMBER
114. Function of x. The value of the expression 3x— 2
depends upon the vahie of x.
Thus, when x= 4, 3, 2, 1, 0, -1, -2, -3, -4,
3a;-2=10, 7, 4, 1, -2, -5, -8, -11, -14.
When the value of an expression depends upon the value
of X, the expression is called a function of x.
Thus, 2a;— 3, 5a;, ^a;+ 1, are functions of x.
What is the value of each of these functions when a; = 2, 1, 0,
-1, -2 ?
Instead of repeating the words " the expression " or " the
function," we might represent the function by a symbol,
say ij.
Thus, if y = 5x-\-\, when x= 1, y = & ; a;=3, y= 16.
If i/ = ^a;+4, what are the values of y when x has the values 6, 3, 0,
-1, -8 ?
115. Variables and Constants. A quantity that has not
always the same value is called a variable, while a quantity
whose value does not change is called a constant.
Thus, the population of a city and the height of the barometer are
variables, while the ntunber of days in a week and the length denoted
by an inch are constants.
Note. — To do the work of this chapter properly, pupils should be
suppUed with squared paper. Paper ruled in tenths or eighths of an
inch will be found most satisfactory.
165
166
ALGEBRA
116. Connected Variables. Two variables may be so
connected that for every change in the value of one there is
a corresponding change in the value of the other.
Thus, if y = 2x + 5, for each value of x there is a corresponding
value of y. Here x and y are variable quantities, but 5 is a constant.
In arts. 20 and 21 we have shown how the changes in two
variable quantities may be represented by a diagram. Those
diagrams show that for each variation m time there is a
corresponding variation in temperature.
117. Graph. A line so drawn as to exhibit the nature of
the relation of two variables is called a graph.
118. Arithmetical Graphs. The solution of many problems
in arithmetic might be rej)resented graphically as follows :
Ex. 1. — The passenger rate on a railway is 3 cents per
mile. Represent graphically the amount charged for any
number of miles from 1 to 10.
In the diagram each iinit on the hori-
zontal line OX represents 1 mile and each
unit on the vertical line OY represents
3 cents.
The point A shows that the cost for
4 miles is 12 cents. What does the point
B show ? The point C ?
Read from the figure the cost for 2
miles, 5 miles, 9 miles. How far can I
travel for 9 cents, 21 cents, 27 cents ?
■~ 15
t-- 12
Y
/
/
C
/
/
/
/
B
/
/
A
/
/
^
X
8 9 10
No. of Miles
Ex. 2. — Represent graphically the
simple interest at 2% on |100 for any number of years from
1 to 6.
Reading from the diagram (on the next page), what is the interest on
SI 00 in 2 years ? In 5 years ? In 4^ years ?
What does the point A show ? The point C ? The point D ? The
point half way between C and D ?
In how many years is the interest $8, $6, $5, $11, $6-50 ?
GEOMETRICAL REPRESENTATION OF NUMBER 167
Place a ruler on the points marked A, B, G, D, E, F. What
peculiarity do you notice ?
Make a similar diagram, on sqviared paper, which will give the
interest on §200 at 4% for any num-
ber of years from 1 to 7. So that
your diagram will not occupy too
much space vertically, suppose each
unit on OY to represent §4 instead
of $1.
Read from your diagram the in-
terest for 3 years, for 5 years. In
how many years is the interest $8,
$24, S4, 812, S44 ?
Q 8
■5 ^
^ 6
Y
/
/
F
/
/
E
/
/
D
/
/
C
/
/
B
/
/
A
d
X
Time in Years
Ex. 3. — A starts at 9 a.m. on
a bicj^cle at 8 miles per hour.
He is followed at noon by B on an automobile at 16 miles
per hour. When and where will B overtake A ?
Each space on the horizontal line represents 1 hour, and on the
vertical line 4 miles.
At the end of successive hours A's position will be C, D, etc., and
B"s will be F, G, etc.
The line OP is the graph of A's journey and
MQ is the graph of B's.
The diagram shows that B overtakes A at
the point R, which is 48 miles from the start-
ing point and that the time is 3 p.m.
How far is A ahead at 12 o'clock ? at 1 ?
at 2 ?
Solve the problem otherwise and compare
the results.
Ex. 4. — A starts from P at 9 A.M. to
go to Q, a distance of 60 miles, travel-
ling at 5 miles per hour. He stops at
12 for one hour for lunch. B starts from Q at 11 A.M. to go
to P, travelling at 15 miles per hour. An accident detains
him from 12 to 2. Where and when will they meet ?
The graph of A's journey is represented by the upward line drawn
from P, and of B's by the downward Hue drawn from C.
The position of each at the end of the successive hours is marked
on the diagram. (See next page.)
K
48
M
40
36
32
za
lA
ZO
16
\l
6
4
0//
P
^R
r»
/
9
/
D
/
1
k
/
c
/
/
1
/
h.
k
c
) 10 II 12 1 2 3 A 5 6 1
168
ALGEBRA
They meet at M at about 3.15 p.m. and at a distance from P of
about 26 miles.
How far are they apart at the end of each hour from 11 to 5 ? When
did B reach P ?
We might solve the problem
algebraically.
Suppose they are together x
hours after 9 a.bi.
Tlien A has travelled x—l
hours at 5 miles per hour, and
B x — 4 hours at 15 miles per
hour,
.-. 5(a;-l) + 15(cc-4) = 60.
Solve and compare with the
■ 9 io 11 12 1 2 3 4 5 results obtained graphically.
119. Graphical Results only Approximate. The last problem
illustrates the fact that the results obtained by graphical
methods are approximate only. When the problem is solved
algebraically we find that they will meet 26J miles from P
at 3.15 P.M.
Q
o
V
P
A
s\
\
\,
V
V
->
\
r^
\
-r
A
-
^
^
V
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^
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-
>
^
^
^
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BXE3RCISE3 82
1. A man walks at the rate of 4 miles per hour. Construct a
graph to show the distance he walks in any number of hours from 1 to
10. Read from the graph the distance walked in 3, 5, 1\, 9^ hours.
(Take two units on the horizontal line to represent 1 hour and one unit
on the vertical line to represent 2 miles.)
2. In Ex. 1, if he rests 30 mmutes after walking each 4 miles,
how long will it take him to walk 8, 12, 14, 7, 17 miles ? How far
will he have gone m \\, 3|, 5|, 8| hours ?
3. A starts running at the rate of 6 yards per second, and
4 seconds later B starts from the same place at 9 yards per second.
Construct a graph to show when and where B will overtake A.
How far apart are they 6 seconds after A started ? When was A
12 yards ahead oi B 1
4. Oranges sell at 19 cents per dozen. Make a graph from which
you can read off the price to the nearest cent of any number from 1
to 12. What is the cost of 2, 5, 7, 8, 10 oranges ? How many can I
buy for 3, 5, 8, 13, 16 cents ?
GEOMETRICAL REPRESENTATION OF NUMBER 169
5. If 8 kilometres equal 5 miles, construct a graph which will
enable you to change into miles any number of kilometres up to 20.
Read the approximate number of miles in 3, 5, 11, 13, 16, 19, 20
kilometres.
6.* A starts from Toronto at 12 miles per hour to motor to Hamilton,
a distance of 40 miles. An hour and a half later B starts from Hamilton
to drive to Toronto at 8 miles per hour. By means of a graph, find
when and where they will meet.
7. The distance from yl to 5 is 10 miles, 5 to C 8, C to D 8, Z> to
E 10 miles. A mail train, which leaves A at 10 a.m., arrives at B at
10.24, C at 10.48, D at 11.12, E at 11.40. An express train leaves E
at 10.24 and without stopping reaches A at 11.28. If the mail train
stops 4 minutes at each station, show graphically :
(a) when and at what point they pass each other,
(6) how far they are apart at 10.30 and at 11.12,
(c) when the express passes through B.
120. The Axes. In the diagram the Hne OX is called
the axis of x and 0 Y the axis of //.
We will call the measurement along
OX, X, and along OY, y.
For the point A the x measure-
ment is 3, and the y measurement
is I. What are the x and y
measurements for the points B,
C, D, E^
Examine the x and y measure-
ments for each point marked on
the line OP. What equation con-
nects the values oi x and y for
each point on the line OP ? For each point on OQ, OR,
OS^
OP is the graph of the equation y=2x, OQ of y^4:X, OR
of y==x, and OS of y—\x.
The X and y measurements of every point on the line OP
satisfy the equation y=^2x. This equation is not satisfied
74^ 7 Z^
^ t J-
fi §t p -.Z
^ 4 T^^ -7^
^ 2 L Z
t ^ 2^.F
. ^tJ5 ^Z ^
^ I^^^ 2 ^
^ i t 7 s
17 2 s
9 tt^^ =.1
^-iJZTt ^^^^
,itz ^^^
^yTZ^^^ -
Jt^ X
Y 1 E 3 4 5 6.
170
ALGEBRA
by the x and y of any point not on the line OP.
hy the x and y of the points A, C, D, E "^
Is it satisfied
121. Equation of a Line. Since the equation y—2x is
satisfied by the values of x and y for each point on the line
OP and by no other points, the equation y—2x is called
the equation of the line OP.
What is the equation of OQ ? oi OR \ of OS ?
122. To Construct the Graph of a given Equation.
Ex. — Construct the graph of y=\x.
Here when
Y
^
■
^^
.>
,''
'!
^
^
^
K
4<-
u
—
LL
—
x = 2, y=\,
a; = 6, 2/ = 3, etc.
To find the point where x = 2, y=l,
count 2 units from O along OX and then
1 unit upwards. Find in a sinxilar
manner the points where a; =4, y = 2;
x=6, y = 3 ; a;=8, 2/ = 4 ; a;=10, y = 5.
Join all the points located. They are all seen to lie on the same
straight line passing through 0.
This line is the graph of tlie equation y = ^x.
123. The Origin. All the lines we have so far considered
have been drawn through the point 0. This point is called
the origin.
The X and y measurements of the origin are x=0, y=0.
These values satisfy the equation ?/=^x, and consequently
the graph of this equation should pass through the origin
as the figure shows.
BXE3RCISE 83
Construct the graphs of :
1. y=x. 2. y=5x, 3. y=^x.
4. y='^x. 5. 4y=x. 6. 5y—2x.
124. Coordinates. In the diagram 0 is the origin, XOX'
is the axis of x and YO Y' is the axis of y. The x measurement
of the point A is 4, and the y measurement is 2.
GEOMETRICAL REPRESENTATION OF NUMBER 171
These are called the coordinates of the point A, 4 being
the oc coordinate and 2 the y coordinate. The coordinates of
the point A are written (4, 2), the
,v
p
1-
n
u
A
7.
K
Y,
Q
"t 7-
P
p
u
'J
;v;
-4, 6).
a; coordinate being written first.
Similarly, the coordinates of B
are (3, 5) and of C (6, 7).
So far all our measurements
have been made from 0 towards
the right and then U2nvards. We
might also measure from 0 to the
left and downwards. When we
do so we indicate the change in
direction by a change . in algebraic
sign.
Thus, to reach the point D we measure 2 units to the left and then
3 units upwards. Therefore the coordinates of D are ( — 2, 3).
Similarly, the coordinates of £• are ( - 6, 5), of i^ ( - 3, - 3), of G (3, - 5),
and of H (0, -4).
What are the coordinates of P, Q, R, S, O 1
Mark on the diagram the points whose coordinates are (2, 2), (■
(2, -5), (-1, -3), (0, 4), (-3, 0), (0, -3).
Using squared paper, take the origin at the intersection of two lines.
Mark the points which would be indicated thus : (2, — 3), (5, 6), ( — 3, 7),
( — 6, —2), (3, 0), (0, 3). Mark any other four points and show how
their positions would be indicated.
125. Quadrants. That part of the plane between OX and
OF is called the first quadrant, between OY and OX' the
second quadrant, between OX' and OY' the thud and between
OY' and OX the fourth.
Thus, the points A and B are in the first cjuadrant, D and E in the
second, F and S in the third and P and G in tlie fourth.
In which quadrant are both x and y negative ?
126. Plotting Points. When we represent the position of
a point with respect to the axes XOX' and YOY', we are
said to plot the point.
When two i)oints are plotted the distance between them
may be obtained by adjusting the points of the compasses
172
ALGEBRA
to the two points and transferring the compasses to the line
OX, or any other Hne, and reading off the distance.
Plot the points (3, 5) and (6, 1), and see if the distance
between them is 5.
EXBRCISE 84
1. In what quadrants are the points (3, 4), (4, —1), (—5, 3),
(-1,-2)?
2. Plot the points (1, 2), (4, -6), (-3, 7), (-5, -2).
3. Plot the points (5, 0) and (—3,0). What is the distance
between them ?
4. Where are the points (0, 0), (0, 2), (-5, 0), (4, 0) situated ?
5.* What is the distance between the points (6, 4), (1, —8) ?
6. What kind of figure is formed by joining the points (0, 0),
(4, 0), (4, 4), (0, 4) m order ? What is its area ?
7. What kind of a triangle is formed by joining the pomts (0, 2),
(2, 6), (2, 2) ? What is its area ?
8. Plot the points (1, 1), (1, 3), (2, 1), (3, 3), (3, 1). Join them
in order. What letter is formed ?
9. What is the area of the figure formed by joining (1, —3).
(-5, -3), (-5, 6), (1, 6) in order ?
10. The angular points of a triangle are (6, 0), (3, 4), (—2, 0),
Construct the triangle and find its area. Measure or calculate the
lengths of the sides.
11. What is the length of the perpendicular from the point (5, 8)
to the line joining (3, 2) and (7, 2) ?
127. Complete Graphs. In art. 122 we constructed the
graph of the equation y=hx,
but only for positive values
of X and y.
The diagram, which is here
repeated, shows that the line
also passes through the points
Y
/^l
^
^
(6,
3)
■^
^
(4,
2)
X
v
^
(2,
1)
X
^
^
^
"(-
2,
-1)
^
'f-
4,-
2)
^
<-
6,-
3)
-Y
GEOMETRICAL REPRESENTATION OF NUMBER 173
(—2,-1), (-4,-2) and (-6,-3). This is as we would
expect because
x= — 2,y= — l; a;=— 4, ?/=— 2; x=-—Q,y= — 3,
all satisfy the equation y=^x.
128. Linear Equation. It is seen that the graphs of all
the equations so far constructed have been straight lines.
This is true concerning all equations of the first degree. For
this reason an equation of the first degree is sometimes called
a linear equation.
Since a straight line is fixed or determined when any two
points on it are fixed, it follows that to construct the graph
of an equation of the first degree, we need to determine only two
'points on it.
129. Lines not passing through the Origin. Every equation
of the form y—mx represents a straight line passing through
the origin, because the equation is satisfied by .t=0, y=0.
If the equation contains a term independent of x and y,
it represents a straight line which does not pass through the
origin.
Thus, y = 2a;+ 1 represents a straight line which does not pass through
the origin, because this equation is not satisfied by x = 0, y = 0.
Ex. 1. — Construct the graph of y=2x-\-\.
The coordinates of two points on the line
are x = 0, y=\ and x= 1, 2/ = 3.
Locate these two points and draw the un-
limited straight line which joins them. This is
the required graph.
The diagram shows that it also passes through
the points (2, 5), (-1, -1), (-2, -3), (-3,
— 5). Do the coordinates of these points satisfy
the equation ?
In constructing the graph of an equa-
tion by locating two points on it, the
pupil should try and determine two points whose coordinates
are integers.
Y
/
/
/
(1.
3)
/
X
J
5"
,1)
X
A
%
y
~
'
-)
/
/
/
^
174 ALGEBRA
Ex. 2. — Construct the graph of 3x4-4?/= 15.
TT 15 — 3a; , , /x=l, w = 3\
Here y = ; — and when ^ „ •
Plot the points (1, 3) and (5, 0) and join them giving the required
graph.
We might have found the points at which the graph cuts the axes.
Thus, when x=0, 2/ = 3f and when y = 0, x — 5. The required line is
then found by joining the points (0, 3|) and (5, 0).
If the latter method is followed and fractions appear in
the coordinates of either of the points found, the unit of
measurement should be changed, in this case, by taking four
spaces as the unit instead of one.
When the unit is not one space, it should be clearly shown
on the diagram what the selected unit is.
EXERCISE 85
1. Find two pairs of values of x and y which satisfy a;+?/=6.
Plot the points whose coordinates are the values found and construct
the graph of the equation x-\-y^6.
2. What are the coordinates of the points at which the graph in
Ex. 1 cuts the axis of x, the axis of y ?
Construct the graphs of the following equations :
3. y=x+3. 4. y=x—3. 5. y=2x—3.
6. ?/=3,r— 2. 7. x+2y=7. 8. x—2y=l.
9. 2a;+3?/=12. 10. 3a;— 4^=16. 11. 5a;+6y=17.
12. Construct the graph which cuts off 4 units from the axis of x
and 6 miits from the axis of y. Find the area of the triangle which this
line forms with the axes.
13. On the same sheet construct the graphs of x—y=lO and
x+2y=7. What are the coordinates of the point at which they
intersect ? Do the coordinates of this point satisfy both equations ?
GEOMETRICAL REPRESENTATION OF NUMBER 175
14. Will the point (3, 4) lie on the graph of the equation 4x+3y=24 ?
Which of the following points lie on it : (2, 6), (0, 8), (6, 0), (9, —4),
(5, 2), ( — 1, 9) ? Verify by constructing the graph.
15. By constructing the graph of 2a;+3?/=24, find three sets of
positive integral values of x and y which satisfy this equation.
16. Why is there an unlimited number of positive integral values
of X and y which wiU satisfy 2,i-— 3^=24, but only a hmited number
which will satisfy 2x-\-3y=24: ?
130. Graphical Solution of Simultaneous Equations. In this
diagram are shown the graphs of the equations x-\-y=^5 and
2x—3y=l5.
The coordinates of the point P, at
which the hnes intersect, must satisfy
both equations.
The coordinates of P are (6,-1).
.". x=Q, y= — l, must be the values
of X and y which satisfy both equa-
tions. We have therefore obtained
the sohition of these two equations
graphically.
Since it is evident that two straight
lines can intersect at only one point,
it must follow that there is only one pair of roots of two
simultaneous equations of the first degree.
In this diagram are shown the graphs of
(a) x-y=3, (6) 2x-y=l, (c) Zx+y=-l.
At what point do the graphs of (a)
and (6) intersect ? (a) and (c) ? (6)
and (c) ?
Is there any point which is common to
the three hnes ? Are there any values
of X and y which will satisfy these three
equations at the same time ?
When three equations in x and y are
all satisfied by the same values of x and
y, what peculiarity will appear iri their
graphs ?
\
Y
~
~
\
\
s.i
\
X
\
■^
^
X
(•\
\
Xj
\
y
R
k-
\
y
^
\
^
IOl^
.^
A
h
y
>>
^
1^
\
(L
,
V -i
^\
t
^
^
\ V
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ii -L^ _^ X
-A ^L
-^ zjr
- ^ y le
- aZ. J.\
0 ^^ tt
^
V \7^
A/
i it
^/
jZ
/
- l\'
. u ^
176
ALGEBRA
131. Special Forms of Equations.
(1) In this diagram are drawn the graphs ot
a;+2?/=6 and x+2y=2.
The hnes which
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•^1
^
^
"^j;,^^
>,
■^
r^
^'?
s
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N
■v
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y
V
N
these equa-
tions represent are seen to be
•parallel, that is, there is no point
at which they intersect. This is
equivalent to saying that these
equations have no solution.
Compare with art. Ill, where inconsistent equations were
discussed.
(2) If we draw the graphs of a;— 3^=10 and 2a:— 6?/=20
on the same sheet, we shall find that they represent the same
straight Ime, so that any points which lie on the graph of one
of them will also lie on the graph of the other.
These equations are indeterminate (art. 111).
(3) An equation like a:=^3 may be written a;+0?/=3.
This equation is satisfied by a;=3 and any value of y.
Thus, a;=3, y—\\ ir=3, y=2 ; a;=3, ?/=10, etc., satisfy
the equation. If we plot the pomts which have these co-
ordinates, we see that .t=3 represents a straight Ime parallel
to the axis of y and at a distance 3 to the right of it.
Similarly, x= —3 represents a line parallel to the axis of y and at a
distance 3 to the left of it. Also y= —4 represents a line parallel
to the axis of x and at a distance 4 below it.
Thus an equation which contains only one of the variables
X and y represents a line parallel to one of the axes.
What Ime does ,t==0 represent ? y=0 ?
EXERCISE 86 (1-4, Oral)
1. What is the graph of x=^ ? Of a;+3=-0 ?
2. What is the graph of 2/-4=0 ? Of 2«/+3=0 ?
3. If a;-)- 3?/= 11, express a; as a function of y and y as a function
of X.
GEOMETRICAL REPRESENTATION OF NUMBER 177
4. If 3x—2i/=6, express x as a function of y and y as a function
of X.
Solve graphically and verify :
5. x+3y=9, 6. a;+ y= 8, 7. .r— 2y= 6,
2a;+ t/=8. 3x-4)/=10. 2.r-3i/=:ll.
8, 2a;+3^= 6, 9. 2i/= a;, 10. x+4y=9,
3x+2y=U. 10y=ix—2. Zx—8y=-—3.
11. Show by graphs that the equations x+y=5, 2x--t-3//=12,
3x—2y=o have a common pair of roots and find them.
12. Show graphically that 2x-{-3y=lS and \x-\-^y=2 are incon-
sistent. What is peculiar about the graphs of these equations ?
13. Show graphically that no values of x and y wiU satisfy all of
the equations a;+?/=4, 2x—y=\\, 4.c+?/=13. What values satisfy
the first and second, the first and third, the second and third ?
14. Show the coordinates of the points where the graph of y—2x+3
cuts (1) the axis of x, (2) the axis of y, (3) the graph of y=6—x.
15. Show by graphs that the values of x and y which satisfy
2x—3y+l—0 and 5x— 2?/— 14=0 will also satisfy 3x— 4^=0 and
x-2y+2=0.
BXBROISB 87 (Review of Chapter XIII)
1. At what point does the graph of x + y = 5 cut the axis of a; ?
The axis of y ? Construct the graph. In the same way construct
the graph of x-{-4:y= —4. At what point do they intersect ?
2. How does it appear geometrically that two equations of the
first degree can have only one set of roots ?
3.* Plot the points (0, 0), (-3, 4), (3, 12), (-2, 0). What is the
distance between each consecutive pair of these points ?
4. From a certain point a man walks 5 miles E., then 4N., then
2W., then 3N., then 3E., then 4S. Using squared paper, determine
by measurement how far he is now from the starting point.
5. A man walks 8 miles W. and then 5S. Find by calculation
how far he must now walk to reach a point 4 miles E. of his starting
point.
N
178 ALGEBRA
6. If 11 lb. equal 5 kilogrammes, make a graph from which you
can express any number of kilogrammes in lb. or lb. in kilogrammes.
Read from the graph 3^ kilogrammes in lb. and 8| lb. in
kilogrammes.
7. What is the perimeter and area of the triangle whose angular
points are (0, 0), (5, 0), (0, 12) ?
8. How do you show that the point (3, —2) lies on the graph,
of 5x — 2y=l9 ? Which of the following lie on it: (6, 5), (1, —7).
(_3, _17), (4, 1), (_2, -12), (5, 3) ?
9. Find the area of the triangle formed by joining the points (5, 9),
(8, -6), (-7, -6).
10. Draw the triangle whose vertices are (2, 0), (10, 0), (5, 6) and
find its area. Why do the points (2, 0), (10, 0), (8, 6) determine a
triangle of the same area ?
Solve graphically and verify :
11. x+2y=l2, 12. 3x-4:y= 0, 13. y-x=4,
x-3y= 2. 4:X-3y=-U. x = 2.
14. y-2x=-Z, 15. 2x+ly = 52, 16. 2/ = ix+4.
x+2y=li. 3x-5y=l6. y=^x + 5.
17. What is the area of the figure formed by the lines whose
equations are: a; = 4, x=—2, y=3, ?/= — 1 ?
18. What are the coordinates of the middle point of the line joining
the points (2, 3) and (6, 5) ?
19. On the same sheet draw the graphs of the equations
y = x + 4:, y = 4:X—2, y = 2x+2.
What peculiarity is presented by the graphs ? What conclusion
do you draw concerning these equations ?
20. Draw the graphs of 2x-\-3y = 20, 4a; +6?/ = 35 on the same sheet.
What do you conclude as to the solution of these equations ?
Determine graphically whether these sets of equations are consistent
or inconsistent :
21. X- y= 4, 22. x+2y=10, 23. 2x+ y= 8,
4x+ y = 26, Sx- y= 9, Sx+2y = l3,
2x-5y= 2. 2x- y= 1. 5x-3y= 9.
24. Describe the triangle whose sides are represented by the
equations: 3x-\-2y=l4:, 5a;— 6?/= — 14, x+10y= — 14:. What are the
coordinates of its vertices ? (Verify by solving the equations in pairs.)
GEOMETRICAL REPRESENTATION OF NUMBER 179
25. At what point do the graphs of 2x-\-Zy=\2, ^x—2y=5
intersect ? At what angle do they seem to intersect ?
26. A teacher's salary is increased by S50 each year. His salary
for the first year is $750. Construct a graph from which you can
read off his salary for any year. What is his salary for the 8th year ?
In what year would his salary be $1300 ?
27. In the process of solving 2a; — 3^=1, 3x+2?/ = 8, by eliminating
y we have
2x—Zy=\, I 4a;-6i/= 2, 1 x=2, | x = 2,
3a;+2?/ = 8, | 9a; + 6i/ = 24. | 2x~Zy=\. \ y=\.
On the same sheet show the graphs of each of these sets of equations,
and thus show that they all determine the same point and that the
four sets are therefore equivalent.
N 2
CHAPTER XIV
HIGHEST COMMON FACTOR AND LOWEST COMMON
MULTIPLE
132. In Chapter IX. we defined the terms highest common
factor and lowest common multiple, and showed how they
were found in simple cases.
When the expressions under consideration can be factored,
the H.C.F. and L.C.M. can at once be written down from
the factored results.
A few examples are here given of a more difficult character
than those previously considered.
Ex. 1.— Find the H.C.F. and L.C.M. of
x^y-\-lxy^-\-12y^ and x^y—x^y^—l2xy^.
x^y + 7xy^+l2y^ =y{x^ + lxy-\-12ij^)=y{x+4y){x+3y).
x^y—x^y*—l2xy^=xy(x' — xy—12y^)=xy{x—4:y){x+3y).
Here the common factors are y and x+3y, and since the H.C.F
is the product of all the common factors,
.-. the H.C.F. =2/(a;+32/).
The L.CM. is the expression with the lowest number of factors which
will include all the factors of each expression,
.-. the 'L.C.M. =xy{x + 4y){x + 3y){x — 'iy)
Ex. 2.— Find the L.C.M. of
x^—1, x^-\-l, x'^—x and rr^+x^+l.
a;2-l = (a;+l)(a;-l).
x^-\-\ = (x-\-\){x^-x-\-\).
x*—x—x{x^— \)=x{x— \){x^-\-x-\- 1).
a;<+a;2+l = (a;2+l)2-a;2 = (a;2+a;+l)(a;2-x+l),
,*. the L.C.M. = .r(a;+ l)(a;- \){x^+x^ \){x^-x-\-\).
180
H.C.F. AND L.C.M. • 181
If the multiplications be performed the L.C.M. will bo found
to be x' — x. It is customary, however, to leave tlio result in tlie
factored form, as it is in this form that it is usually made use of.
EXERCISE 88
Find the H.C.F. and L.C.M. of •
1.* 4.x^yH, %xyh^, \2axyH.
2. x^ — y^, xy—y^, x'^—xy.
3. a^-h\ ab+b\ a^+2ab+b^.
4. a;2-7a;+12, .t2+2x-15, x^-9.
5. a'+8a+15, a--2a-35, a^+3a-lO.
6. 3z2-12a;+12, 3.r2-I2, 3x^-Bx-6.
7. x^—xy-\-xz—yz, xy—y^.
8. m^— 8, m*H.2— 4to^w^ 4)?i^— l&»i -|-16.
9. 6a3_663, 2a3+2a26+2a62.
10. a--\-ab—ac, a^-{-b-~c^-\-2ab.
11. a^-b^-c^-2bc, b^-c^-a^-2ca, c^-a^-b^-2ab.
12. x^+y^, x*+x^y^+y*.
13. 3a;2+7a;-6, 3x^-Ux+6, 6a;2_ 13:^+6.
14. 10ax—2a+15cx—3c, 2oa;2— 1, 25a;2-10a;+l,
15. x^-5x^-^6x, x^-3x^+5x-l5.
16. M* — U*, M"' — V^, U^ — V^, U — V.
17. a;3+2a;*-8.r— 16, a'3+3a;2-8.r— 24.
18. Show that the product of x-— 8.r+15 and x^+a;— 12 is equal
to the product of their H.C.F. and L.C.M.
19. The L.C.M. of a^— 5a+6 and a- — 6 is a^—^a^—^a+\2
Supply the missing term.
20. Find two trinomials whose H.C.F. is x—2y and whose L.C.M.
is x^—lxy^-\-%y^.
182 ■ ALGEBRA
Ex. 1.— Find the H.C.F. and L.C.M. of
x^-\-2x—3 and a;^— 8a:+3.
Here a;2-f2x — 3 is readily factored, but none of the methods
previously given will apply in factoring x'— 8x4-3, except by using
the factor theorem of art. 101.
The difficulty is, however, easily overcome thus :
z^ + 2x-3 = {x-l){x+3).
If the expressions have a common factor it must evidently be either
x—l or x-{-3.
By using the factor theorem, find if x—l or x+3 is a factor of
x^-8x+3.
When .^-1 = 0 or x=l, .'c3_8a;+3= 1-8 + 3= — 4,
.'. x—l is not a factor.
When x+3 = 0 or a;=-3, x3-8a;4-3= -27 + 24 + 3^0,
x+3 is a factor.
How can we obtain the other factor of x' — 8x+3 ?
We now have x2 + 2x-3 = (x— l)(x + 3),
and x3-8x+3 = (x+3)(x2-3x+l).
.-. theH.C.F.=x+3,
and the L.C.M. = (x+3)(x- l)(x2-3x+l ).
Ex. 2.— Find the H.C.F. and L.C.M. of
x^—lx-{-lO and a;^— 6a;2+lla:— 6.
The factors of x- — 7x+10 are (x — 5)(x — 2).
Here it is evident that x — 5 is not a factor of the second expression,
since its last term is — 6, which is not divisible by 5.
Is X— 2 a factor of x^ — 6x2+ llx— 6 ?
Complete the solution.
SXERCISEI 89
Find the H.C.F. and L.C.M. of :
1.* x2-3x+2, x3— 6.r2+8x-3.
2. a2-6a+5, a^-19a^-\-na+l.
3. x3-2x2+4x-8, 2x3-7x2+12.
5. .t3+3x2-4x, x3-7x+6.
H.C.F. AND L.C.M. - 183
6. If x—2 is a common factor of
a;H3a;--ar-2 and .x-3-4.r2+3.r+2,
find their L.C.M.
7. Reduce to lowest terms :
a3-19a62+3063 2x^-Ux'^—\2x
8. Find two expressions of the third degree in x, whose H.C.F.
is a;2— 5a;+6 and whose L.C.M. is a;*— 10,r3+3o.c--50x-+24.
133. Method of finding the H.C.F. of two expressions which
can not be factored by the usual methods. From the preceding
it is seen that the chief difficulty in finding the H.C.F. of
two expressions is in factoring the given exj)ressions.
If neither of the expressions can be factored by the usual
methods, another method may be used which depends
upon the same principle as that of finding the G.C.M. of two
numbers in arithmetic.
134. Fundamental Theorem. This method of finding H.C.F.
depends upon the following theorem :
If X is a common factor of any two quantities, then ■dc, is also a
factor of the sum or difference of any multiples of those quantities.
Thus, X is a common factor of mx and nx.
Then mx-[-nx, mx—nx, pmx-\-qnx, rmx—snx, are each the
sum or difference of multiples of mx and nx.
It is evident that each of these is divisible by x, the
quotient in each case being found by division, thus :
x\mx-\-nx x\mx—nx x\pmx-\-qnx x\r^3: — snx
m +n m —n ' pm -\-qn rtn —sn
The way in which the theorem is applied is shoAvn in the
following examples.
Ex. 1.— Find the H.C.F. of
.r3+4x2+4.r+3 and x-^-f 3x2+4a;+12.
Any common factor of these is a factor of their difference, which
is x^ — 9.
But a;2-9 = (a;-3)(a;+3),
.-. the H.C.F. is a;- 3 or 03+3 or (a;-3)(a;+3).
184 ALGEBRA
It is evident that x— 3 is not a factor of either expression, since their
terms are all positive. Therefore if they have a common factor it
must be x-\-3.
By applying the factor theorem, or by division, we find that a;+3
is a factor of each, and since it is the only common factor, it must be
the H.C.F.
Ex. 2.— Find the H.C.F. of
3x^-nx^~5x-\-10 and 3:^3-23^2+ 23a:— 6. •
Their difference = 6a;2 - 28a;+ 16 = 2(a;-4)(3x- 2).
Now 2 is not a factor of either and may be discarded, also x — 4 is
not a factor, since 4 is not a factor of 10 nor of 6. Therefore if there is
a common factor it must be 3a;— 2.
Divide 3a;— 2 into one of them and see if it divides evenly. If it
does not there is no common factor but unity.
If it does divide evenly into one of them, it is not necessary to divide
it into the other, for if it is a factor of one of them and also of their
difference it must be a factor of the other.
Ex. 3.— Find the H.C.F. of
3a;3-13a;2+23a;-21 and 6x^-\-x^-44:X-{-21.
Multiply the first by 2 and subtract the product from the second
and we get
21x^~90x + G3 = 9{x-l)(Sx-l).
Now since 9(a;— l)(3a;— 7) is the difference of two multiples of the
given expressions, it must contain all their common factors. Which
of these factors may be discarded ? Complete the solution.
We nfiight have obtained the H.C.F. thus :
The sum of the expressions is
9x^-12x'^-2lx=3xix+l){3x-l).
This expression contains all the common factors of the given
expressions.
Complete the solution by this method.
The object in each case is to obtain from the given
expressions an expression of the second degree. If this
expression can not be factored, it must be the H.C.F., if
there is any common factor other than unity. If it can be
factored the H.C.F. can then be found either by the factor
theorem or by ordinary division.
H.C.F. AND L.C.M. 185
In obtaining the expression of the second degree, the last
problem shows that it is sometimes easier to ehminate the
last terms than the first terms.
Ex. 4.— Find the H.C.F. and L.C.M. of
6a:3_5a;2_8x+3 and 4a;3— 8a;2+a;+3.
Eliminate the absolute terms and show that 2x — 3 is the H.C.F.
Since 2a;— 3 is a factor of each, the other factors may be found by
division, then
6a;3 - 5x2 - 8a; + 3 = ( 2a: - 3)( 3a;2 + 2a; - 1 ),
4:x^-8x^+x+3 = {2x-3)(2x^-x-l),
:. the L.C.M. = (2a;-3)(3a;2 + 2a;-l)(2a;2-a;-I).
Why is it unnecessary to factoi- Zx^-\-2x— 1 and 2a;^ — a;— 1 ?
Ex. 5.— Find the H.C.F. of
a-4—4r3-l- 10x2-1 la;+ 10, (1)
and x*-x^-^x^\-\^x-\5. (2)
Subtract (1) from (2), and we get
3a;»-14a;2 + 30a;-2o. (3)
Multiply (1) by 3 and (2) by 2 and add to eliminate the absolute
terms. Remove the factor x and we obtain
5a:3-14a;2 + 22a;+5. (4)
The common factor we are seeking must be a factor of both (3)
and (4).
Eliminate the absolute terms from (3) and (4) and show that the
H.C.F. is a;2-3a; + 5.
Find also the L.C.M.
Ex. 6.— Find the H.C.F. of
8:r4-f 4a;34-4a;2— 4x and Qx'^+^x^+^x^—^x.
Here 4a; is a factor of the first expression and 2a; of the second, and
therefore 2a; is a common factor. Remove these simple factors and
find the H.C.F. of the quotients, and show that the H.C.F. is
2a;(a;2 + a;+l).
135. Product of the H.C.F. and L.C.M.
Suppose that x is the H.C.F. of mx and nx, so that m and
n have no common factor.
186 ALGEBRA
Then the L.C.M. of mx and nx is mnx.
But XX mnx = mx x nx,
therefore the 'product of any two quantities is equal to the
product of their H.C.F. and L.C.M.
Is a similar theorem true concerning any three quantities
mx, nx and px ?
If the H.C.F. of two quantities has been found, we might
therefore find their L.C.M. by dividing their product by the
H.C.F.
BXERCISB 90
Find the H.C.F. of :
1.* x^—lx^^lZx-l^, a;3— 6a:2+a;+20.
2. a3-10a2+33a-36, a3-2a2-23a+60.
3. 6a;3+ 10x2+8x4-4, 6x3-2x2-4.
4. 2x3— 5x2— 20x+9, 2a;3+x2— 43x— 9.
5. 263+562-86-15, 463-4&2_96+5.
6. 3x3+ 17x2?/ -44x1/2 -282/3, Qx^-^x^y-ZZxy^-\-2%y\
7. 2a3-3a2-4a+4, 3a*-4a3-i0a+4.
8. 2x*-12x3+19x2-6x+9, 4x3- 18x2+ 19x-3.
9. 18a^6-3a'»6-12a36-3a26, 12a5c-6a*c-9a3c+3a2c.
10. x3-x2-2x+2, x*-3x3+2x2+x-l.
Find the L.C.M. of :
11. x3— 7x— 6, x3— 4x2+4x— 3.
12. x3+6x2+llx+6, x3+7x2+14x+8, x3+8x2+19a;+12.
13. 2x3+9x2+7x-3, 3x3+5x2- 15x+4.
14. x3-6x2+llx-6, x3-7x2+14x-8.
15. 20x*+x2— 1, 25x«— 10x2+1, 25x*+5.r3— x— 1.
16. Find a value of x which will make x3 — 13x+12 and
x3— 6x2— x+30 gach equal to 0.
17. The L.C.M. of two numbers is 70 and the H.C.F. is 7. If one
of the numbers is 14, find the other.
H.C.F. AND L.C.M. 187
18. The H.C.F. of two expressions is x—2, the L.C.M. is x-^— 39a; +70.
If one of the expressions is a;^— 7a;+10, find the other.
19. Two integers differ by 11. If they have a common factor,
other than unity, what must it be ?
EXERCISE 91 (Review of Chapter XIV)
Find the H.C.F. and L.C.M. of
1.* a;2-20a;+99, a;2-24x+143, a;2-2Lr+110.
2. x^-l5x+36, x^-21,x^-3x^-2x+6.
3. a^-b^,a^ — 2ab + b^,a^-b^.
4. x^ — 2x'^-l5x,x^ + x^—l4:X—24:.
5. 4a3-12a2-a + 3, 2a3-|-a2-18a-9.
6. x^—ax—bx+ab, x^ — bx—cx+bc.
7. x^ — 6x^+Ux—6,x^ + 4x^+x-6.
8. a;*+3a;3 + 3x2 + 5a;-12, a;*-4x3- 19x2+ 10a;+ 12.
9. 2a4+15a3 + 39a2-(-40a+12, 2a«+9a3_2a2-39a- 18.
10. x*—6x^y+l3x^y^-12xy^ + 4y*,x*+2x^y-3xhj^-4:xy^ + iy*.
11. x* + x'y'^+y\ x*—2x^y + 3x'y'^~2xy^+y\
12. Show that two consecutive integers can have no integral
common factor except unity.
13. Two odd integers which differ by 6 have a common factor
other than unity. What must it be ?
14. Find the H.C.F. of a;« + a« and x« + x*a* + a^.
rth
15. If the H.C.F. of a and b is d, show that the L.C.M. is -7 •
a
16. If a is the H.C.F. and b is the L.C.M. of three quantities, show
that the product of the quantities is a^b.
17. For what common values of x will
x^-3x^-x + 3 and a;^-4a;3 + 12a;-9
both vanish ?
18. Find two expressions of the second degree in x, whose H.C.F.
is x—l and L.C.M. is a;'-8x2+17a;- 10.
.„ „ , 18x'-3a;*+2aj+8 . ,
19. Reduce X2^3^8a?'-7x+12 *° ^°^^^^ *®"^-
CHAPTER XV
FRACTIONS
In Chapter IX. fractions were introduced and simple
examples of operations upon fractions.
In this Chapter the subject is extended and applications
made to more complicated forms.
136. Changes in the Form of a Fraction. Both terms of a
fraction may be multiplied or divided by the same quantity
without altering the value of the fraction. As previously
stated, the only exception to this rule is, that the quantity
by which we multiply or divide must not be zero.
The rule might be stated in the symbolic form :
a _ ma na _ a
b ~ inh nb ~ b
The case in which the terms are multiplied or divided
by ^ 1 deserves special attention.
From the rule of
sig:
ns for division
a
-b
is
seen
to be
the
a
same as — 7- ,
0
so also
is'
—a
b
—a
b ~
a
-b^~
a
b'
Similarly,
—a
-b
a
-a
b~ ~
= —
a
It is thus seen, that the value of a fraction is not changed
by changing the signs of both of its terms ; or by changing the
sign of one of its terms and at the same time changing the sign
before the fraction.
FBACTI0N8 189
Since (a— 6)x( — 1) = — a+6 or h—a, it is seen that a—h
and h—a differ only in sign, or that each one is equal to the
other multiplied by — 1.
That is, a—h= — {h—a) and h—a= — {u—h).
rp, a — b _ (a—b) x(— 1) _ b — a __ a — b _ b — a
c—d~{c—d)x{ — i.)~d — c~ d—c~ c—d
Also, since ( — a) x ( — 6) = ( + a) x ( + 6)=a6,
it follows that {a — b){c — d) = {b — a)(d — c),
m _ m _ —m m
(a-b){c~d) ~ (b-aj(d^) ~ (^'bj(d-c) ^ ~{b-a)(^^f
(^"Z^K^y] = (a^-q)(y-fc) ^ {a-x)(y-b) ^ ^^^
(b-xj{a-y) {x-b){y-a) (b-x)(y-a)
EXERCISE] 92 (1-29, Oral)
Express these fractions in their simplest forms with no negative
signs in either term :
-2
^- 4 •
4
- -2 ^•
-6
-9'
4.
— 3a
—a
— a
5.
y
5
6. 7.
—m
-iab
26
8.
—ax
-bx
-3x-5
»• 7 •
—axb
10. _^ . 11.
—a. —b
—c
12.
—X. —y
—a. —b
Express with the numerator a—b ,
»
b—a
13. 3 .
b—a
14. - — r • 15.
— 0
b—a
x-y
16.
b-a
—3{c—d) '
Express with the denominator c—d:
-5
d—c
—a;. —V
18. , ' ■ 19.
rf— c
a—b
d—c
20.
—m{x—y)
d-c^
Express with the positive sign before the fraction
-4
21.- ,
4
22. - — • 23.
a
~b'
24.
a—b
c
X
25. - ;•
a—b
26. -^-,- 27.
o— 0
x-2
x—y
28.
c—d
c+d
29. What is the relation between
and
x—y
a a+b , 6+
» V and ,
y—x a—b b—
a b—a
a' -3
and -
7''
190 ALGEBRA
(p — q)(q — f)
30. Write -. r^ — -^ , in four equivalent forms, with the positive
(x-y){y-z)
sign before the fraction.
31. Which of the following are equal in value :
(a— 6)(6— c)(c— a), {b—a){c—b){a—c), {a—b){c—b){a^c),
{u-b){b-c){a-c), {b-a){b-c){a-c) ?
137. Reduction of Fractions to Lowest Terms. The formula
, - = 7 may be used to reduce a fraction to its lowest terms,
ox o
by dividing both terms by all the common factors.
Ex. 1.— Reduce - , , , -l , .
x^-\-y^ = {x-\-y){x^—ocy-\-y^).
x'^ + x'^y^+y^ = {x^-\-y^)^ — x^y^={x^+xy-\-y'^){x'^—xy-\-y^),
x+y
Ex. 2. — Reduce
the fraction - „ , , „
(»~+xy+y^
a2 + 62-c2 + 2a6 = {a + 6)2-c2 = (a+6+c)(a+6-c).
Complete the reduction.
T. o T, T "a;2-lLr+28
Ex. 3.— Reduce ^r-^ — „ „ , _ rn,
2x^—Qx^-{-Tx—Q0
x^-nx+2S = {x-4)(x- 7).
Which of these factors can not divide into the denominator ?
Complete the reduction.
Ex. 4.-Reduce e.3+3,._5,+ i
Here the factors of neither term can be readily obtained, so the
common factor must be found by the method of art. 134.
Eliminate the x^ and we obtain
33x2 + 33a;-ll or ll(3x2 + 3x- 1).
This expression must contain any common factor of both terms.
Since 3a;- |-3a;~l can not be factored, what conclusion can be
drawn ? Complete the reduction.
FRACTIONS
191
EXERCISE 93
Reduce to lowest terms :
a-4-3a+2
^' 4^2+ 8a; +3 '
a^-\-a'^b+ab^+b^
«3+2a2+2ar+l "
a2-4a+3
4a3_9a2_l5a+i8'
2x3-x-2+2a;-3
7 *
6.
8.
11.
13.
2.^3+3x2+4x4-3
_ «^+a2-3a-3
a5_a4_2a3+2a2_3(j_5 '
10.
12.
14.
a;2+7xy— 8y2
a;2+ 5x2/— 24^2
x2+2xy+y2-za
2-3.v-2y2
4— 5?/— 6?/2
x^— x2— 2x
x3— 3x2+4 "
3x2-3x-18
6x5-12x4-18x3"
3x3+4x2_6x-8
36x3+27x2-40x-16
2x«— 4x3— 2x2— 12x
4x4+2x3+6x2— 4x '
138. Addition and Subtraction. In adding or subtracting
fractions we should be careful to note whether any of the
given fractions can be reduced to lower terms. When the
result is obtained we should examine it to see if it can be
reduced.
2y_ _ xy^-\-y^
Ex. 1.— Simplify - J^ +
^ -^ X x—y
The expression
x'^—xy
yH^+y)
x-y , _2y ^
X x—y x(x—y){x+y)'
x—y
+
'■iy
y
x—y x{x y)
= (^^^0H-_2xy-^ _
X(x-y)
_ X2 _ X
x{x—y) x—y
The form of the last fraction in the given expression shovild prompt
the pupil to examine whether it can be reduced.
192 ALGEBRA
Ex. 2.— SimpUfy —^ + ^
x—2 ' a;2-3a;+2 a;^— 4x+3
The expression
x~2 ' (x-l){x-2) (a;-3)(x-l)
{x-l){x- S)+x- 3- 2(a:-2)
(a;-2)(a;-r)(x-3)
a;*— 5a;+4 _ (x— 4)(a;— 1) a;— 4
(a;-2){a;-l)(a;-3) (a;-2)(a;-l)(a;-3) (x-2)(x-3)
Simplify ;
EXERCISE 94
l.*_^__L_L. 2. -i L.. 3. a;+y a;— y
4.
a-4 a-7 ^ 2a^ 2a „ a^
a
a-2 a-5 a^-b^ a+b a-a^ l+a^
2x2 2a;2 a^— .V ,1 <. x^— 4?/^ x—2y
x^—y^ x^-\-xy ' x^—y^ 2x—y ' x^-}'2xy x
10 1 1 11 J^ 3?/ x^+y"
* a;2_[.9a;_^20 x2+12a;+35 ' x+?/ x—yx^—y^
a;2— 3x— 10_a;2+2.T— 3 a:— y 2.r x^+x^y
a;^— 8x+15 x^— 3x+2 ' y x—y x^y—y^
a^—ab-^-b"^ a^-\-ab-\-b^ x-\- • 4xy _y—x
a—b a-\-b ' x—y x'^—y^ x-\-y
a-b a+b _ a^+b^ ^^ 3x^-8 _ 5x+7 _2_
* 2{a+b) 2{a—b) a^—b^' ' x^-l a;2+x+l a;— l'
18 1 1 2a+96 1 ^g Jl 2 1
■ 2a-36 4a2-962^2a+36 ' x-1 x-2"^x-3
20. — i_+ 3 *
21.
x2-3x+2 ' x2-7x+10 x2-6x+5
(a+6)(6+c)(c+a) _ a-\-b _ 6+c _ c-\-a
abc cab
22.
23.
FRACTIONS 193
a2-62+26c-c2 _ c2+2ca+a2_62
62_c2+2ac'_:o- 62:p26c+c2— a2 '
(a+c)2-62 (6+a)2_c2 + (^qi^pITaa
g. o^— 2ffl 3a 5a
a2-a-2 6a-4 ^ 6a2+2a— 4
25. -i— 1-and ^ ^ ^^
26.
a;-?/ a;+?/ a;-?/ x+y x^-^-y^
1 1 22/ 4?/3
27.
a;—?/ a;+«/ a;2+?/2 x*-i-y*
_l 1_ _ 2x
3—x 3+a:~9+a;2'
28 -^ 4- -^ ^ 2a2 4a262
29 ^ ^ 1 ^ 3^
4— 4.r 4+4a; 24-2a;2~iT^*
30. Solve -^- 1 1^ ^^ _1 /Verifv-k
139. Special Types in Addition and Subtraction.
We have already seen that
b—a= — {a—b) and a—b= — {b—a),
or (a— 6)-f(6— a)= — 1.
When a—b and b—a occur in the factored denominators
of different fractions, which are to be combined, it is not
necessary to inchide both of them in the L.C.D.
Ex. 1.— Simplify -^ + -^ .
^ -^ a—h b—a
Here only one factor is required in the L.C.D. and we may use
either a — b or b — a.
If we decide to use a-b, then it is better to change the second
fraction into the form .
a — b
Then -^ + ^ = ^ ^ = ?Z1^=1
a — b b — a a — b a—b a — b
194 ALGEBRA
0,^4 3 , x—S
Ex. 2.— Simplify r + :j . •
^ x—l x-\-l l—x^
The denominator of the last fraction should be changed to x^ — 1,
so as to be the product of a;— 1 and a;+ 1.
4 3 x-3
The expression
x—l x+1 x^—V
4(a;+l)-3(a;-l)-(a;-3) 10
(x-l){x+l) (a;-l)(a;+l)
Ex. 3. — Simplify
1,2 3
{x-l){x-2) ' {x-2){S-x) {x-3){i-x)
Here there are only three factors x—\, x—2, x—3, required in the
L.C.D.
We therefore change the second and third fractions so that the given
expression
+
{x-l){x-2) {x-2)(x-3) ' (a;-3)(a;-l)
Complete the simplincation.
140. Cyclic Order, Suppose we wish to simplify
b-\-c c-\-a a-\-b
{a—h){a-c) "^ {b-c){b-a) "^ {c—a){c-b) '
The L.C.D. in this case will contain three factors and it
might be written in different forms as
(a— 6)(6— c)(c— a), {a—b){a—c){b—c), etc.
The pupil is advised to write the factors in what is
called cyclic order.
If we arrange the letters on the circumfer-
ence of a circle, as in the diagram, and follow
the direction of the arrows we see that a is
followed hy b,b by c, and c by a.
Thus, if we write a — b as the first factor, then
changing ato b, b to c, and c to a, we write the second
factor b — c and the third c — a.
If we write the L.C.D. as (a— 6)(6 — o)(c — a), we should change the
fractions so that these factors appear in the denominators.
FRACTIONS 195
The given expression then
b+c c+a a-{-b
ia-b){c-a) (6-c)(o-6) {c-a){b-c)
-{b+c){b-c)-{c+a)(0-a)-{a+b)(a-b)
{a-b){b-c){c-a)
(a — b)(b — c)(c — a)
= 0.
„ CI- IT ^c ca ^ ab
Ex. — Simplify j-- + -pr r-r : +
{a—h){a—c) {b—c){b—a) {c—a){c—b)
Proceed as in the preceding example and you should get the restilt
— b^c-'rbc^~c^a-\-ca^ — a^b-\-ab^
(a — b)(b — c)(c — a)
This fraction is equal to unity, for the numerator is equal to the
denominator. Prove that this is true.
EXERCISE 95
Simplify :
1 1
1 *
a^+ax x^^ax 2a— Zb 36— 2a
„ x+3 x—S . ^ . 1
x—2 2—x x^—9y^ 3?/— a;
5 2 3 g x+a x'^-aP-
a^—ab 62_^" ' x—a a^—ax'
y _2 3 x2-3 ^ _5 4 16
x-l x+1 l-a;2' ' x-2 2+x^ 4:-x^'
3 3 24
9 _^ - J^ 4- y^ 10
x—y x+y y^—x^ a;+4 x— 4 16— a;^
11. ^ ^ + -^,
^ 3a+2a; _ 3a— 2a; 16a;2_
3a— 2« 3a+2x "^ 4a;^— 9a2
j3 _1 4 8_ ^ 3a;+7
a;— 1 1— a; 1+x a;^— 1
02
106 ALGEBRA
a c
14. — — .
c(a—b) a{b—a)
.5 1 I 1
* {a-b){c-a) "^ (b-a)(c-6
16. , ^_^+ ^
{x—a){a—b) {x—b){b—a)
2 2 1
17. - +
18.
y^ — x^ x^ — 2xy-\-y^ x^-\-'xy
3—x 3+x l-lQx
r=^ 1+3^ 9a;2-l '
2 2 4
19 I _)_
' x^-Sx+ 15 ^ a;2-4a;+3 ^ 6a;-a;2-5
20 ct+^ I *^+c c+a
(6— c)(c— a) (c— a)(a— 6) (a— 6)(6— c;
(a—b){a—c) {b—c)ib—a) (c— a)(c— 6)
22. ,-^! ,+,^ ^; .+ '^^
(a— 6)(a— c) tb—c){b—a) {c—a)(c—b)
23. , -f, , + , I ,+ ^
(a:— 2/)(x— 2) {y—z){y—x) (z-x){z—y)
ax— be , bx—ca , cx—ab
24. 7 tTT ^ + /IT- \/T ^ + '
(a— 6)(a— c) (6— c)(6— a) (c— a)(c— 6)
• /_■> 19\/ .0 .9x "t" /to -9\/l,'> _9\ '
„ 6c(a+(i) , ca{b-\-d) ab(c-\-d)
^"' ; — iT^/ ^ ~r 77 — \/£ ; +
27.
(a— 6)(a— c) (6— c)(6— a) (c— a)(c— 6)
a^ 2(6+0.) 2(a2"+p) ^ 6* - a ' '
x-5 X+5J \x+3
'»• J^ + iis-^-^ (Check when .=2.)
FRACTIONS 197
141. Multiplication and Division. The ordinary cases in
multiplication and division of fractions have been treated in
art. 74. Some special forms which appear are illustrated
in the following examples.
Ex. 1. — Multiply a -\ by a ,
^ -^ a—x "^ a-\-x
Here the mixed expressions shovild be reduced to the fractional
form before multiplying.
r_, , , a^—ax+ax a^-\-ax—ax a*
The product = X
a-\-x a^—x^
Ex. 2.— Multiply ?+^ + l by J+^-1.
Multiply this in the ordinary way, by multiplying each term of the
one by each term of the other.
We should recognize that the first expression is the sum of 7- H —
and 1 and the second is the difference.
The p,„duct ^ (» + 5)'_ .= »_v 2 + ^^: _.=«_!+,+ ^^:.
Or, we might proceed as in Ex. 1, thus :
ah ab a^b- a-b^
a^
fe2
q2 f)2
This result is seen to be the same as ^ + 1 H — ~, and the answer
may be given in either form.
x^ 1 r 1 1
Ex. 3.— Divide ^ + - by 4-- + -.
y6 X y^ y X
The dividend = ^!±^ the divisor = ^^-^V+V^
xy^ xy^
the quotient = ^3 x j
x^-\-y^ xy^ _ x-\-y
xy^ x^—xy-\-y^ y
Divide in the ordinary way and get the quotient - + 1.
We must not make the error of thinking that we can invert
the divisor, or take the reciprocal of it, by inverting
198
ALOEBRA
each term of it, and change the problem to one in multi-
pUcation, thus :
-^ is a-\-h, but the reciprocal of
The reciprocal of
U--T-0'
- + Y- is not a-\-h.
a 0
_ 1,1 a-\-h J .^ . , . ab
For - + r = — .— and its reciprocal is — -y
a 0 ab a-\-b
Miiltiply :
EXERCISS 96
1*
3.
-x—6
c^-2x-S
a;2+4a;+4' x^-lx+12'
a;^4-2a;-15 x^+lx-U
x^+Sx-dB' a;2+9a;+20 '
^' 1 n,' ^ + — A •
a-\-o a—o
7. :^2+i+l x2-l + i
a2_2a6+62' a^-^-ab'
xy
^ y
2 — 3:8
6. y + -^^, y- , , 2 , ,
1/— a; a;+2/ ^z^ + aj^
8. a2 _(_ 2 + ,
r2'
2 +
9.
J . hx ^
6 + — , 1
a
a a
a-\-x bx
Divide :
a:2_lla;+30
11.
13.
14.
15.
by
x^—6x
a;2— 6a;+9 '' x'^—Zx
a2-j-62_c2_i_2a6 a+6+c
^6+c— a'
X y
c^—a^—b^+2ab
X y
X
X . y
y^ x^ y X y X
y^ x'^ y X
Simplify :
a+l
17.
a+2
a^— 2a a^—a ' a^—a^
10.
12.
16.
18.
t3_53 (j+ft a^^ab+b"^
as+fcs' cj_6' a2_^a6+62
^ by 2'
x^—xy xy—y^
x^'—y"
x*^—y*
by
x^—y^
x—y
a2-4 ^ a^+2g
aH-5a ■ a2-25'
FRACTIONS 199
' x~-4x+3 ^ 6X-+X-2 • 3.r2-7x-6'
20. — + -^ - (a+a; 2.
21. ri+'Yi+-Ufi+?X'+f).
22.
a/\ aj ' \ b,
(a4-6)2-(c+<?)^ . (a-c)2-((Z-6)2
(a-j-C)2_(6+d)2 • (o_6)2_(rf_c)2
g2-64 a2+12g-64 _^ a2-16ft+64
■ a2_f.24a+i28 ^ a3-64 " a2+4a+16 '
„^ / 5a 26 \ / 2a 26— a
24.
a— 66 3a— 26/ " Va+26 26— 3a
142. Complex Fractions. A complex fraction is one
which contains fractional forms in either the numerator or
denominator or both.
Thus, - is a complex fraction and is, of coxirse, only another way of
writing r -i- ;; • It is simplified in the usual way by changing it into
a d , . , .ad
r X - which equals r~ -
A complex fraction may sometimes be easily simplified
by multiplying both terms by the same quantity.
Thus, — ~- = —TV — on multiplying each term by 4.
a+26 a
a+6 "*" b _ 6(a+26)+o(a+6) _ a^ + 2o6 + 26' _ ,
0+26 ^«_ ~ (o+26)(a+6)-a6 " a2+2o6 + 26* "
b a-^b
Here both terms were multiplied by b{a -\- b).
200 ALGEBRA
If the L.C.D. is not the same for both terms of the
fraction, it is usually better to simplify the terms separately.
x+l x—l
_,. ... x+2~^^
Ex.-Simplify ^_^ ^_^^.
The numerator =
The denominator =
x+2 x—3
{x+l){x-2)-{x-l)(x+2) -2x
(x+2)(x-2) (x+2)(x-2)
(a;-2)(a;-3)-(a;+2)(a; + 3) _ - lOa;
(x+2){x-3) ~ {x+2){x-3)
-2x {x+2)(x~3) cc-S
the fraction = (^^2){x-2) ^ -lOx == 5{^^ '
EXERCISE 97
Simplify :
6a ^ 1_1
^ ^ 2 Qa_ g g— 6 ^ :» y
12c * 56 * a-\-b ' x—y
12c
1 11 , a;2+w2
1+a; ^ a+6'^a-6 „ x+y
1+a; a+6 a— 6 .a;+?/
„ a^— 5a
''• _3c^ • (3x+2/)2-(3:i:-2/)2
■^a-3
10. ,y-l • a + b- '' '
a^-ab+b^
c d
12. ^±L_i±£. ^ 13. i
a 6
6+c c+a
2x + 3 -
1__j!_
a;+6
FRACTIONS 201
14. "! +— ^ 15 ^"^^^
^!+i"_6 ^!±^^-a A-Jl + L
a+6 a+6 «^ «& b^
16. Find the value of when x= ~ , y— — — .
x+y a—b a-\-b
17. Find the vahie of — ■- — ^ when a = —-^, b= ) — --^.
l+2a+6 x+y {x^y)^
EXERCISE 98 (Review of Chapter XV)
Simplify -.
a
^ * a a_ , 2a- „ a _ &
o __«^ _ y _ jx-y)- , 2 _^ x__
"** x+Sy Sy-x a;2-92/2" ' 1_1 j_^ j_?
X y X y
1 1 2 4 , X 11
{'+^}-('+'S0-
(l+a;)2-{l-a;)2
26c \/, 26c \/, 6c-c2\ 62
/, 26c \/, , 26c \/, 6c-c2\
62 h^+c^
^^' x+1 a;«+3x+2"^ a;3 + 6x2+llx+6'
o^-o* a6 + 62 2a26
-'■■'■" _a ta ~r _ a , j, a i
o«-6« cH^ ct+6
a2+2a6 + 62 -^ o2_52 -^ „4^.„2ft2 + 54
ifi «+a: g-a; 2x»
a* + oa;+x2 "•" a^-aaj + x*"*" o* + o2a.^_|_^4*
14. (l-o2) ^ |(l-a)2 - (a^- 1) ^ (a + ^)}'
._ 6+c— g c+g— 6 a+6— c
• (g-6)(g-c) "•' (6-c)(6-g) "^ (c-g)(c-6) *
202 ALGEBRA
16. 7^7 , + , 1^ +
(x-y)(x-z) (y-z){y-x) {z-x)(z-y)
ix+e 3x—9 15a;- 10
6a;2+5a;-6 ' 2x^-Zx—^ 9x2-12a;+4
18. Express the product of
8-6a;+a;2 , 1 1
and
1+x (a;-l)(2-a;) (a;-2)(4-a;) (cc-4)(l-a;)
as a fraction in its lowest terms.
19. How can you show mentally that 3 is the sum of
a; XX ^^ ah c_ ^
x-\-a x-\-h x+c x+a x-\-b x-\-c '
20. Divide -^ + -^ ^, by -^ + '^
x-\-y x—y x^—y^ x-\-y x^—y^
22. Divide^^ + '-^+'-^-l by 2-(l+Ul).
a b c \a 0 c/
23. Show that (i_^^),_(^^y), - (i_^.)(i_y2)_4^ '
24. If a=2^^, 6^-24,. c=^^, d=^^, prove a=a:.
25. Find the product of
l+x
and l-{-x-\--
1 + x^ 1 , a;
a;— 1
26. Subtract - ,^ from r and determine which fraction is the
6+10 b
greater if a is greater than b and if both a and 6 are positive.
(l+o6)(l + ac) (l + fec){l + 6a) (l+ca)(l+cb)
27. Add
(a— 6)(a— c) ' (6 — c)(6 — o) ' (c — a)(c — 6)
28. Show that 1 r has the same value when a;=a+6 as it
x—a x—o
has when x— — ~r •
a+b
29. Prove that the product of any two quantities is equal to their
sum divided by the sum of their reciprocals.
FRACTIONS 203
30. If x= , y = — T — , 2= , prove that
a 0 c
ocyz-\-x-\-y-\-z = 0.
x—y , y—z z—x , ^, ,
31. If a=—~, b=^-~—, c=— — , show that
x+y y+z z+x
(l_a)(l_6)(l-c) = (l+a)(l+6)(l+c).
32. If a and b are positive, which is the greater
a4-36 a+2b „
— ■ or — '
a+26 a + b '
_1 1 a+Sb 463
a-6 2(a+6) 2(0^+62) a*— 6«
x—y y—z z—x
33. Simphfy
34. Add
■(a;-2/)2' a;2-(2/ — 2)2' ^/S— (z — a;)2
35» Simphfy -^ % + — ^ ^.
36, When x= , find the value of
a-\-c
x—2a x + 2a ^ac
x-\-2c x—2c ^ a;2-4c*
y—x y—x
37. Simphfy \ — 4±^, ^^ I - f ^ - -Y
^ x{y-x) ^ y{y-x)l • \x y!
1+xy l — ocy
CHAPTER XVI
FRACTIONAL EQUATIONS
143 If an equation involves fractions, the fractions may
be removed by multiplying every term by the same quantity.
In Chapter VI. simple examples of fractional equations
were given.
The case in which two fractions are equal deserves special
attention.
CI c
Thus, a T = -, and each side is multipUed by bd we have
0 a
- X bd = J X bd.
b d
ad=bc.
144. Cross Multiplication. It is thus seen that when two
fractions are equal, we can remove the fractions by multiplying
the numerator of each fraction by the denominator of the
other and equating the results. This operation is sometimes
called cross multiplication.
Ex. 1.— Solve
a;+3
^9
Cross multiply {x-5)(x + 9) = {x+3)(x-l),
.'. x^-\-4x — 45 = x^ — 'ix — 2\,
8a; =24,
x=3.
Verify by substitution.
This method is applicable only when a single fraction
appears on each side of the equation.
FRACTIONAL EQUATIONS 205
Ex. 2.— Solve — hr- = 7 t- •
x-\-3 X—4:
Simplify the right-hand member and we have
4x4-17 _ 4a;— 18
x+Z' ~ x—4: '
Now cross multiply, complete the solution and verify.
EXERCISE 99
Solve and verify :
^ 3a:+l_a:+12 2 3a:-5 3a--l
2 3~" * 5x-Z~ 5x-\'
* 3 II ~ ■ * x-S .T-5"~
g x+2_x-2_x-l g a;+l a:-3_a:+30
* ~5 ~2~ ^7~' ' 2 ~3 13
7 7a;-3 _ 2a:+5 ^ x^+Tx-G _a:+l
2 3' ' a;-+5.r— 10 "x— l"
9. ^±^ = 2-^+1. 10. -L+l^_l_.
X— 1 X- — 3 X— 1 .T x+l
x+2"^x-2 x-3" *" 3x-2 6x-l
13. 2x+38^6x+8_j ^^^ 2/-8 _ y-l2
15.
19.
x+12 2x+l 2/2-8^+15 2/^-122/+30
2x+7 8x+19_5a;+ll j^. x-1 •-?:— 5 _ ^
3 "12 7x+9 '* .T-2 x-3
6-8x _3_ ^ ^ 2x+7 3x-5 ^ 5x-f9
^^^^r^~ * ' x+l x+2 ~ x+3 '
4x3+ 4x2-(-8x+J ^ 2x2-{-2x+l
2x2+2x+3 ~' x+1
20. ?^=:B=,^=f+i.
X— 3 X— 5
21 ^—^^ I ^^ _ a;-2 _ 2x— 3 „ -
' 1-5 ^1-25 T8"~ 9 ^'^'
206 ALGEBRA
22. Solve — - — '- = ^ by first reducing each fraction
3a;— 4 4a;— 5
to a mixed expression.
23. Find three consecutive numbers so that the sum of ^ of the
first, i of the second and \ of the third may be 30.
24. Divide 300 into two parts so that if one be divided by 5 and
the other by 7, the difference of the quotients will be 18. Give two
answers.
25. How much water must be added to 100 lb. of a 4% solution
of salt to make a 3% solution ?
26. A pupil was told to add 3 to a number and to divide the result
by 5. Instead of doing so he subtracted 3 and multiplied by 5 and
obtained the correct answer. What was the number 1
27. A man bought 180 lb. of tea and 560 lb. of coffee, the coffee
costing -^ as much as the tea per lb. He sold the tea at a loss of 25%
and the coffee at a gain of 50%, and gained $62-60 on the whole. What
did the tea cost per lb. ?
28. I sold some butter at 25c. a lb. If I had received 5c. more
for 1 lb. less, I would have received 2c. more per lb. How many lb.
did I seU ?
29. If I walk to the station at the rate of 11 yards in 5 seconds
I have 7 minutes to spare ; if I walk at the rate of 13 yards in 6
seconds I am 3 minutes late. How far is it to the station ?
145. Fractions with similar Denominators.
^ , ^ , a;+6 2x-18 , 2x+3 16 , 3.T+4
Ex. l.-Solve — 3- + ^_ = - + -^.
Here we might multiply each term by the L.C.D., which is 132.
It will be found simpler, however, to remove all the fractions but the
first, to the same side of the equation, as they are easily reduced to a
common denominator.
g!+6 _ 2a;- 18 _ 2a;+3 ,16, 3a;+4
11 ~ 3 4 "^ 3 "^ 12 '
a;-f-6 _ 4(2a;-18)-3(2x+3) + 64+3a;+4
■ 11 ~ 12
Now simplify, cross multiply and complete the solution. The
correct answer is a; =5.
FRACTIONAL EQUATIONS 207
This problem shows that the denominators of certain
fractions are such that these fractions can be conveniently
combined when they are grouped on one side of the equation.
Ex.2.-Solve __^^— ^ + __^.
Since 4x+4 = 4(a;+l), it is seen that it is simpler to combine the
second fraction with the first than with the third.
2x+3 Jx+5_ _5x+A
Ic+T ~ 4(a;+l) ~ 5x+ 1 '
Subtract the first two fractions, complete the solution and verify
the restilt.
Ex. 3.— Solve — _ H _ = 0.
x—2 x—3 x—5 x—o
Here it is too laborious to multiply all the fractions by the
L.C.D. It will be found easier to change the equation so as to have two
fractions on each side, then simplify each side and cross multiply.
Solve by transposing the last two fractions, also by transposing
the second and fourth, and compare the results.
Ex. 4.-Solve -^„ + h> = "L + - Tq *
x-\-5 x-\-2 x-\-4 x-{-3
AAA- u -A 2x4-7 2a: + 7 ...
Adding on each side, , , _ — -^rx = „ , „ — -^5 ' (1)
2a;+7 2a;+7 _
x^ + lx+lO a;2+7a;+12
.-. {2x+l) (^2^7^^ 10 - a.2+7a;+i2) = ^'
x=-3^ or x2 + 7a;+10=a;2 + 7a;+12.
Since the equation a;* + 7a;-f 10 = a;* + 7x+ 12 is impossible, the
only root of the given equation is a;= — 3i. (Verify this root.)
If in line (1) we divide each side of the equation by 2x-\-l,
an impossible equation will result. It is not allowable to
divide both sides of an equation by a common factor unless we
know that the factor is not zero. Here 2a; +7 might be equal
208 ALGEBRA
to zero, and, in fact, would be if a;=— 3|. If x= — 3| the
equation in line (1) is satisfied as each side becomes zero.
Solve the equation by writing it in the form
^ ]__ J^ |_
x-\-5 x+4"~a;+3 x-\-2'
EXERCISE 100
Solve and verify :
1 2a:+l 6a;— l_3.r-2 2x+3 _ x—l_ _ x-\-2
5 15 " Qx+3 ' ' 4 6a;— 8 ~ 2 '
3 a:+3 3a;+5_2a:+l 6a;+l 3a:— 1 2x—1_q
7 6a;+2"~ 14 * 4 2 3a;-2
5. ^ + -A- = ^1+ ''
8.
10.
11.
12.
14.
a;-8 2a;-16 24 3a;-24
A-L 5a;— 5 _6x+7
12"^12a;+8~9a;+6"
13a:-10 4a;+9 7a;-14 _ 23a:-88
36 "^ 18 12 ~17a;-66"
3a:— 4 _ 1 6a;-5
6a;-9 ~ 12 8a;— 12 "
„ 5a:- 17 , 2a;- 11 23 3a;- 7
13— 4a; 14 42 21
1 5a;-7 4a;-3
10 10a;-5 4a;- 2
J- 1___1 1_
a;— 1 a;— 2 a;— 3 a;— 4
1111
a;— 10 a:— 5 a;— 7 a:— 2
13. -^-+ 1 3 1
3a;+12 6a:+24 2a;+10 a;+6
_3 ^__8 \_
x—5 x—1 1—x 5—x
._ a:— 8 , a;— 4 a;— 5 , a;— 7 ,, , .
FRACTIONAL EQUATIONS 209
,„ 2x-21 , x-1 x-\2 , 2a;- 17
lb, =
a-U a;-8 .r-13 x-9
2x 27 . 1
17. Solve the preceding example by changing — - into 2 -|
and making similar changes in the other fractions.
4a:- 17 10a:- 13 _ 8a:- 30 5a;-4
' a;-4 "^ ~2^^^ ~ '2a;-7 ^ r-l '
5a;-64 2a;-ll 4a;-55 a:-6
19.
a:— 13 a;— 6 a:— 14 a;— 7
20. ^ + ^i + ?z:? = 3.
a;+l X — 2 X — 1
c 1 x—a , x—b , x—c „
21. Solve 1 1 = 3.
o + c c-\-a a-\-b
22. If a-—b'-=a—h, does it follow that a must be equal to 6 ?
What is the alternative conclusion ?
146. Literal Equations with one Unknown. Equations often
occur in which the known quantities are represented by
letters instead of numbers.
These are called literal equations.
The same methods are used in solving them as were used
in solving equations with numerical coefificients.
Ex. 1. — Solve ax=bx-\-c.
ax—bx=c,
.'. x{a—b)=c,
_ c
a — b
Solve 8a:=3.r+20.
8a;- 3a; =20,
.-. 5a; =20,
.-. a3=\'^ = 4.
Here the letters a, b, c represent some known immbers
whose values, however, are not stated, while x represents
the unknown whose value is to be found in terms of a, b and c.
Usually the earlier letters of the alphabet are used to
represent known quantities, and the later ones x, y, z to
represent unknown ones.
Compare the two solutions given. They are practically
identical. When we work with numerical coefficients the
result can usually be expressed in a simpler form.
P
210 ALGEBRA
Note. — The pupil must not make the mistake of giving a;= ■
as a solution of ax = bx-j-c.
This statement is true, but it is not a solution, since it does not
give the value of the iinknown in terms of known quantities only.
Ex. 2.— Solve a{x—2)—b=a—2x.
Removing brackets, ax—2a~b=a—2x.
Transposing, ax-{-2x=a-\-2a-^b,
:. x{a+2) = 3a+b,
_3a + b
The result should be verified by substitution, but this will
frequently be found more troublesome than the solution.
When it is not verified in the usual way, the pupil should
review his work to ensure accuracy.
T-, ,» o, 1 ^—^ x—a
Ex.3. — Solve =-,
a—x b—x
Cross multiply, bx—b^—x^-\-bx=ax—a'^ — x^-\-ax,
:. 2bx-2ax=b^-a^,
_b^-a^ _ b + a
^~2{b-a)~ 2
Verify by substitution,
BXERCISE3 101
Solve for x, verify 1-12 :
X
1. mx-}-a=b. 2 ax=bx+2. 3. a-\-- = c.
0
x-\-a _1 x—a_^a „ x—c a—h
x—a 3 * x+b b ' x-\-c a-\-b
_ a_ 6 x—a_x—b x—ax—b_,
X x—a-\-b. ' x—b x—a ' 2b 2a
10. ^-^ = -_^. 11. aix-a)+b{x-b)=0.
b a a c
12. l--^-=:--l. 13.* {a+x){b+x) = {c-\-x){d+x).
FRACTIONAL EQUATIONS 211
14. {ax-h){hx+a)^a{hx^-a). 15. ^fc^) _ ^i^±£) =a;.
') a
a b _a—h
x—a x—b x—c
17. x{x—a)-\-x{x—b)=2{x—a){x—b).
18. {x—a){x—b)^{x-a-b)^
19. J^ L_^_i L_.
x—a x—2a x—3a x—4a
20. ix—a){x—b)—(x+a){x+b)=(a+b)K
21. (o+a;)(6+a:)-a(6+c) = ^^ + a;2.
0
22. {a+x){b-x)+x^^b{a+x) — —'
23. a2^+63+a6a;=a3_62^.,
24. The excess of a number over a is three times its excess over b.
Find the ntimber.
25. Divide the number a into two parts so that one part may
contain b as often as the other will contain c.
26. Divide a into two parts so that m times the greater may exceed
n times the less by b.
27. A rectangle is a feet longer and b feet narrower than a
square of the same area. Find the side of the square.
28. If a number be divided by a, the sum of the divisor, quotient
and one-third of the number is 6. Find the number.
29. A man sells a acres more than the mth part of his farm and
has b acres more than the nth part left. How many acres were in the
farm ?
30. Solve {a—x){b—x)=ic—x){d—x).
Check by putting a=l, 6=6, c=2, d—3.
31. If 5 =: - {ci-{-l), solve for n ; for a ; for L
I
32. If 5 = , solve for a ; for I ; for r.
r-1
33. If s=at + ^gt-, solve for a ; ior g .
p2
212 ALGEBRA
147. Literal Equations with two Unknowns. Every simple
equation in x and y may be reduced to the form ax-\-by=C,
where a, h, and c represent known quantities.
If two equations in x and y with Uteral coefficients be given,
the equations may be solved by the same methods as were
used with equations with numerical coefficients.
Ex. 1.— Solve
ax-\~by=c,
ax—by=d.
Adding,
2ax=
=c+d, ..X 2„ •
Subtracting,
2by =
c—d
-0 d, .: 2/ - 26 •
Verify in the usual
way.
Ex. 2.— Solve
ax-\-hy=c,
mx-\-ny=k.
Multiply (1) by n,
Multiply (2) by b.
Subtracting,
nax-\-nby = cn,
bmx-{-bny=kb.
x(na— bm) =cn— kb,
cn — kb
na — bm
(1)
(2)
We might substitute this value of x in either of the given equations
to find y, but it is simpler to solve for y in the same way as we did
for X.
, „ , cm — ak ak — cm
Ehminate x from the two equations and nnd y = r or r —
Ex. 3. — Solve a^x-[-h{y=c^,
a2X-\-h.yy~c.,.
Here the symbols aj, Uo, etc., are used to represent known quan-
tities. They are read " o one, a two, b one, etc." There is no relation
in value between a^ and a^, nor Oj and 6]. The notation is used to
obviate the necessity of employing many different letter forms.
Solve these equations as in the preceding example and obtain
afi<^—a^i ' b^a^—b^Oi
FRACTIONAL EQUATIONS
213
EXERCISE 102
Solve for x and y, verify 1-12 :
1. mx-{-ny=a, 2.
'mx—ny^=h.
4. ax-\-hy=a'-\-b", 5.
x-\ry=a+h.
7. ax-hy=2a'^+'ih\
hx-{-ay=—ah.
9. a"x^h'^y=a-—ab+h'^,
ax—by=a—b.
lx-\-m,y^m,
mx-{-ly=l.
ax-{-by=2ab.
6.
bx—ay=b^—a^.
8. ax—by=2ab,
2bx+2ay=Sb'^—a
px+qy=r,
x+y^O.
ax-{-by=2,
a^x—b^y=a—b.
10.
a b
x_y
a b
= 3,
= 1.
13.* a^x-{-b^y=a^-¥
bx-\-ay—0.
11.
14.
a 6
ax—bi/=a-—h'^.
a^x-{-biy=Ci, 15.
.r+?/=l.
2, 12. a2-c_52y^c^3_f.i,3^
2/.
^ ^ = 2
a 6
- + - — — -J.
a; y
3a 4&
5A.
//
M. (a+b)x-{a-b)y=a^~+b^
X — y=a — b.
18. x +
20.
a?/
ax , ,
0 a+o
17.
19
a; y
a., , fto
^ + -= = c..
X y
l-5a , 3-26 _„^
X y
•5a 'ZSb _ ,„_
a; y
If ax-\-by=c and x— ?/=l, prove that x(c— a) = ?/(6+c).
21. If y=ax+b and x=py—q, prove that y{q—bp)=x{aq—b).
22. If 6a;+a2/+c2=ac+6c, ax+by=ac, cy-\-dz=ad, solve for
a;, 2/ and z.
23. What is the value of m, in terms of a and 6, if the following
equations are consistent
ax+Zby=a'^+Zb^,
Zx+y=^a-\-b,
4:X—3y=m ?
3a;- 2
2x-3
x+ll ^
a;+10 ''■
2a; + 7
9a;- 8 a;- 11
7
11 2
a;+l
2
3 X 5—x
a; 3 6
214 ALGEBRA
BXERCISE! 108 (Review of Chapter XVI)
Solve and verify 1-24 :
1. §(7-a;)-4(ll-a;) = ^(a;-8). 2.
"*• 2 ^ 3 ^ 3 ^ 2 6.
2a;- 3 3a;- 2 5a;g-29a;-4
• a;-4 + a;-8 ~a;2-12a;+32' *
2 3 21 5 ■ ^"' 2a;+l ^ 3x + 5~ -'•
11. a{x—a)—b{x—b) = {a+b){x—a—b), 12. aa;+6 = 6a;+a.
.„ 8a;+5 9a;-3 4a;-3 ^^ 4-a; 2+4a; ^
^^- -10--7^+2 = -5-- 1*- ^3 7- = ^-^-
-,= ««, 2/, a; ?/k ^c «'^+& ca;+d
a b 3a 46 ax—b cx—d
5 2 7 1 -r 1 '>r
3a; ' 5y ' 6a; ' 1% ' * a-26 2 "^ 2a-6
19. ox+% = 2o6, a2/-6a:=a2-62. 20. ^ — = 1 ^
x-\-b—a x+b — c
21 ?-±^ _ ^±^ = ?±? _ ^±^ 22 3 2 ^ _5_
• a; + 3 x+4 x+6 a; + 7' * a;+10"^a;-10 x-2'
a;-l a;-3 x-5 x—1 ^ „^ a;+l 1 , 8x-3
X--2 x-4 a;-6^a;-8 ' 3a;-4 5 "^ 16a;- 20
25.* (3a-a;)(o-6) + 2aa; = 46(a-fa;).
26. ^ + 2b{a-c) + ~ = c(a+b) + ~'
y , 2y-3x Ix-Zy
^^' 1-^1= -'^' ^x-2y^-\(iTy.
FRACTIONAL EQUATIONS 215
30. ^ + ^=2^xy, 5y-2x=2^xy. ^
31. What value of y will make
x+5 , y—l , , x—l v+11 „
-T-+V equal to ^-+^-±-?
32. Two sums of money are together equal to $1000, and 5|% of
the larger exceeds 6^% of the smaller by 16 cents. Find the sums.
33. Find a fraction such that if 4 be added to its numerator it
becomes equal to |, but if 4 be added to its denominator it becomes f .
34. If ax + b = cx + d, give the argimient which leads to the con-
clusion that X = , indicating at what point it is assumed that a
a~c
and c are unequal.
35. Take any two proper fractions whose sum is unity. Add
unity to the difference between their squares. Show that the result
is always twice the greater fraction.
36. A man has $30,000 invested, part at 4|% and the rest at 5J%.
He receives $65 per annum more income from the former than from
the latter. How much is invested at each rate ?
37. The sum of three numbers a, b, c is 3036; a is the same
multiple of 7 that b is of 4, and also the same multiple of 5 that
c is of 2. Find the numbers.
38. If 5 = H(2a + nd — d), solve for ct ; for rf.
39. If ax — by = a^ + b'^, x — y=2b and x^->ry' = c, find c in terms of
a and b.
40. A man can walk 2J miles an hour up hill and 3| miles per hour
down hill. He walks 56 miles in 20 hours on a road no part of which
is level. How much of it is up hill ?
41. A farm cost 3 J times as much as a house. By selling the farm
*•■* "77-% gain and the house at 10% loss, $2754 was received. Find
the cost of each.
42. In 10 years the total population of a city increased 11%. The
foreign population, which was originally -^j of the total, decreased by
1160 and the native population increased by 12%. Find the total
population at the end of the period.
CHAPTER X\T:I
EXTRACTION OF ROOTS
148. Square Roots by Inspection. In art. 65 we have seen
that the square root of any trinomial, which is a perfect
square, may be written down by inspection.
We have also seen that every quantity has two square
roots differing only in sign.
Thus, the square root of a^-{-2ab-\-h^ is ±{a-\-b),
and of a'^—2ab-\-b^ is ±{a—b).
± {a-\-h)=a-{-b or —a—b; ±{a—b)=a—b or b—a.
If we had written a^—2ab-}-b- in its equivalent form
b^—2ab-^a^, it is seen that b—a is a root.
It is usual, however, to give only the square root which
has its first term positive, and we say that the square root of
a^-\-2ab-\-b'^ is a-\-b and of a^—2ab-'rb" is a—b or b—a.
EXERCrrSE 104 (Oral)
State the square of :
1. —abc. 2.
4. 2a— 36. 5.
7. a+b+c. 8.
State the square root of :
10. IGx^y^. 11,
13. a^—2a+l. 14.
16. x^+x+i. 17.
19. x^+y^+z^+2xy-}'2xz+2yz.
20. a^+b^+c^-2ab-2ac+2bc.
21. 4a2+962+l + 12a6-4a-66.
216
x+l.
3.
-x-1.
a^+1.
6.
x^—x.
a+b—1.
9.
2a+b—c.
la%\
12.
x^+2ax+aK
4a2-12fl6+962.
15.
9x^-30xy+25y'
a*+2a^+a^.
18.
\6x*-i8x^+S6.
EXTRACTION OF ROOTS 217
149. Formal Method of Finding Square Root. When the
square root of an expression of more than tliree terms is
required, it is not always possible to write down the square
root by inspection.
Thus, to find the square root of
9^4- 12a;3+ 10a;2-4a;-|-l.
Here we could say that the first term in the square root is
3x^, and that the last term is either +1 or —1, but it is
evident that there must be another term as well.
Let us again examine the square of the binomial a+^>
which is a^-^2ab-\-b'^.
The first term of the square root is a, which is the square
root of a^. The second term of the square root, b, may be
obtained in two different ways, either from the last term, b^,
or from the middle term, 2ab.
Let us now see how we could obtain the second term in the
square root from the middle term 2ab. This term is twice
the product of a which is already found, and of the last term
of the square root which is still to be found.
If twice the product of a and the last term is 2ab, then we
can find the last term of the root by dividing 2ab by 2a, which
gives b.
The quantity 2a which we use to find the second term in
the square root is called the trial divisor.
Since a^-\-2ab-{-b^=a^-\-b{2a-\-b), we see that the complete
divisor is 2a-\-b, that is, the trial divisor with the second term
in the square root added to it.
The steps in the process are : a^-\-2ab-\-b^ \a-\-b
(1) The square root of a^ is a. The a^
square of a is subtracted from the expression ~~
leaving 2a6 + 6^. _2a + 6 | + 2«6 + 6^
(2) The trial divisor for obtaining the 2ab-{b^
second term in the square root is 2a.
When 2a is divided into 2ab the quotient is b, the second term in the
root.
(3) The com.plete divisor is 2a-\-b, and when this is multiplied by b
218 ALGEBRA
and the product subtracted from 2o6 + 6- there is no remainder. The
square root is then a + b.
It might be thought that step (3) is imnecessary, as the root has
already been found in (I) and (2). It is unnecessary if we take for
granted that the expression is a perfect square.
If you attempt to find the square root of a'^-\-2ab-\-4b^ and do not
go beyond steps (1) and (2), you would get the result a+b, as before.
This, however, is not the correct result. Why ?
We can now extend the method to find the square root of
a quantity of more than three terms.
9x«- 12.^3+ 10a;2-4a;+ 1 1 3a
9x*
;2-2a;+l
3x^-2x \ - 12x3 + 10x2- 4a; + 1
-12a;'+ ix-
6a;2-4a:+l 1 6a;2-4.-r+ 1
Qx^-ix+l
After finding the first two terms in the root, as in the previous
example, the 3a;2 — 2a; is treated as a single quantity and the second
trial divisor is twice Sx^—2x or Qx^ — 'ix
The square root is Sx^—2x-\-\.
150. Verifying Square Root. We might verifj^ the result
in the preceding example by Avriting down the square of
3x2— 2.'r+l. Verify in this way.
A simple method of checking is to substitute a particular
number for x.
When a;=l, 9.i;^- 12.'(;3+ 10a;2-4a;+ 1 = 9- 12+ 10-4+ 1 = 4,
and 3a;2-2a;+ 1 = 3-2+1 =2.
Since the square root of 4 is 2, we presume the work is correct.
EXERCISE 105
Find the square root, by the formal method, and verify the results :
1. a;2+12a;+36. 2. Oa^-Ga + l.
S. 9.^2+24x2/+ 16?/2. 4. 25x^-10xy+y\
5. l-18ab+8la%^. G. 49a*-28a^~b^+4:b*.
7. a4+2a3-3a2-4a+4. 8. 'ix*+^x^+5x^+2x+l.
EXTRACTION OF ROOTS 219
9. x«-6x3 + 17a;2-24.r+16. 10. 9o4-12a''+34a2_20a-}-25.
11. a^-4a36+6a262_4a63+64. 12. a*-4a3-f 8a+4.
13. 9a4+12a36+34a262^20a63-f-256^
14. x«-4x5+6a;3+8a;2+4a;+l.
15. a:*— 2x3+2x2— X-+1
a* 4a3 2a- 4a
17. a--4a6H-6ac-f462-i26c+9c2.
18.* Simplify a(a+l)(a+2)(a+3)-f-l, and find its square root.
19. By extracting the square root of x*+4x'+6x2+3x+7, find a
value of X which will make it a perfect square. (Verify by substitution.)
20. If the square root of .r*— 8x3+30x2-56.r+49 be x^-\-mx-\-l,
what is the value of m ?
21. Using factors, find the square root of
(x2+3x+2)(.r2+5x+6)(x2+4x+3).
22. Find the first three terms in the square root of 1— 2x— 3x- and
of 4-12x.
23. When x= 10, the number 44,944 may be written
4xH4x3+9x2+4x+4.
Find the square root of the latter and thus deduce the square root of
44,944.
151. In algebra, an expression of which the square root is
required is usually a perfect square. When such is the case
the formal method may be greatly abbreviated.
Ex. 1. — Find the square root of
a;4_4a;3_|_ io.x2_ 12.r+9.
The first term is x* and the last is + 3 or — 3.
The trial divisor for obtaining the second term of the root is 2x*,
therefore the second term is — 4x'-h-2x^ or — 2x.
.•. the square root is x*— 2x+3 or x*— 2a;— 3.
220 ALGEBRA
If we square x^ — 2x-\-S, the term containing x will be twice the
product of —2x and 3 or —12a;. If we square a;^— 2a;— 3, the term
containing x will be + 1 2a;.
We thus see that if the expression is a perfect square, the square
root is x^ — 2x-\-3.
Check this by putting x= 1.
Ex. 2. — Find the square root of
4a;''+20x3+13a;2_30a;+9.
What is the first term in the square root ? What is the trial divisor ?
What 'is the second term in the root ? What may the last term be ?
What is the square root ? (Verify your answer.)
Ex. 3. — Find the square root of
-3a3 + -25^^4_5^^6 7^2,
Write the expression in descending powers of a.
a«-3a3-|-«|o2-5a+V--
The first term in the root is a~. The trial divisor is 2a^, therefore
the second term is — 3a^-H2a- or — #a.
.'. the root is a^— la + f or a^ — fa — |.
Which is it ? (Verify by squaring.)
Ex. 4. — Find the square root of
4 4
4a2_4a + 9 h-^'
a a'-
Here the terms are already arranged in descending powers of a,
the term +9 coming between a and -.
The first term in the root is 2a, the second is — 4a -^ 4a or — 1, and the
2
last is + -.
a
Complete and verify.
It will be recognized that it is only in the most complicated
cases that it is necessary to use the formal method in full.
It is advisable to use the contracted method whenever
possible.
EXTRACTION OF ROOTS 221
E3XBRCISB 106
Find the square root, using any method you prefer. Verify the
results.
1. a;*+2a;3— a;2— 2x-+l. 2. x'^—Ax^+Qx-—'^x+\.
3. a*— 6a3+5a2+i2a+4. 4. a;*+8a;3+12a;2— 16a;+4.
5. 9a*— 6o3+13a2_4a+4. 6. x'^+&xhj+lx"y-—&xif-iry\
7. 4x^+20x3— 3x2— 70a;+49. §. l-10x+27x-2-10.i-3+x«.
9. 67x2+49+9x4— 70a.--30.i-3. ^q. ai2_8^9_(_i8^6_8(^3_f_i^
11. x*+2x3-x+i. 12. a-*-2x3+fx2— Jx+Jg-.
13. c*-6a2+ll --^ -f 1. 14. l^*_^^_^ + ?^+l.
a- a* ?/* 2/^ y^ y
15. --3x3+a^x2-2x+i 16. ^* + — + 21^' + ^+1.
4 ^ -^ ^'^ 25 ^ 5 ^ 45 ^ 3 ^
-,r- e I 4 4 o 1 •> -fo ^'' 2x , „ 2w , w2
17. 5H — 2x+x2. 18. + 3 --\-^.
x^ X y^ y X x2
19.* (a;+2/)4-4(x+?/)3+6(x+2/)2-4(x+y) + l.
20. x2(x-5a)(x-a)+a2(3x-a)2_3a2.r2.
21. (a-6)2{(a-6)2-2(a2+62)j.+2(a4+6*).
22. (a+6)*-2(a2+62)(a+5)2_|_2(a4^54),
24. If x*+6x^+7x2+ax+l is a perfect square, what is the value
of a?
25. If the sum of the squares of any two consecutive integers be
added to the square of their product, prove that the result will be a
square.
26. If 4x*+12x3y+A:x2y2_j_6_^^3_j_y4 jg J^ perfect square, find k.
27 . Ji m^x and n= y — - , show that
X ■ y
mn+ V(m2+4)(n2+4)=2x«/+ — •
xy
28. Find the square root of 4x*+8x='+8x2+4x+l. Check when
x=10.
222 ALGEBRA
152. Cube of a Monomial. When three equal factors are
multipUed together, the product is called the cube of each of
the factors.
Thus, the cube of 2a or {2a)^ = 2a.2a. 2a=8a'^,
the cube of a^ or {a^)^ =a^ . a- . a^ = a^,
and the cube of 3a» or (3a3)3 = Sa^ . Sa^ . ^a^ = 21a^.
The cube of a monomial is found by writing down the cube of
each factor of it.
Thus, the cube of 5ab^x is 125a*6^x*.
153. Cube of a Binomial. Fmd the a^+2ab+b''
cube of a-\-b by multiiDlying its square by " +^
a-\-b. Find also the cube of a—b. o*+2a26+ afe^
{a—b)^=a^—2a^'b+Zab^—b^ a^+3a^b+3ab'~+b^
Note that m each case the cube contains four terms, in
descending powers of a and ascending powers of b, and the
numerical coefficients are 1,3,3, 1 .
The cube of a—b is the same as the cube of a-\-b, except
that the signs are alternately plus and minus.
From the forms of these two cubes, the cubes of other
expressions may be written down.
Ex. 1. (x + 2y)'> = x^ + 3x^2y) + Sx{2yy~ + (2y)^,
=x^ + 6x^y+l2xy^ + 8y^.
Ex. 2. (3x-2y)!' = (3a;)3-3(3x)2(22/) + 3(3x)(2!/)2-(2y)3,
= 21x^-54:X^y + 3Qxy^-8y^.
Ex. 3. {a+b + c)^ = {a + b + cy,
= (a+6)' + 3(a+6)2c + 3(a+6)c2 + c',
= a^+3a^b + Sab^ + b^ + 3a^c + 6abc + Sb^c + 3ac^ + Sbc^ + c^,
= a^+b^ + c^+3{a^b+ab' + b^c+bc^ + c'^a+ca-) + 6abc.
EXTRACTION OF ROOTS 223
EXERCISE 107 (1-12, Oral)
Find the cu
[be of :
1. -1
2
-2a.
3.
-3o62.
4.
—x^yz^.
5. x+y.
6.
x—y.
7.
m-\-n.
8.
p-q.
9. x+l.
10.
x-\.
11.
a2+6.
12.
\-a\
13. a;+3.
14.
2x—y.
15.
2a + 36.
16.
I -2a.
17. a-46.
18.
\-a\
19.
a-\-b—c.
20.
a—b—c.
21.* Simplify {a+b)'^+{a-bf and {a+bf-ia-b)"^.
22. Show that {x-{-y)^=x^-\-y^-'r3xy{x-\-y) and write a similar
form for (x — y)^.
23. Simplify ia+b+c)^+{a+b-cf.
24. Show that {a-b)^+{b-c)^+{c-af=3{a-b){b-c){c-a).
25. Show that the difference of the cubes of any two consecutive
integers is greater than three times their product by unity
26. When x=y-\-z, show that x^—y^—z^—Sxyz.
27. Two numbers differ by 3. By how much does the difference
of their cubes exceed nine times their product ?
28. Three consecutive integers are multiplied together and the
middle integer is added to the product. Show that the result must
be the cube of this middle integer. What is the cube root of
241x242x243+242?
154. Cube Root of a Monomial. The cube root of any
quantity is one of the three equal factors which were
multiphed to produce that quantity.
Thus, the cube root of 8 is 2, of a' is a, of 8x' is 2x, of a* is a^, of
27a36« is 3ab\
The cube root of any power of a letter is obtained by dividing
the index of the power by 3.
The symbol indicating cube root is ^"~
Thus, -^^125=5, ^?^=a», VSx^»^2x'y.
224 ALGEBRA
155. Cube Root of a Compound Expression.
The cube root of a^-{-3a^b-{-3ah"-\-b^ is «+&,
and of ci^—Sa^b-{-3ab^—b^ is a— b.
Therefore, when an expression of four terms is known to be
a perfect cube, its cube root can at once be MTitten down
by finding the cube roots of its first and last terms.
Ex. 1. — The cube root of x^ — 6x^y -\- 12xy^ — 8y^ is x—2y, since the
cube root of x^ is x and of — 8y^ is —2y.
Ex. 2. — The cube root of a^—2a^b-\-^b^—^%b^ is evidently o— §6.
In the cube of a-\-b, the second term is 3a^h. After finding
the first term a of the cube root, we might have found the
second term of the root by dividing 3a -6 by Sa^, that is, by
three times the square of the term aheady found.
Thus, the second term of the cube root in Ex. 1 is
— Qx'^y-^Sx^ or —2y,
and in Ex. 2 is —2a^b-^3a' or — 16.
Here three times the square of the first term of the root is the
trial divisor, corresponding to twice the first term in finding
the square root.
Ex. 3. — Find the cube root of
8x^+l2x^—30x^—35x^+45x^-\-21x-21.
The first term in the root is 2x^ and the last is — 3.
The trial divisor for finding the second term of the root is 3(2x")*
or 12a;*.
.•. the second term of the root is 12a;5^12a;* or x.
.'. the cube root is 2x^-\-x—3.
It is thus seen that it is easier to find cube root by inspection
than to find square root, as in finding cube root there is no
ambiguity as to the sign of the last term in the root.
156. Higher Roots. Since {x^)^=x^, we may find the fourth
root by taking the square root and then the square root of
the result.
EXTRACTION OF ROOTS 225
Also, since {x'^)^=x^ and {x^)^—x^, we can find the sixth
root by taking the square root of the cube root, or the cube
root of the square root.
Thus, the square root of x* + 8x^ + 24x2-)- 32a; + 16 is x' + 4a;4-4,
therefore the fourth root is x-\-2.
The cube root of a;« — 6x*+ 15a;* — 20a;3+ ISx^ — 6a;+ 1 is cc* — 2a;+l,
therefore the sixth root is x— 1.
EXERCISE 108 (1-15, Oral)
State the cube root of :
3. -125a%3, 4^ _8(a_ft)3
6. x^—Zx'^y-\-^xy'^ — y^.
8. 8.T='-12.c2+6.r-l.
10. 64a3- 144^2 + l08a-27.
12. 21x^—21xhj+^xy--y^.
16. ^ - 6.r* + '\-2xhf — 8j/«.
Ill, lit- y^
17.* In finding the cube root of x^+^x^+&x^+lx^+Qx'^-\-Zx-\-\.
what is the first term in the root ? What is the last term ? What is
the trial divisor for finding the second term ? What is the cube root ?
Check by substituting a;=l.
Find the cube root and check :
18. l-6a;+21a;2-44a;H63a;*-54a;5+27a;«.
.„ x3 a:2 ^ 18 27 , 27
'"• 27-y+2--^ + -.-^+^-
20. 27a«- 108a5 + 171a*- 136a3+57a2- 12a+l.
21. (l+3x2)2-a;2(3+a;2)2.
22. For what value of x will x^-\'Zcx'^-\-2c^x-\-5c^ be a perfect cube ?
23. Find the fourth root of a;*— 4a;3+6a;2— 4a;+l.
Q
1.
-64. 2. 27a3.
5.
a:3-f-3.r-+3a;+l.
7.
a3+6a24-i2a4-8.
9.
a:V+3a;2t/2+3a;y+l.
11.
125a;3-75a;2+15a;-l.
x3 3a;2 , 3a; ,
13.
8 4+2-'-
15.
3 „ , 27 27
226 ALGEBRA
24. Find the fourth root of a'^-l2a^+5'ia^-\08a+81.
25. Find the sixth root of
a;«-12x5 + 60a;*-160a;3+240a;2-192a-+64.
E3XBRCISE 109 (Review of Chapter XVII)
Find the square root of :
2. x^ + 4x^-2x^—l0x^+13x^-6x+l.
3. x^' + 6x'^'> + 5x»-8x^ + 16x^-8x^ + 4:.
5. 12a^x-2ea-x^ + 25x*+9a*-20ax^.
6. 4a;2(7 + a;2 + 3a) + (3«+7)'.
7. (a;2 + 5a;+6)(a;2 + 7a;+12)(x2+6a;+8).
8. (2x^-x-S)(x^-4x-5)(2x--13x-ir 15).
9. 4x4 _ 20a;3 + 33^2 _ 32a; + 34 - ^^ + -^ .
a; a;^
What is the cube root of :
10. 27- 1 35a; + 225a;2- 125x3.
11. 8a;8-12x5+18a;4-13a;3 + 9a;2-3a;+l.
12. (a-6)3 + 36(a-6)2 + 362(a-6) + 63.
13. Find to three terms, the square roots of :
l-2a;, l-a, 4 + x.
14. Find the value of y for which x'^ — 2{a — y)x+y^ is a complete
square and prove by trial that your result is correct.
15. The first two terms of a perfect square are 49x* — 28x', and the
last two are +6a;+|. What must the square root be ?
16. Prove that the product of any four consecutive integers
increased by unity is a perfect square.
17. Find the square root of a* + 4a^ + 6a^-\-4a-\-\ and deduce the
square root of 14,641.
18. By finding the cube root, simplify
{a + b)^ + ^a + b)^a-b) + ^a + b){a-b)2 + {a-b)^
19. Tf a = 6+l, show that a»- 63-1 = 3a;;.
EXTRACTION OF ROOTS 227
20. Show that the product of any four consecutive even integers
increased by 16 is a perfect square. How might the result be deduced
from No. 16 ?
21. By inspection, find the values of
(a-6)2+(6-c)2 + (c-a)2 + 2(a-6)(6-c) + 2(6-c)(c-a) + 2(c-a)(a-6),
(2x-y)^~3{2x-yn2x+y) + 3{2x-ij){2x+yy--(2x + y)\
22. To the square of the double product of any two consecutive
integers, add the square of their sum. Prove that the result is always
a perfect square.
23. Express in symbols : The difference of the cubes of any two
numbers exceeds the cube of their difference by three times their
product multiplied by their difference. Prove that this is true.
24. The expression
8x9-36a;8 + 66x'-87x6+105x5-87a;* + 61x3-42x2+12j;-8
is a perfect cube. Find its cube root by getting two terms from the
first two terms of the expression and the other two from the last two
terms. Check when x=l.
25. What number must be added to the product of any four
consecutive odd integers so that the sum may be a perfect square ?
26. Show that the sum of the cubes of three consecutive integers
exceeds three times their product by nine times the middle integer.
27. Find the cube root of
(4a;-l)3 + (2a;-3)3 + 6(4a;-l)(2a;-3)(3a;-2).
[Note that it is of the form a^-\-b^+'3ab{a + b).]
28. If 4cc*+ 12x^4- 5x* — 2x* are the first four terms of an exact
square, find the remaining three terms.
Q 2
CHAPTER XVIII
QUADRATIC SURDS
157. Surd. When the root of a number cannot be exactly
found, that root is called a surd.
Thus, we cannot find exactly the number whose square is
equal to 2, and we represent the number by the symbol V2
and we call V'2 a surd.
If no surd appears in any quantity, it is called a rational
quantity.
By the arithmetical process of extracting the square root of 2, we
can obtain the value of '\/2 to as many decimal places as we please,
but its exact value can not be found.
To four decimal places the value of V'2 is 1-4142. Find the square
of 1-4142 by multiplication and see how closely it approximates to 2.
We can find geometrically a line whose length is V2 units. In this
square, whose side is 1 imit, draw the diagonal BD.
Then, from geometry, we know that
:. BZ)2=12+12 = 2,
.-. BD =V2.
On squared paper mark the corners of a square
whose side is 10 units. Measure the diagonal and
thus estimate as closely as you can the value of V 2.
Make a diagram like this to show how to represent
graphically lines whose lengths are V2, Vs, V 4, V5,
etc. Take the unit line 1 inch in length. What test
have you of the accuracy of your drawing ?
158. Quadratic Surd. A surd like V2 in which the square
root is to be found is called a quadratic surd. In this Chapter
quadratic surds only are considered.
22S
QUADRATIC SURDS 229
159. Multiplication of Simple Surds.
Since V 2 represents a quantity whose square is 2,
:. V2xV2=2=V4:,
also VlxVd^VSQ, because 2x3=6.
Similarly, we might expect that V2xV3=VQ. That this
is true may be shown by finding the square of y/2x Vs.
('\/2x v'3)2=a/2x V3x V2x V3, [Just as {ab)^^a .b .a.b.]
= V2xV2xV3xVS,
=2x3=6.
.'. V2xV3=V6.
Similarly, VSxV5=VT5,
and Va x Vb— Vab.
Therefore, the product of the square roots of two numbers is
equal to the square root of the product of the numbers.
Since Vab=VaxVb, .\ \/l2 = \/4x V3=2a/3,
and
V56= V25 X V2=o V2 ; VlSa^= VWa- x V2a=3aV2a.
Thus, we see that if there is a square factor under the radical
sign, that factor may be removed if its square root be taken.
Conversely, 5^3= V25 xV3=Vl5,
aVb= Va^ X V^= Va^,
axVmy = Va^x^ X Vmy = Va^x'^my.
160. Mixed and Entire Surds. When a surd quantity is
the product of a rational quantity and a surd, it is called a
mixed surd. If there is no rational factor it is called an
entire surd.
Thus, oVS, aVb, {a — b)Vx — y are mixed surds, and VS, V50,
Vax + b are entire surds.
In the preceding article we have shown that a mixed surd
can always be changed into an entire surd, and an entire
surd can sometimes be changed into a mixed surd.
230 ALGEBRA
A surd is said to be in its simplest form when the quantity
under the radical sign is integral and contains no square
factor.
Thus, the simplest form of VSO is 5V2.
EXERCISE 110 (1 29. Oral)
Find the product of :
1. V2, Vs. 2. V5, Vs. 3.
V2, Vs.
4. sVl, 2\/7. 5. VS, VS, V2. 6.
Vs, VI
7. Va, Vg, Vf. 8. (Vabc)K
Express as entire surds :
9. 2V3. 10. 3v/2. 11. W5.
12. aVb. 13. SaVl. 14. b ./-.
^ b
/^
-6
16. SVa-b. 16. (a+6) /v/ — rr •
Simplify, by removing the square factor :
17. Vs. 18. V12. 19. V2O. 20. V75.
21. V27. 22. V56. 23. Vl62. 24. V2a^
25. VIOOO?^. 26. |V32. 27. V{a-bf. 28. -Va^b.
a
29. Solve x^=2 ; 3^2=27 ; ^x-=9.
30. Show by squaring that
V3xV7 = V2T and VaxVbxVc^Vabc.
31. Show that V8=2V2, by extracting the square roots of 8
and 2 to three decimal places.
32.* Describe a right-angled triangle whose sides are 2 inches and
3 inches. Express the length of the hypotenuse as a surd.
33. By using a right-angled triangle, how could you find a line
whose length is VlO inches ?
QUADRATIC SURDS 231
34. If the area of a circle is 66 square inches, find the length of the
radius (7r = 31).
35. The sum of the squares of two surds, one of which is double
the other, is 40. Find the surds.
36. The length of the diagonal of a square is 10 inches. Find
the length of the side.
37. One side of a rectangle is three times the other and the area is
96 square inches. Find the sides.
161. Like Surds. In the surd quantity SVS, 5 is a
rational factor and VS is called a surd factor.
When surds, in their simplest form, have the same surd
factor, they are called like surds or similar surds, otherwise
they are unlike surds.
Thus, 3V'2, 5\/2, aV2 are like surds.
I „ _ _
and 2V3, 3V2, W5 are tmlike surds.
162. Addition and Subtraction of Like Surds. Like surds
may be added or subtracted, the result being expressed in
the form of a surd.
Thvis, 3^/2 + 5^2 = 8^2, just as 3a+5a=8a.
7^/3-4^3 = 3^3, just as 7a-4a = 3«.
a/75- 2V3 = 5\/3- 2^3 = 3^3.
V50 + \/32-V'l8 = 5\/2 + 4V2-3V2 = 6\/2.
The sum or difference of unlike surds can only be indicated.
Thus, V2+\/3 can not be combined into a single surd, but the
approximate values of V'2 and Vs may be found and added.
Show that V2+V3=V5 is not true, by finding the square roots
of 2, 3 and 5 each to two decimals.
Is it true that V 4 + Vo = V 13 ?
EXERCISE 111 (1-8, Oral)
Express as a single surd :
1. 3V2+5\/2. 2. 5\/7-3a/7. 3. 2Va+3Va.
4. 2Vx+5Vx—Vx. 5. \/8-|-V2. G. \/l2+\/3.
232 ALGEBRA
7. Vis- Vs. 8.
9.* \/75+Vl2+3\/3. 10.
11. V45-V20+V80. 12.
13. 4\/r28+4V50-5\/l62. 14.
15.
16.
2\/18+3-s/8-5a/2.
2V63-5\/28+a/7.
10\/44-4a/99.
V45-\/20+V'80.
4 a/ 128 + 4 V50 - 5 V 162.
Vis + V'20 - VSO + V 180.
Vl2 + ^98 - V 1 28 - \/32 - VSO.
Simplify the following and find their numerical values, correct to
two decimal places, using the square root table :
17. V75. 18. -\/63. 19. V60+a/15.
20. VU'l-2Vl2. 21. Vl28-Vl62. 22. V56+Vf2+V9b.
Solve, finding x to three decimal places :
23. a:2^37. 24. 3x^-+5=50. 25. Ja;^— 4=19.
26. 31x2=132. 27. i(3x2-ll)=53. 28. Jx2=^x2-47.
29. The area of a circle is 176 square inches. Find its radius.
Square
Roots
OF Numbers
FROM 1 TO
50.
n
V'n
n
Vn
n
Vn
n
v^
n
v/^
1
1-0000
11
3-3166
21
4-5826
31
5-5678
41
6-4031
2
1-4142
12
3-4641
22
4-6904
32
5-6569
42
6-4807
3
1-7321
13 3-6056
23
4-7958
33
5-7446
43
6-5574
4
2-0000
14 3-7417
24
4-8990
34
5-8310
44
6-6332
5
2-2361
15 3-8730
25
5-0000
35
5-9161
45
6-7082
0
2-4495
16 i 4-0000
26
5-0990
36
6-0000
46
6-7823
7
2-64.58
17 4-1231
27
5-1962
37
6-0828
47
6-8557
8
2-8284
18 4-2426
28
5-2915
38
6-1644
48
6-9282
9
3-0000
19 4-3589
29
5-3852
39
6-2450
49
7-0000
10
31623
20 4-4721
30
5-4772
40
6-3246
50
7-0711
163. Multiplication of Surds.
3\/2 X 4\/3=3 X V2 X 4 X V3,
=3x4x-\/2x\/3,
= 12\/6.
QUADRATIC SURDS 233
It is thus seen that the product of two surds is found by
multiplying the product of the rational factors by the product of
the surd factors.
5\/3x2V3=10.3=30,
also aVc X b Vc = abc.
It, therefore, follows that the product of two like surds is
always a rational quantity.
Ex. 1.— Multiply VEb by VtS.
Here the siirds should be simplified before multiplying.
Since a/50 = 5V2 and \/75=5V'3,
.-. V50xVlB=5V2x5Vl=25V6.
Ex. 2.— Multiply 2 + 2\/3
2+2 V3 by 3-\/2. ^~ ^^
Here the multiplication is performed in a 6 + 6v3
manner similar to the multiplication ofa + 6 — 2v2 — 2v6
by x+y. ~ ~ ~
6+6\/3-2\/2-2\/6.
164. Conjugate Surds. If we wish to multiply
5\/3 + 2\/2 by 5\/3-2\/2,
we may follow the same method as in the preceding example.
These expressions, however, are seen to be of the same form as
a-\-b and a—b,
:. (5\/3+2V'2)(o\/3-2\/2) = (5\/3)2-(2V2)2=75-8=67.
Similarly, (3+ \/2)(3- V2) = 9-2 = 7,
and (2-Vl0)(2 + Vl0) = 4-10=-6.
Such surd quantities as these which differ only in the sign
which connects their terms are called conjugate surds.
Note that the product of two conjugate surds is always a
rational quantity.
234 ALGEBRA
EXERCISE 112 (1-12, Oral)
Find the product of :
1. 2\/3, 3\/5. 2. 5\/2, WS. 3. aVb, bVa.
4. 3\/2, \/3, Vs. 5. (2\/3)2, (\/2)2. 6. V2+1, V2.
7. VS + a/S, ^2. 8. Va+Vft— 1, Vc.
9. a/3+V'2, ^3-^2. 10. VrO-3, VIO+3.
11. Vx—Vy, Vx+Vy. 12. 2\/2+\/.3, 2V2--V'3.
13.* 3\/6, 4V'2. 14. 3\/2, 4\/7, J\/2.
15. (v'3+\/2)2. ^Q {2V5-Vl)^-
17. (3\/2+2V3)2. 18. (Va-f\/6)2.
19. 4+3 \/2, 5-3\/2. 20. 3a/2+2a/3, 5\/2-3\/3.
21. 3^5-4^2, 2V5+3\/2. 22. 3\/a-2v'6, 2\/a-3'\/6
23. V'5+V'3 + V'2, ^5+^3-^2.
24. \/7 + 2a/2-V3, V7-2\/2+\/3.
25. Va+6— 3, •\/a+6+2. 26. Va+Va—1, Va—Va—l.
27. (\/3+V2+l)2. 28. (V5+2V2-V3)'-
29. (V^6+\/a'^)2. 30. (3\/.^^-2\/^+^)^-
Simplify :
31. (6-2\/3)(6+2\/3)-(5-a/2)(5+\/2).
32. (v/3-\/2+1)2+(v'3 + a/2-1)2.
33. (\/50-\/l8+\/72+\/32)xiV3.
34. 2(4V3+3V'2)(3\/3-2V2) + (5\/2-3\/3)(4V24-2V'3).
35. (\/3+V2)(2V3-\/2)(\/3-2\/2)(V3-3\/2).
36. By squaring VlO+VS and VS+Vl, find which is the greater.
37. The product of 5^3+3^7 and SVS—Vl lies between what
two consecutive integers ?
38 . Find the area of a rectangle whose sides are 5 + V2 and 10— 2^2
inches.
QUADRATIC SURDS 235
39. The sides of a right-angled triangle are 7+4\/2 and 7 — 4^2
inches. Find the hypotenuse.
40. The base of a triangle is 2\/3+3\/2 inches and the altitude
is 3v'3+2\/2 inches. Find the area to two decimal places.
165. Division of Surds.
Since VaxVb=Vab, :. Vab-^Va^J^^Vb.
y a
„. ., , /- /r Va la
Similarly, ■\/a^vo= = a / - ,
Vb ^ b
and 3Vl5^2V'5=|V3.
Ex. 1. — Find the numerical value of V5-^\/2 or — -_•
V2
(1) We might find the square roots of 5 and 2 and perform the
required division.
V'5-:--\/2 = 2-236h- 1-414= 1-581
(2) V5^\/2=V'f=V2^= 1-581.
, VB VBxV'2 ViO 3162 , _-,
yo) —j^=—^ -^ = -— — = — - — =1-581.
V2 \/2xa/2 2 2
Here the third method is at once seen to be simpler than
either of the others.
V5 VlO
In (3) we changed — - into , that is, we made the
^ V2 2
denominator a rational quantity. This operation is called
rationalizing the denominator.
Ex. 2.— Find the value of V> if V2= 1-4142.
V2
Here, instead of dividing 1 by 1-4142, we first rationalize the
denominator.
T>,«r, 1 lxV2 V2 1-4142
Ihen -^^ = —— ~ = -— - = — — = -7071.
V2 V2xV2 2 2
Ex. 3.— Divide QVs by 10\/27.
6V8 6x2V'2 2a/2 2xV2xV3 2V6 2x2-4495
IOV27 10x3\/3 5V3 5xV3xV3 15 15
= -3266.
236
ALGEBRA
Ex. 4. — Rationalize the denominator of
2+\/5
3+V5
We have already seen that the denominator will be rational if we
multiply it by its conjugate 3— V5.
2+V5 _ (2+V5){3-V5) _ l + \/5 ^ I + a/5
3+V'S ~ (3 + V'5)(3-V5) ~ 9-5 4
Ex. 5.— Divide 5+2^3 by 7-4\/3.
5 + 2^^3
Write the quotient in the fractional form j^t rationalize the
^ 7-4V3
denominator and simplify.
Divide :
1. 3^/27^ Vs.
3. V72^3\/8.
5. A/l8 + \/i2byV3.
Rationalize the denominator of :
EXERCISES 113 (1-12, Oral)
2. Vi2^\/3.
10.
13.*
16.
2
V3
a
Vb
i-Ws
Va
Va+Vb
8.
11.
14.
17.
4. VaFc-
^Va.
6. V
a6+\/ac
by Va.
of:
10
V5
9.
V5
3\/5
12.
1
V6
V2-1
12
3\/2-2a/3
15.
V3+\/2
V3-V2
5\/3-3\/5
18.
V7-\-V2
Vs-a/s
9+2 Vl4
Find the value to three decimal places, using the table :
1 15 2\/3
— ^ • '*{) — = • 21 ^ .
a/3 " ' Vis ■ 3V2
19
22.
25.
V3+\/2
V'3^\/2.
23.
26.
17
3\/7+2\/3
2^63-^3^35.
24.
Vi-Vs
Vt+Vs'
l^(7+4\/3).
QUADRATIC SURDS 237
Solve, giving the value of x to two decimal places, using the table :
28. a;V2=3. 29. xV3=V2. 30. xV3=V2+l.
31. a;\/3-.rV'2=l. 32. a;\/5-5=2a;- v/5.
33. x%V3-l)=2{VS+l).
34. The area of a triangle is 2 square feet. The altitude is
Vs+VS feet. Find the base to three decimals.
35. Simplify 2+VlO
4\/2+-v/20-V'l8-\/5
166. Surd Equations. A surd equation is one in which
the unknown quantity is found under the root sign, in one or
more of the terms.
Thus, Vx+7 = 4c, Vx-\-Vx—5 = 5, are surd equations.
Ex. 1.— Solve Vx^=2.
Square both sides, a;— 3 = 4,
x=7.
Verification: Vx—'3=Vl — S=Vi = 2.
Ex. 2.— Solve V5x- 1 - 2 Vx+S = 0.
Transpose 2\/.t + 3, VSa;— l = 2\/a; + 3.
Squaring, 5a; — 1 = 4a; + 1 2,
.-. a;=13.
Verification : V5^^^-2a/x + 3= \/64-2\/l6 = 8-8 = 0.
Note that in verifying we have taken the positive square root only,
as defined in art. 63.
BXE3RCISB 114 (1-8, Oral)
Solve and verify :
1. 2Vx=6. 2. Vx-5=4. 3. 6-^^=1.
4. a/x+2=4. 5. Wx=V2b. 6. Vx-b=a.
7. m+Vx=n. 8. 1 — Vx^=3.
9. Vx^+9=9—x. 10. Vx^^Ux+3=x+5.
11. V9x^-nx^5=3x-2. 12. 2a;-V'4a;2_i0a;-t-4=4.
13. 2a+Vx+a^^b+a. 14. V{x—a)^+2ab+b^^x—a-\-b.
238 ALGEBRA
EXERCISE 115 (Review of Chapter XVIII)
Simplify :
1.* Vs+VIS+VgS. 2. V 500 +V 80-^20.
3. 5\/3 + 3V'27-\/"i8. 4. (4\/5+ \/T8)(4\/5- VlS).
5. (6\/6-5)(6\/6+5). 6. (V'6+ V2 + 2)( V6- V2-2),
7. (a/8+ ^2-2)2. 8. (V'3-2\/2-l)2.
9. 5\/27^6\/75. 10. (\/5-2)^( V5 + 2).
11. (\/l25+V'i5)-^V'320. 12. (5+ \/3)(5- \/3)-f-(A/l3- \/2).
13. Multiply 3\/8 + 2a/3-\/2 by 2V8- \/3 + 4"v/2.
. 14. By how much does the square of \/3 H — ■j= exceed the square
of V2 + 4- ?
V2
15. Show by multiplication that the value of Vs lies between
1-732 and 1-733. Which of these is the closer approximation to V3 ?
16. Which is the greater, Vu+Vs or y/ \2 -\- \/ \b "i
17. The product of 3\/2 — 2\/3 and 2^3— V2 lies between what
two consecutive integers ?
18. Rationalize the denominators of :
4 3\/2 3 /5 /s-^ 2^6-2
A. 3V2 3 /5 .3:5 2
a/2' 2V3' 2V 6 "^ ' Q^
V2 2V3' 2V 6 ' 3\/3 + V'2
Solve and verify :
19. ^^+3 = 4. 20. V'ix^^=2\/x-2.
21. Vx^^5+l=x. 22. V2x+'l = ^Vx.
23. Vx235a;+ll = x+2. 24. Vx^-2=l-x.
25. Using the table, solve: a;2=75, a;2=63, Ja;2 = 49, x\/3=V5,
.-cV'2 + l = V'3.
26. Find to three decimal places the values of :
2 1 Jl_ 2ViO-V5 3\/2-2
vl' \7l' vl+r vro+V5' 4V2+1'
QUADRATIC SURDS 239
27. Find the value of
(2V'2+ \/3)(3\/2- V'3)(3\/3- V2).
28. If the sides of a right-angled triangle are VS+l and VS— 1,
what is the length of the hjrpotenuse ?
OQ «• vf ^5-1 V5-3 . V3+V2 \/3-V2
29. Simplify —7= y= and -^= 7= 7= ;;:: .
V5-2 •\/5 + 3 \/3-V2 V3+V2
30. Find the value to two decimal places of
^±^ + ^^, whenx=2+\/3, 2/ = 2-\/3.
31. Multiply 2V30-3V'5 + 5V3 by \/3 + 2V2-\/5.
32. Multiply V7'+2V6 by Vt-WB.
33. The area of a rectangle is 16\/l() — 25 and one side is sVo — ^2.
Find the other side to two decimal places.
CHAPTER XIX
QUADRATIC EQUATIONS
167. A quadratic equation has already been defined in
art. 104. In the same article we considered the method of
solving some of the simpler forms of it.
Quadratic equations frequently occur in the solution of
problems as shown in the following examples.
Ex. 1. — Find two consecutive numbers whose product is
462.
Let the numbers be x and .1;+ 1-
.T(a;+1) = 462,
.-. x^ + x-462 = 0.
Ex. 2. — The length of a rectangle is 10 feet more than the
width and the area is 875 square feet. Find the dimensions.
Let x = the number of feet in the width,
.'. a; + 10 = the number of feet in the length,
a;(a;+10) = 875,
.-. a;2+ 10a; -875 = 0.
Ex. 3. — Divide 20 into two parts so that the sum of their
squares may be 36 more than twice their product.
Let a;=one part,
20— a; = the other part,
a;2 + (20-a;)2 = 2a;(20-a;) + 36,
a;2 -f 400 - 40x + a;2 = 40a; - 2a;2 + 36,
4a;2- 80a; +364 = 0,
a;2-20x + 91 = 0.
240
QUADRATIC EQUATIONS 241
EXERCISE 116
Represent the number to be found by x and obtain, in its simplest
form, the quadratic equation which must be solved in each of the
following :
1.* The sum of a number and its square is 132. Find the number.
2. Find the number which is 156 less than its square.
3. The sum of the squares of three consecutive numbers is 149.
Find the middle number.
4. The product of a number and the number increased b\- 6 is
112. Find the number.
5. The length of a rectangle is 6 feet less than five times the width.
The area is 440 square feet. Find the width.
6. The average number of words on each page of a book is 6 more
than the number of pages. The total number of words is 9400. Find
the number of pages.
7. The area of a rectangle is 88 square inches and the perimeter
is 38 inches. Find the length.
168. Standard Form of the Quadratic Equation. Every
quadratic equation may be reduced to the form
in which a, b and c are any known numbers, except that a
can not be zero.
The term not containing x is called the absolute term.
It is frequently necessary to simplify equations to bring
them to the standard form, and thus determine if they are
quadratic equations.
Ex. 1. (a;+l)(2.r+3)=4:r2-22,
2x2 + 5a; + 3 = 4x2-22,
.-. -2a;2 + 5.T + 25 = 0,
2x2 -ox -25 = 0.
Here the equation is seen to be a quadratic. The coefficient of x* is 2,
of X is — 5 and the absolute term is — 25.
Or, o = 2, 6= -5, c= — 25.
It
242 ALGEBRA
Ex. 2. —H =1,
4 a;
.-. 7x2 + 4(a;-7) = 4a;,
7x2-28 = 0,
a;2-4 = 0.
Here a= 1, 6 = 0, c= —4.
T-, <, 2a; , a;+l
a;— 1 a;+2
.-. 2x(x + 2) + (x+l){x-l) = S{x-l)(x+2),
2x- + 4x + a;2-l = 3x2 + 3x-6,
a;+5 = 0.
Here a = 0, 6=1, c = 5, and the equation is not a quadratic, since
the coefficient of x^ is zero.
EXERCISE 117
Reduce to the standard form and state the values of a, b and c,
in which a is alwa\'S positive :
2, 25a:=6.r2+21.
4, 2=Ux—12x^-.
'• " + . = 2-
8. {3x-5){2x-5)=x~+2x-3.
^ 3A--8_5a:-2 IQ * 5 3
1.*
6x2=a;+22.
3.
19a;= 15-8x2.
5.
— -_5.r.
7.
a;2
8 — a; = — .
4
x— 2 a; 4-5 .r— 1 x-\-2 x
x+l 3a; _. 2a; _^— 3_,
a;+2 a;— 1 ^ ' ' a;— 3 x ~
169. Solution by Factoring
(1) When the absolute term is zero, the equation can
always be solved by factoring.
QUADRATIC EQUATIONS 243
Ex. 1.— Solve 2a;2-3x=0.
a;(2a;-3) = 0,
.-. a;=Oor 2x-3 =0,
.". a;=0 or |.
Verify both roots.
Ex. 2.— Solve ax^+bx=0.
x{ax + b) = 0,
.". a;=0 or aa;+6=0,
b
.-. x=0 or — - •
a
(2) When the middle term is zero, the equation can always
be solved by factoring, or by extracting the square root.
Ex.— Solve 3x2-27=0.
3(a;-3)(a;+3) = 0,
.-. a;-3 = 0 or a;+3 = 0,
x=±3.
Orthiis, 3x2-27 = 0,
.-. x^ = 9,
.-. a;=±3.
(3) The equation is a complete quadratic when none of the
coefficients a, b, c is zero. If the quadratic expression,
ax^-\-hx-\-c, can be factored by any of the methods previously
given, the solution is then easity effected.
Ex. 1.— Solve 3a;2_lla;=14.
3x2- llx- 14 = 0,
.-. (x+l){3x-14) = 0,
x= — 1 or V.
Verify both of these roots.
Ex. 2. — Solve x'^—mx-\-nx—mn={).
x(x — TO) + n(x — m) = 0,
.". (x — m)(x + n.) = 0,
x = m or —n.
R 2
244 ALGEBRA
exercise: 118
1-12. Solve the equations in the preceding exercise and verify.
13-19. Solve the problems in the first exercise in this Chapter.
(Verify the results.)
Solve by factoring and verify :
20. x^—3ax+2a^=0.
21.
a;2_52^0.
22. a;2— ma;— 6m2=0.
23.
x^—ax—bx-\-ab=0.
24. .-c2+2x(a+6)+4a6=0.
25.
2ax^+ax~2x=l.
26. {x—a)(x—b)=ab.
27.
x"—a^—{x—a){b-\-c)
170. Consider the problem : Find two numbers whose
sum is 100 and whose product is 2491.
Let ;c=one number,
100— .T= the other number,
.T(100-a:) = 2491,
.-. .t2-100x+2491-=0.
To solve this equation by the preceding method, we must
find two factors of 2491 whose sum is 100, but this is exactly
what the problem requires us to find.
The necessity is therefore seen for another method of
solving the quadratic equation when the factors of the
quadratic expression cannot be obtained readily by inspection.
171. Solution by Completing the Square. We know that
(^x-{-a)^=x^-\-2ax+a^, the middle term being twice the product
of X and a.
If the first two terms of a square are x^-\-2ax, we know that
it must be the square of x-\-a, and, therefore,
a'^ must be added to x^-{-2ax to make a
complete square.
What is the area of the shaded portion in
•^' " the diagram ?
Similarly, x^-\-4x must be the first two terms in the square of x-\-2.
To make x^-\-Ax a complete square we must add 2^ or 4. Also, x^ — Sx
are the first two terms in the square of a; — 4, and, therefore, 4* or 16
must be added.
QUADRATIC EQUATIONS 245
To complete the square, it is seen that the quantity to be
added is the square of half of the coefficient of x.
Ex. 1. — Factor .r'^+Bx-— 40.
Add 9 to a;- + 6x to make a complete square.
Then x^ + 6x-4:0 = x^ + Qx + 9-d-40,
= x- + 6x+9~49,
= (a;+3)2-72,
= (x+3 + 7)(a; + 3-7),
= (a;+10)(x-4).
Ex. 2.— Factor a;2+5x-806.
Add (f;)^ or -/' to x^-\-5x to complete the square.
Then x^ + 5x-SQ6 = x^ + 5x+\''-^^-806,
= x^ + 5x+ -Y- — -¥-»
= (x+g)2-(V)'',
= (a;+|+V)(a;+i--V-),
= (x + 31)(.T-26).
Ex. 3.— Solve a;2_ioOa;+2491=0.
Add 50^ or 2500 to complete the square.
.r2-I00a;+ 2500 -2500 + 2491 = 0,
a;2- 100^ + 2500-9 = 0,
(x-50)2-32 = 0,
.-. (a;-50 + 3){a;-50-3) = 0,
(a:-47)(x-53) = 0,
a;-47 = 0 or a;-53 = 0,
x = 47or5.3.
The solution might be contracted by writing it in the
following form :
a;2- 100^+2491 = 0.
Transpose the absolute term, .'. x^— 100a;= —2491.
Add 2500 to each side, .". x^- 100a; + 2500= -2491 + 2500=9.
Take the square root, ,*. x— 50= + 3,
a; = 50±3,
= 53 or 47.
246 ALGEBRA
Here the solution depends upon the same principle, but
assumes a simpler form.
It is thus seen that we effect the solution of a quadratic
equation hy finding and solving the two simple equations of
which it is composed.
Thus by the first method of solving x-—\00x^24Q\=0,
we obtained the two simple equations a;— 47=0 and x— 53=0,
and by the second .r— 50=3, and a;— 50= — 3.
Ex. 4.— Solve 3:r'-+.r=10.
Divide by 3 to make the first term a square.
Add ( ,1 ) '' to each side, .-. a;^ + Jx + ^V = ¥- + uV = W-
Take the square root, .'. ^+i= + -V>
x=+-V — i = tor— 2.
Verify both of these roots.
The steps in this method are :
1. Reduce the equation to the standard form and remove the
absolute term to the right.
2. Divide by the coefficient of x- if not unity.
3. Complete the square by adding to each side the square
of half the coefficient of x.
4. Take the square root of each side.
5. Solve the resulting simple equations.
EXERCISE 119 (1-8, Oral)
What must be added to each of the following to make a complete
square ?
1. x^-\-2x. 2. a;2— 4a;. 3. a;2+10a;. 4. x^—Ux.
5. x^-\~3x. G. x^—bx. 7. x^-{-^ax. 8. x'^—\x.
Factor, by making the difference of squares, and verify :
9. a;2+4.T— 77. 10. a;^— 54a;+713. 11. a;^— 2a;— 899.
12. a;2— a;— 1640. 13. a;^— i.ix+¥. 14. 3a;2+16a;— 99.
QUADRATIC EQUATIONS 247
Use the method of completing the square to solve the following and
verify the roots :
15. a;2+8x=9. 16. x^-()x=l. 17. a-^-lOx-f 9=0.
18. a;2-9x+18=0. 19. .t2+7x+10=0. 20. x-^x=2.
21. 2.t2— 3x=2. 22. 2a:2+.r=1081. 23. 6x^+ox=6.
1
24. If x-+x=lh, find the values of .r + - •
' •' X
172. Equations with Irrational Roots. In all the quadratic
equations we have solved, we fomid tliat when we had
completed the square on the left side, the quantity on the
right was also a square. This would not alwaj's be the case.
Ex. 1.— Solve .T2-6.r-l-0.
X^ — ()X=l,
:. x''-6a;+9=10,
a;-3=±\/T0,
a;=3±V'IO.
The two roots are S+VlO and 3— ViO. _
We might go a step further and substitute for V 10 its approximate
value 3-16.
The two roots would then be
3±3-16 = 6-16 or -16.
If we substitute either of these values for x in x^ — 6x—l, the result
will not be exactly 0, as we might expect, because \/lO is not exactly
3-16, but tlie difference between 0 and the value found for a;^ — Ox— 1
will be very small.
Ex. 2.— Solve 2.r2+a;=2.
X^ + ^X=1,
:. x^+^,x+^, = H,
x+k=±vn=±ivv7.
x=-i±kVvi.
The two roots are -l + iVT?, — ^-^VT?, or -781, -1-281, on
substituting V 17 = 4-123.
248 ALGEBRA
173. Inadmissible Solutions of Problems. When a problem
is solved by means of a quadratic equation, it does not follow
that the two roots of the equation will furnish two admissible
solutions of the problem.
Ex. — A man walked 25 miles. If his rate had been one
mile per hour faster he would have completed the journey
in 1^ hours less. What was his rate ?
Let his rate be x miles per hour.
At the supposed rate his tinae = hr.
■■ X x+1 *
Simphfying, x^ + a; — 20 = 0,
Solving, a; = 4 or —5.
Therefore his rate was 4 miles per hour, the other root giving a
solution which is inadmissible.
EXERCISE 120
Solve, finding the roots approximately to three decimal places,
using the table :
1.* a;2— 4.T=1. 2. x^—10x+ll=^0.
4. a;2+8.r=19. 5. .r(a:+3)=f.
Solve, expressing the roots in the surd form :
7. x^-6x=2. 8. .r2+8a;=ll.
10. 4x2-8a:=37. 11. 3.r2-5x-ll=0.
The following problems reduce to quadratic equations. In solving
the equations factor by inspection where possible and verify the
results.
13. The sum of two numbers is II and their product is 30. Find
the numbers.
14. The sum of the squares of two consecutive numbers is 85.
Find the numbers.
3.
.r2+2.r-6=0.
6.
2.r2+3.i-4=0.
9.
4a;2-4a;=7.
L2.
ix^+kx=l.
QUADRATIC EQUATIONS 249
15. The difference between the sides of a rectangle is 13 inches
and the area is 300 square inches. Find the sides.
16. Find two consecutive numbers such that the square of their
sum exceeds the sum of their squares by 220.
17. A merchant bought silk for §54. The number of cents in the
price per yard exceeded the number of yards by 30. Find the number
of yards.
18. The area of a rectangular field is 9 acres and the length is
18 rods more than the width. Find the length.
19. The three sides of a right-angled triangle are consecutive
integers. Find the sides.
20. How can you form 730 men into two solid squares so that the
front of one will contain 4 men more than the front of the other ?
21. The owner of a rectangular lot 12 rods by 5 rods wishes to
double the size of the lot by increasing the length and width by the
same amount. What should the increase be ?
22. If .r+2 men in .t+o days do five times as much work as x+1
men in x— 1 days, find x.
23. A rectangular mirror 18 inches by 12 inches is to be surrounded
by a frame of uniform width whose area is equal to that of the mirror.
Find the width of the frame.
24. What must be the radius of a circle in order that a circle with
a radius 3 inches less maj' be ^ as large ?
25. One side of a right triangle is 10 less than the hypotenuse
and the other is 5 less. Find the sides.
26. A man spends $90 for coal, and finds that when the price is
increased .$1-50 per ton he wiU get 3 tons less for the same money.
What was the price per ton ?
27. A man bought a number of articles for 8200. He kept 5 and
sold the remainder for S180, gaining S2 on each. How many did he
buy?
28. The sum of the two digits of a number is 9. The sum of the
squares of the digits is 5 of the number. Find the number.
29. A number of cattle cost 8400, but 2 having died the rest
averaged 810 per head more. Find the number bought.
250
ALGEBRA
30. How much must be added to the length of a rectangle 8 inches
by 6 inches in order to increase the diagonal by 2 inches ?
31. In the figure, the rectangle AO . 05= rect-
angle CO . OD.
(1) If .40=16, B0=3, C0= 8, find OD.
(2) If .40=10, 50=4, CD=IS, find OD.
32. In the figure, when 0.4 is a tangent to the circle,
OA^=OC .OD.
(1) If 00= 4, CD= 5, find 0.4.
(2) If 0.4= 8, 0Z)= 10, find OC.
(3) If 0.4 = 15, CD=m, find OD.
33. I sold an article for $56 and gained a per cent,
equal to the cost in dollars. What was the cost ?
34. The denominator of a fraction exceeds the
numerator by 3. If 4 is added to each term the
resulting fraction is | of the original fraction. Find the fraction.
35. An open box containing 432 cubic inches is to be made from a
square piece of tin by cutting out a 3 inch square from each corner
and turning up the sides. How large a piece of tin must be used ?
36. A and B can together do a piece of work in 14':- days, and A
alone can do it in 12 days less than B. Find the time in which A could
do it alone.
EXERCISE 121 (Review of Chapter XIX)
1 . What is a quadratic equation ?
Is (x+ l)(.'c — 2)(a;4-3) = (a;— 4)(a; - l)(a;+7) a quadratic equation?
Solve it.
3. The sum of a positive number and its square is 4. Find the
number to two decimal places.
5 8 „ 1 J_ _ J.
a;-8~ 12'
4. Solve -_ -- + -^- = 3 ; — -_
o—x o — x x—9
5. If x^j/^ — 6a;?/ — 7 = 0, what are the values of xy ?
6. Are x — 4 and a;2=16 equivalent equations, that is, have they
the same roots ?
7. Solve x- — xy-{-y~ = ^9, when y = '7.
QUADRATIC EQUATIONS 251
8. Divide 14 into two parts so that the sum of thoir 8([uare3
may be greater than twice their product by 4.
9. If {x~2){x-3) = l{x-S), does it follow that a;-2 = 7 ? Wliat
is the proper conclusion ?
10. The distance (s) in feet that a body falls from rest in
t seconds is given by the formula s=16-l<2. How long will it take
a body to fall 6440 feet ?
11. Solve 3a;2- 4a;- 1 = 0.
12. Ten times a number is 24 greater than the square of the
number. Does this condition determine the number definitely ?
13. Solve ^-±| = U^f^.
X— 1 3 2it;
14. Find two consecutive odd numbers whose product is 399.
15. Solve(2a;+3)2-2(2a;+3) = 35.
16. The units digit of a number is tlie square of the tens digit
and the sum of the digits is 12. Find the number.
^„ „ , .-c+10 10 11 7 15
17. Solve = = TT ; —r-k s = o *
x~5 X 6 a;+5 x—Z 3
18. If a train travelled 10 miles per hour faster it would require
2 hours less to travel 315 iniles. Find the rate.
19. Solve (3.r-7)(2a;-9)-(5.c-12)(a;-6) = (x-2)(2.r-3).
20.* Find, to three decimal places, the positive number which is
less than its square by unity.
21. If 4x2-3a;2/ + 2/2=i4, find a; if ?/ = a;+3.
22. The perimeter of a rectangle is 34 feet and the length of the
diagonal is 13 feet. Find the sides.
23. Solve a;^ + (a; — 4)2 = 40. State the problem, the condition in
which is expressed by this equation.
24. A line 20 inches long is divided into two parts, such that
the rectangle contained by the parts has an area of 48 square inches
more than the square on the shorter part. Find the lengths of the
parts.
25. Solve x^ + 2/' = 9, when y = Z — x.
26. The diagonal of a rectangle is 39 feet and the shorter side is yV
of the longer. Find the area.
27. If 5 is one root of x^ — lx^-\-Q>x-\-20 = Q, find the other roots
to three decimal places.
252 ALGEBRA
28. Find the price of eggs per dozen when 10 less in a dollar's worth
raises the price 4 cents per dozen.
29. The length of a field exceeds its breadth by 30 yards. If the
field were square but of the same perimeter, its area would be ^^
greater. Find the sides.
30. If 8a; = 4, find x to three decimal places.
X
31. The cost of an entertainment was $20. This was to be divided
equally among the men present. But four failed to contribute anything,
and thereby the cost to each of the others was increased 25 cents.
How many men were there ?
32. If a man waliied one mile per hour faster he would walk 36 miles
in 3 hours less time. What is his rate of walking ?
33. A polygon with n sides has ^n(n — 3) diagonals. If a polygon
has 20 diagonals, how many sides has it ?
34. Solve a2(a;_a)2 = 62(a; + a)2.
35. A can do a piece of work in 10 days less than B. If they work
together they can do it in 12 days. In what time could each do it
alone ?
36. If x^+ 0 = 84, find the value of x^ and of x.
x^
37. The length of a rectangular field is to the width as 3 to 2
and the area is 5-4 acres. How many rods longer must it be to contain
6 acres ?
CHAPTER XX
RATIO AND PROPORTION
174. Methods of Comparing Magnitudes. When we wish to
compare two magnitudes, there are two ways in which the
comparison may be made.
(1) We may determine by how much the one exceeds the
other. This result is found by subtraction.
(2) We may determine how many times the one contains
the other. Here the result is found by division.
Thus, if one line is 6 inches in length and another is 18 inches, we
may say that the second is 12 inches longer than the first, or that tho
second is three times as long as the first.
Neither method of comparison can be used, unless the
magnitudes compared are of the same denomination, or
can be changed into equivalent magnitudes of the same
denomination.
Thiis, we can compare 3 lb. and 10 lb. ; 2 yd. 1 ft. and 2 ft. 9 in. ;
but we can not compare 5 lb. and 4 ft.
175. Ratio. When two magnitudes, of the same kind,
are compared by division, the quotient is called the ratio
of the magnitudes.
Thus, the ratio of 3 to 4 is the same as the quotient of
3-1-4, which is usually \^Titten f .
The ratio of 3 to 4 is written 3 : 4,
.-. 3 : 4=3-4=|.
Similarly, a:fe=a-i-6 = ^.
253
254 ALGEBRA
It will thus be seen that all problems in ratio may be
considered as problems in fractions.
176. Comparison of Ratios. To compare two ratios we
simply compare the fractions to which these ratios are
equivalent.
Ex. 1. — -Which is the greater ratio, 3 : 4 or 7 : 9 ?
The problem is at once changed into : " Which is the greater fraction
f ©r i ? "
To compare the fractions we reduce them to the same denomination
in the forms |^ and f |, and it is seen that the latter is the greater. We
might also comiDare them by reducing the fractions to equivalent
decimals.
Ex. 2. — Which is greater, a : a+2 or a + 1 : a+3.
a _ a(o + 3) _ a^ + 3a
0^+2 ~ (ra + 2)(a+3) ~ (o + 2)(a+3) "
o+l _ (a+l)(a+2) _ a^ + 3a+2
a+3 ~ (a+2){a+3) ~ (a+2)(a + 3)'
What is the conclusion ?
177. Terms of a Ratio. In the ratio a .b, a and b are called
the terms of the ratio, a being called the antecedent and b
the consequent. The antecedent corresponds to the numerator
of the equivalent fraction, and the consequent to the
denominator.
a _ antecedent _ numerator _ dividend
' b consequent denominator divisor
178. Equal Ratios. Since a ratio is a fraction, all the laws
which we have used with fractions may also be used with
ratios.
(X 771(1/
Thus, since r = — r , it follows that a : b=ma : mb.
b mb
Hence both terms of a ratio m,ay be multiplied or divided
by the same quantity {zero excepted) without changing the value
of the ratio.
Thus, 6 : 9 = 2 : 3, ^ : ^ = 3 : 2,^ : ^ = a^ : b^
RATIO AND PROPORTION 255
EXERCISE 122 (1 15, Oral)
Simplify the following ratios by expressing them as fractions in
their lowest terms :
1.
10: 15.
2.
2^:5.
3.
45 : 63.
4.
15 : 10.
5.
$2 : S6.
6.
$2-50: 810.
7.
2 ft. : 3 yd
8.
2 days : 12 hr
9.
2 ft. 3 in. : 3 ft. 3 in
0.
24rt : 8a.
11.
5ah : lOa^.
12,
a + b:u--b-.
3.
a—h : a^—
h^.
14.
x—y : x—y.
15.
X X
-6.
1
1
_.
17.
a-2+2.w+?/2 : x-3+w3.
x—2 ' x'^—5x-\r&
18.* If 12 inches= 30-48 centimetres, find the ratio of an inch to
a centimetre and of a metre to a yard.
19. The edges of two cubes are 2 inches and 3 inches. Find the
ratios of their volumes.
20. If 25 francs =S4-80, find the ratio of a franc to a dollar and
of a quarter to a franc.
21. If a metre=39-37 inches, find the ratio of a kilometre to a mile.
22. Which is the greater 2 : 3 or 4 : 5, 15 : 37 or 11 : 27, a: a-\-2
or a+3:a + 5 ?
23. Arrange in descending order of magnitude :
2:3, 3:5, 11 : 15, 13: 18.
24. What is the effect of adding the same number 5 to both terms
of the ratio 7 : 15 ? What is the effect of subtracting 5 from each
term ?
25. What is the effect of adding 5 to each term of 15:7? Of
subtracting 5 ? Compare your results with the results of Ex. 24.
26. Separate 360 into three parts which are in the ratio 2:3:4.
(Let the parts be 2x, Zx, 4a;.)
27. Divide 165 into two parts in the ratio of 2 : 3 ; 510 in the ratio
3:7; 36 in the ratio 1^ : 2^.
256 ALGEBRA
28. When a sum of money is divided in the ratio 1 : 2, the smaller
part is $20 more than when it is divided in the ratio 2:7. Find the
sum.
29. What number must be added to both terms of f to make it
equal to 4 ?
30. What number subtracted from each term of 7 : 10 will produce
13 : 19 ?
31. What must be added to each term of a : 6 to produce c : d 1
What is the conclusion when c=dt
32. If a is a positive number which is the greater ratio,
l+2a l+3a„
— ■ or — ?
l+3a l+4a
33. The rate of one train is 30 miles per hour and of another is
55 feet per second. What is the ratio of their rates ?
34. Divide a line a inches long into two parts whose lengths are in
the ratio b : c.
35. A's income : JB's income=3 : 4, and ^'s expenditure : B's
expenditure =5 : 6. If ^ spends all his income, what per cent, of his
income does B save ?
36. Divide $315 among A, B and C, so that ^'s share will be to fi's
as 3 : 4, and 5's to C"s as 5 : 7.
37. A line is divided into two parts in the ratio of 5 : 7 and into
two parts in the ratio 3:5. If the distance between the points of
division is 1 inch, find the length of the line.
38. Two numbers are in the ratio of 3:5, but if 10 be taken from
the greater and added to the smaller, the ratio is reversed. Find
the numbers.
39. Two bodies are moving at uniform rates. The first goes
VI feet in a seconds and the second n yards in b minutes. What is
the ratio of their rates ?
179. Proportion. A proportion is the statement of the
equality of two ratios.
Thus, 3 : 4=15 : 20, since |=if.
Therefore, 3 : 4 = 15 : 20 is a proportion, or 3, 4, 15, 20 are
eaid to be in proportion, or they are said to be proportionals.
RATIO AND PROPORTION 257
If a, b, c, d are in proportion or a -.b—c : d,
a c
then - = -,
0 d
/. ad— be.
In the proportion a:b=c:d, a and d are called the
extremes and b and c the means.
Since ad=bc, it is seen that the lyroduct of the extremes is
equal to the product of the means.
180. Fourth Proportional. When a : b=c : d, d is called
the fourth proportional to a, b, c.
Thus, if the fourth proportional to 10, 12, 15 is x,
then 10: 12=15:a;or !^= '^,
12 X
.-. 10a;= 12x15,
x=lS.
181. To find a Ratio, by Solving an Equation. From certain
types of equations in x and y, the value of the ratio oi x : y
may be found.
Ex. 1. — If 5x—6y, find the ratio of x : y.
Since ox = 6y,
^ = ly.
x 6
y~5'
the ratio of x : y = ^ or 6 : 5.
Ex. 2.— If 3x-{-4:y=3y~lx, find -
3a; + 7a; = 3y — 4y,
x_ l^
" y~ To'
If each term in the equation is of the first degree in x or y,
the ratio oi x : y can be found, but it can not be found if
there is a term not containing x or y.
Thus, from 2x — ly= 10, the value oi x : y can not be found.
S
258 ALGEBRA
Ex. 3.— If 2.r2-7a:?/+6?/2=0, find x:y.
Factoring, {x-2y){2x-Sy) = 0,
.-. a;-2t/ = Oor 2a;-3i/ = 0,
- = 2 or ^ .
y 2
Here there are two values of -. If we divide each term of the given
y
equation by y^, we get
+ 6 = 0.
r-<i)
In this form we see that the equation is a quadratic in -, and we
might naturally expect to find two values for the required ratio.
Ex. 4.— If 2x-5ij+ z=0,
3x-\-2y—2z=0,
find the ratios of x, y, z.
If we eliminate z in the usual way, we get
7a;- 8!/ = 0,
a; _ 8 _ X y
y-r •■■8 = 7'
If we eliminate y we get
a; _ 8 . X _^ z
z ~ 19' • • 8 ~ T9 ■
We can combine these results in the convenient form :
X y z
8 ^ 7 "" 19"
EXERCISE 123 (1-21, Oral)
Find the value of x in the proportions :
1.
24
S^x'
2.
39
7~x'
3.
2 X
5~3'
4.
x_5
6 4"
5,
3 6
X -12*
6.
5
7.
a c
b x'
8.
a X
b~~c'
9.
4_ X
x~W
10.
3: 7 =
12;
; X.
11.
2:3=a;:9.
12.
X : 5=7 : 10.
BAT 10 AND PROPORTION 259
Find the value of x : y,
13.
2x=ly.
14. 3y=l2x.
15. 2x-y=0.
16.
^X = il/.
17. 2x= — 3y.
18. 3y +11x^0.
19.
x'^=iir.
20. ix'^f))/-.
21. {x-3y)(x-5y)=0.
22.
'<l-l-
show that - = -, and - =
c d a
d
c
23.
Find a
fourth
proportional to :
. 2
3, 18; 5, —7, -10;
h h
i ; a, b, c
; a, 2b
, 3c.
24. Find a fourth proportional to: a—b, {a-{-b)", a''—b^; and
to a^— 3a+2, a^— 5a+6, a^— 5o+4.
25. What number must be added to each of the numbers 2, 4, 17, 25
so that the results will be proportionals ? (Verify.)
2«j. If a^x, b+x, c-\-x, d-\-x are proportionals, find x. What
does the result mean when be— ad ?
27. Find a in order that « + 3:a+15=3:4.
28. ^'s age is to B's as 4 : 5. Five years ago the ratio was 3 : 4.
Find their ages.
29.* In an equilateral triangle the ratio of the altitude to either
of the equal sides is a/3 : 2. If the altitude is 10 inches, find the side
to two decimal places.
30. When a line is drawn parallel to the base of a triangle it divides
the sides in the same ratio. In the figure, .45=20, AD=14: and
AC= 15. Find A E and EC.
31. In the figure, the triangles ADE and ABC are
similar. W^hen triangles are similar their correspond-
ing sides are in the same ratio, so that D/. \E
AD : AB=DE : BC=AE : AC. ^ ^
If AB=8, BC=10, AC=d, AD^Q, find the lengths of all the other
lines in the figure.
32. In the same figure, the areas of the similar triangles are in
the same ratio as the areas of the squares on their corresponding sides
If AD=20 and AB^35, find the area of ADE if the area of ABC is
735.
s2
260
ALGEBRA
33. The side of the square ABCD is 10 inches and EF is parallel
to DC. If the length of AE is 8 inches, find the
length of FC to three decimals.
34, If the bases of two triangles are in the ratio
3 : 4 and their heights in ratio 8 : 9, find the ratio of
their areas.
From these equations find x : y,
13x+5?/=9a;+13// ; ax-\-bi/=cx-\-dy ;
mx—ny=nx-\-my ; px-\-qy=Q.
36. Find two values oi x : y when
6a;-— 13.i-^+6^-=0 ; x'=4iXy-\-5y^.
37. If 5rt— 36+2c=0 and a+6+c = 0, find the ratios
of a :h, a : c, a : b : c.
38. Find the lengths of all the other lines in this
figure.
39. If a pole 10 feet high casts a shadow 17 1 feet
long, what will be the length of the shadow cast, at the same
time, by a monument 84 feet high ?
3 D 9
:0 in a form showing -
y
40. Write the equation 3x^—10xy-\-3y-
as the unknown, and find x : y.
41. A number of two digits bears the ratio 7 : 4 to the number
formed by reversing the digits. If the sum of the numbers is 66,
find them.
42. The length of a room is to the width as 6:5, and the length
is to the height as 3 : 2. If the area of the floor is 187J square feet,
find the dimensions.
43. If 4 men and 3 women earn as much as 16 boys, and 6 men
and 5 boys earn as much as 10 women, find the ratio of the earnings
of a man, woman and boy.
44. If 3ab+2b^ : 2a^—ab=9 : 5, find a : b.
45. When the angle A is bisected,
AB : AC=BD : DC.
(1) If .4jS=10, AC=8, BC^12, find BD and DC.
(2) If AB=c, AG=b, BC=a, find BD and DC.
BAT 10 AND PROPOBTION 261
46. The ratio of the area of a rectangle to the area of the square
described on its diagonal is 6 : 13. Find the ratio of the sides.
47. The sides of a triangle are 7, 10 and 12. The perimeter of a
similar triangle is 72|. What are its sides ?
48. If ~ == ^ = — — •i-' — , find x : y: z.
5 7 9
182. Mean Proportional. When three numbers form a
proportion, it is understood that the middle number is to be
repeated. The three numbers are said to be in continued
proportion, and the middle one is called the mean proportional
between the other two.
Thus, 4, 6 and 9 are in continued proportion, since 4:6 = 6:9.
Here 6 is the mean proportional between 4 and 9.
If X is a mean proportional between 3 and 27,
3 X
- = ^, .-. a;2=81, .-. a;=+9.
X 21 -
.'. the mean proportionals between 3 and 27 are +9.
3 9 3 —9
Since q = :j^ and -^ = -^- it is seen that these are the correct
results.
Similarly, if x is the mean proportional between a and b, then
ax . /-L
- = -, .-. x= + Vab.
X 0
Therefore, the mean proportional between any two quantities
is the square root of their product.
183. Third Proportional. Tf a, h, c are in continued
proportion, c is called the third proportional to a and h.
Thus, if X is the third proportional to 6 and 1.5,
4 = -, .-. 6a;=22o, .-. x=37i.
15 a; ^
262 ALOEBRA
EXERCISE 124
1. Find a mean proportional between 4 and 16 ; 2a and 8a ;
4a6^ and da^b ; (a— 6)^ and {a+b)'^.
2. Find a third proportional to 2 and 4 ; 3 and 30 ; f^t and 10«6 ;
x"—y'^ and x—y.
3. The mean proportional between two numbers whose sum is
34 is 15. Find the numbers.
4. Three numbers are in continued proportion. The middle
one is 12 and the sum of the other two is 51. Find the numbers.
5. What number must be added to each of the numbers 3, 7, 12
so that the results will be in continued proportion ?
6.* In the figure, the angle BAC being in a
semicircle is a right angle. When AD is drawn
perpendicular to the Iwpotenuse it is proven in
" D "* geometry that
AD is a mean proportional between BD and DC,
AB between BD and BC, and AC between CD and BG.
(1) If BD= 4, DC= 9, find AD.
(2) If BD= 5, AB= 8, find DC.
(3) If BC = n, AC=\2, find DC, AB, AD.'
(4) If AB^ 3, AC= 4, find BC, AD, BD.
7. How would you use the preceding to find
(1) A line whose lengtii is V6 inches ?
(2) The side of a square whose area is 12 square inches ?
8. Find two numbers such that the mean proportional between
them is 4 and the third proportional to them is 32.
9. Divide a line 21 inches long into three parts such that the
longest is four times the shortest and the middle one is a mean
proportional between the other two.
184. The following examples will illustrate a method which
has many appplieations to problems with ratios or fractions.
RATIO A:ND proportion 263
Ex. l.-If ^ = j^, prove that -^^-^ = ^^-^ •
Since 7- = -,, let each fraction = k.
0 d
Then r = fc, .". a=bk, -, = k, .'. c=dk.
b a
Substitute these values of a and c in each side of the identity to be
proven.
3a2 + 262 3h2i2 + 262 b^{^k^ + 2) Zk^ + 2
a^-5b^ b^k^-5b^ b^k^-5) k^-5
3c^+2d^ _ 3d^k^ + 2d^ _ d^(3fc^+2) _ 3A:'' + 2
c^-od^ ^ d^k^-5d^ ~ d^k^-d) ~~ k^-5'
3a2 + 262 _ 3c^ + 2d^
a^ — 56^ c^— 5f/-
Ex 2.— If - = ^ = "^ , prove ^''^+-^'''+^'^ ^ix±y^\^^
Let x = ak, y = bk, z = ck. Substitute as before and show that each
of the fractions is equal to k^.
.0^ ,.. « f .^ a+& e-\-d
185. If 1- = ^ , then — ~ = :^ •
0 a a—o c—a
Prove this by letting a=^bk and c = dk, as in the preceding examples.
Here the fraction — , was obtained by adding and siib-
a—b -^ ^
trading the terms of the fraction -, and — ^ was obtained
in a similar way from -^ ■
This principle is sometimes useful in simplifying equations.
Ex. 1. — Solve
Adding and subtracting,
4a;- 3
3a -46
Sx
6a
6
~ 8b'
. 646a;
= 36a,
X
9a
~ 166'
Solve also, in the usual way, by cross multiplication.
264 ALGEBRA
Ex. 2.— If —^ — ^ = —r-> ^ ' prove j- = -
a—b—c-\-d a—b^c—d be
Adding and subtracting, ^^-j-^^ = ^b:-^^.
Adding and subtracting,
a-\-d
6 + c ""
a — d
' b-c
a-\-d
a—d
b + c
^ b-c
2a
26
2d"
~ 2c'
a
b
'• d
c
a
d
EXERCISE 125
1. If - = ^, prove that - — -r, = ^ — --,.
6 d^ 2a+3b 2c+3d
2. li a : b=c : d, show that
ma-{-nb : ma—nb^=mc-\-nd : nic—nd.
3. If a : 6 = c : d, prove a2ftf;_|_ft2g_|_5g_(jj2g_|_gf5(^_|_fj()(
4.* If - = -, find the value of 2^±^f .
y 4 6x2/+?/2
(Here x=fy, substitute for x and simplify.)
. rr --c 2/ c J ., 1 t {2x-{-3y){3x-\-2y)
5, If - =:^ ■^, find the value of ^ ^r Iw— -^. •
2 3 {5x-3ij){3x-5y)
r. Tt X n 1 « 2 ,. J ^, , J. \2ax—by
i\. If ^ 3 and - = -, find the value of ^ -,^ .
y b 5 2ax-\-6by
7, If - = ■ = -, prove that each fraction is equal to ^ , that
a b c a+b+c
. . sum of numerators
IS, to
sum of denominators
8. If -— = ^- = ^, prove :r{bi-c)+y{c^a)+z{a+b)=0.
b—c c—a a-b
RATIO AND PROPORTION 265
9. If a, h, c, d are proportionals, prove that a^c^ac^, bH+b(P,
u-c- and bH- are proportionals.
10. If = = , prove that a=b-\'C
11. If a : b — b : c, show that a : c = a- : b'~.
12. If a—b:a^b=c—d:c^d, prove a : b=c : d.
Ts: a+b+c+d a—b+c—d , ^u i. ^ c
13. If — ^ — — = -, show that - = -•
a+b—c—d a—b—c+d b d
14. Solve ■ — - = ■ — - •
3a;- 46 5a -36
15. If the sum of two numbers is to their difference as 7 to 4, find
the ratio of the numbers.
16. If '^^ — '- — = — ^ , show that each fraction
6— c+2 c-rt+4 a— 6+6
equals |. (Use Ex. 7.)
._ o , ax+bA-c bx-\-c-\-a
17. Solve ■ — ~= — — ■ —
ax—b-\-c bx-\-c—a
18. If a : 6=3 : 5, 6 : c=7 : 9, c : rf==15 : 16, find the ratio of a : d.
19. If ■- = "' = - , show that each fraction equals * ^^ , and
a b c ^ 5a-36+2c
, , mx+7iy-~pz
also equals ^ — i- •
'ma-\-nb — pc
20. Find two numbers such that their sum, difference and product
are proportional to 4, 2, 9.
21. If a, 6, c are consecutive numbers and if c-—b- : 6'-— rt-=41 : 39,
find the numbers.
22. The length and breadth of a room are as 3:2, and if 2 feet
be added to each, the new area of the floor is to the old as 35 : 27. Find
the dimensions.
23. If a: b=c : d, prove a : a+6=a+c : a+6-(-c+d.
„. „ lOa+6 I2a+6 , ... a c
24. If '—^ = — -— . show that - — -,■
lOc+d I2c+d 6 d
25. If a : b — b : c, then a'^-'rub : b^=b^-\-bc : c^.
266 ALGEBRA
EXERCISE 126 (Review of Chapter XX)
Write as fractions in their simplest forms :
1- 7^ : 8|. 2. a;2-t/2 : (x-y)'^. 3. a^ + b^ : {a + b)K
4- 1 s : 1-1 5. a- :1 » 6. a :c
x^ X a a c a
^- x^-5x+6'-x^ + x-l2' ^- "'+l+^ = «-l+a-
9. Divide 144 into three parts proportional to 3, 4, 11.
10. What must be added to each term of 4 : 7 to make it equal
to 6 : 7 ?
11. Write as a proportion in two ways :
3. 6 = 2. 9 ; 2 . 5 = 3x; ab^cd;
{a + b){a-b) = 3 . 4; a^-5a+6 = a^ + 5a + 4:.
12. If the means are 7 and 12 and one extreme is 3, what is the
other extreme ?
13. Find a fourth proportional to : 7, 15, 35 ;
a, a^. a^ ; x-\-y, x — y, x'^ — y^; tj r . a^ — b^.
^ " ^ a—b o + ft
14. Find two numbers in the ratio 9 : 5, the difference of whose
squares is 504.
15. Two numbers are in the ratio of 5 : 8, and if 8 be added to the
less and 2 be taken from the greater, the ratio is 14 : 15. Find the
numbers.
16. Find two numbers in the ratio 6 : 5 so that their sum is to the
difference of their squares as 1:3.
17.* If the ratio a — x : b~x is equal to the square of the ratio a : b,
find X.
18. If2a;+3?/ : 3a;-5?/=9 : 11, find a; : y.
19. If (5x-72/)(2a;-32/) = (4a;-52/)(a;-2/), find .r : y.
20. If 4a; -5y = 2a; +22/, find 3a;+2?/ : 2x^%y.
21. If 6.^2+ 15y2= 19x2/, find x : y.
22. Ifa;2 + a;+l : 62(a;+ l) = a;2-a;+ 1 : 63(a;-l), find x.
23. If 2x^y — 2z = 0 and 7x + 6j/ — 92 = 0, find x -.y, x : z and x : y : z.
24. If - = ? find the value of ^f!^ •
y 3 3x+l\y
RATIO AND PROPORTION 2G7
25, Find a mean proportional to a;^ ^ and y^ ^ .
y '('
26, If ax -{-by : bx'\-ay = 9 : 11 and a : 6 = 3 : 2, find the ratio of
X to y.
27, If r = -, = ^, show that each of these fractions is equal to
b d f
ma—nc—pe
mb—nd—pf'
28, Find two numbers whose sum, difference and product are
proportional to 5, 3. 16.
29, If a:b = c: d, show that - = — — ; „ -2~ro»,2 = ^ 2 i o 72 '
(ab + cd)- = (a^ + c^){b' + d%
30, If ; = — ; , = ; , prove that
o-j-c — a c-\-a — o a-\-b — c
x{b~c) + y(c-a) + z{a-b)^0.
31, If any number of ratios are equal, show that each ratio is equal
to the ratio of the sum of all the antecedents to the sum of all tiie
consequents.
32, If 3x- 2^ + 42 = 2a;— 3^ + 3 = 0, find the ratios of x, y, z. If
also, x^ -{- y^ -\- z^ = 150, find the values of x, y and z.
33, The hypotenuse of a right-angled triangle is to the shortest
side as 13 : 5. If the perimeter is 120, find the sides.
34, The length, width and height of a room are proportional to
4, 3, 2. If each dimension be increased 2 feet, the area of the four
walls will be increased in the ratio of 10 to 7. Find tlie dimensions of
the room.
ace a^ + c^-l-e^ ace
^'*- ^H = d= r b^w+p = bdf •
36, If the sides of a triangle are 6 and 8 and the base is 4§, find
the segments of the base when the bisector of the vertical angle is
drawn,
Q- re y — z-\-x x-~y-\-z .1x0 », 2
.i<. If ; — = — , - , show that z^ — x--\-y^.
z—x-\-y x+y+z
38, The incomes of A and B are as 2 : 3 and their expenses are as
f) : 7. If A saves 25% of his income, what % does B save ?
39. Find three values of the ratio x : y if
3{x^-4x^y + 5xy~-2y^)==2{x^~2x-y-2xy^ + 3y^).
CHAPTER XXI
THE GENERAL QUADRATIC EQUATION
186. Type of the General Quadratic. The equation
is called the general quadratic equation, because evei-y
quadratic equation may be reduced to this form.
If the factors of ax'^-\-bx-\-c can be obtained, the roots
of the equation can be found by solving the two equivalent
equations.
187. Solution of Literal Quadratics. The method of
completing the square may be applied to the solution of
quadratic equations with literal coefficients.
Ex. 1. — Solve x^-\-2mx=n.
Complete the square by adding ?n- to each side,
x^-\-2tnx-\-'m^ = 7i-\-m^.
Take the square root, x + m=± V'n + m^,
.'. x=—'m+Vn-\-m''.
The two roots are —m.+ Vn-\-m^, —m~Vn+m^.
Ex. 2. — Solve x^^px-\-q=0.
Transpose the absolute term, x^-\-px= —q.
7)" ^2 rtj2 ^2 — - 4o
Add '. to each side, x^ -\- px -\- -^ = — 7 + "^ = ^ ■
Take the square root, a; + ^ = ± ~^ '
~p Vp" — 4^
••■• =^ = -2"- 2
THE GENERAL QUADRATIC EQUATION 269
Ex. 3.— Solve aa:2+6:c+c=0.
Divide by a to make the first term a square,
b c
:. x^ + -x + - = o.
a a
.be
Irauspose, a;- -| — x =
. , , 62 , , 6 62 62 c 62_4cjc
Add -p- „ to each, x- + -x + j-^^ .-- = . „ •
Take the square root, x + ^r- = ±
h , \ 7>2 — 4ac
'2a 'la
b Vb^-iac
la 2a
_ -b±Vb'^-4ac
~~ 'la
The roots of the general quadratic equation are
2a
188. The roots of the general quadratic might also be
found by factoring as in art. 171.
ax^-\-bx-\-c = 0,
f , b , \'b--iac\
( . b c\
a x- + -a; + -
\ a a/
= 0,
(/ by- c 62)
«lV^+2aj +a-4^/
= 0,
(f , 6\2 62-40C)
= 0,
'""^ex 1 ^ V62-4acl
J r ^ 2a 2a 1
= 0.
2a
Since the product is zero, one of the^ factors must be zero. But
a is not zero, as the equation would not then be a quadratic.
6 V62-4ac ^ , 6 v'62-4ac ^
.'. ^ + o- -\ f. = 0 or a; + ;r- — 5 = 0,
'2a 2a 2a 2a
_ — 6±\/62— 4ac
270 ALGEBRA
BXERCISB 127
Solve by either of the preceding methods :
1.* x"-2ax^3a^.
2.
x2+46a;-562==0.
3. x^—Qmx+3m^=0.
4.
x--{-ipx—p^=0.
5. x^—2ax+b=0.
6.
x^+2bx—c=0.
7. ax^+2ax=b.
8.
ax^'+2bx+c=0.
9. ax^—bx—c=0.
10.
px^—qx-\-?-=0.
189. Solving by Formula. The roots of any particular
quadratic equation may be found by substituting the values
of a, b and c in the roots of the general quadratic.
Ex. 1.— Solve 6x-2-7.i;+2=0.
Here a = 6, b= — l, c = 2.
a u ^-^ ^ ^u 1 • -6±V62-4ac
feubstitute these values in x= ~ ?
2a
+ 7+\/49-48
•■• ^= -^12
7+1 8 6 2 1
= "12~= 1^ "'' r2 = 3 °'' 2°
Verify by substitution.
Ex. 2.— Solve 5a;2+6x-l=-0.
Here a=5, 6 = 6, c= — 1.
-6+V'36-(-20) -6+A/56
•■• ^ = —^^10 = — I^
-6+2\/l4 -S+Vli
10 5
In this case the roots are irrational, but, if necessary, we may
substitute for v/l4 its approximate value 3-742, when the roots become
-3 + 3-742 -742 -6-742 ,,„ , „,„
— = _ or = — = -148 or —1-348.
Note. — The pupil is warned to be careful of the signs when
substituting, particularly when c is negative.
THE GENERAL QUADRATIC EQUATION 271
Ex. 3.— Solve 2a:2-5^+6=0.
a=2, 6= -5, c = 6.
+ 5+v'25-48 5+\/^^23
4 4
190. Imaginary Roots. In the preceding result the numeri-
cal value of the roots cannot be found even approximately,
for there is no number whose square is negative.
Such a quantity as V— 23 is called an imaginary quantity,
and the roots in this case are said to be imaginary. This is
merely another way of saying that there is no real number
which will satisfy the equation 2x2— 5a;+6=0.
191. Methods of Solving Quadratic Equations. When a
quadratic equation has been reduced to the standard form, it
may be solved :
(1) By factoring, by insjiedion or by completing the square.
(2) By substitution in the general formula.
The pupil is advised to try to factor by inspection, and if
this method is unsuccessful, then substitute in the general
formula.
As the general formula will be used very frequently, it is
absolutely essential that it be committed to memory.
The roots of ax^-\-bx-\-e=^0 are —
2a
BXE3RCISB
128
Solve, using the formula :
1.* 3a:2-5x+2=0.
2.
24a;2-46a;+21==0.
3. 575a;2-2a;=l.
4.
2a;2— 6.T— 1=0.
5. 24:lx^+5x=l2.
6.
2x2- 13a;+ 10=0.
7. 391a:2+4a;=35.
8.
1200x-2-10a;=l.
9. x^+xC3b-2a) =
■ 6ab.
10.
2a;2-25x+77=:0.
11. 6a;2-a;-l=0.
12.
1800x2_5a;-l=0.
272 ALGEBRA
Solve by any method. Verify 13-18 :
13. 27a;2-24a;=16. 14. 15a;2-f7a;-2=0. 15. V2x^-x-^=0.
16. 4d;2-17a;+4=0. 17. A%Qx^-^x=l. 18. 5~26x+5a;2=0
19. 9a;+4 = 5a;2. 20. 3x2+2^9a;. 21. 'ia-^— 2a;=f
22. 4a;2-4a;=79. 23. l-y=y\ 24. 1^-- = ?.
3 9 a;
25. x-^-~=Q. 26. ^ + 1 = J-. 27. a;^--^a;=l
a;+2 2^23^ 12
28. (x--4)2-3(a;-9) = 15. 29. (.r-2)(a;+3)-x(5a;-9)-2.
30. 2aa;2+x(a-2) = l. 31. ac^- = -^cx.
a x
32. 2a;(x-2)=a2_2. 33. ^-f?==^ + -.
^ X o ^
3*- -4t + -4o = ^- 35. (x-+2)2+(a:+3)2=:(x-+6)2.
a;+l a;+2 a;+3
36. x^-xy-Zy^=-\2. If ?/=2, find a:.
37. a;2— 4a;y+x3+«/2+5=0. If a:=— 3, find ?/.
x x-\- 1
38. If = , find x to three decimal places.
x+\ 2x ^
39. Find the sum of the roots of a;^— 3a;=20.
40. The area of a square in square feet and its perimeter in inches
are expressed by the same number. Find the side of the square.
41. The length of a rectangular field exceeds the width by 16 rods
and the area is 32 acres. Find the length.
42. Find three consecutive even numbers whose sum is ^ of the
product of the first two.
43. A line 10 inches long is divided uito two segments so that the
square on the longer segment is equal to the rectangle contained by
the whole line and the shorter segment. Find the segments to two
decimal places.
44. Find two numbers whose difference is 3 and the sum of whose
squares is 317.
45. The area of a square is doubled by adding 5 inches to one side
and 12 inches to the other. Find the side of the square.
THE GENERAL QUADRATIC EQUATION 273
46. Three times the square of a number exceeds eight times the
number by unity. Find the number to three decimals.
47. Mr. Gladstone was born in the year a.d. 1809. In the year
A.D. x'^ he was x— 3 years old. Find x.
48. The area of a rectangular field is half an acre. The perimeter
is 201 yards. Find the sides.
49. One root of x^—bx-\-d=0 is 8. Find the value of d and the
other root.
50. If a train travels 10 miles per hour faster than its usual rate,
it will cover 480 miles in 4 hours less time. Find its usual rate.
51. Divide 3 into two parts so that the sum of their squares may
52. I buy a number of articles for 84"80 and sell for S5"95 all of
them but 2 at 6 cents a dozen more than they cost. How many did
I buy?
53. A straight line AB, 12 inches in length, is divided at C so as
to satisfy one of the following conditions. Find, in each case, the
length of AC to two decimals :
(1) AC^^2BC\ (2) AC^=2AB . BC.
(3) 3.4C'2=4.4£ . BC. (4) AC^+^BC-^=2ABK
(5) .4C'2-5C2=10sq. in. (&) AC{AB+BC) = 2 sq. it.
54. I buy a number of books for $6, the price being uniform. If
they had been subject to a discount of 5 cents each, I could have
bought 6 more for the same money. What did each cost ?
55. Solve the equation ax^-^bx+c^O by multiplying by 4a and
completing the square of 2ax-\-h.
o6. Solve —J — -\ „ — — = - •
2—x 2+x a;2— 4 3
Verify the roots obtained.
192. Equations Solved like Quadratics. There are certain
types of equations of a higher degree than the second, which
may be solved by reducing them to the form of quadratics.
T
274 ALGEBRA
Ex. 1.— Solve x^-lOx^+d^-0.
This is an equation of the fourth degree, but we might write it in
the form of a quadratic, tlius :
(a,.2)2-10(a;2) + 9 = o,
or if we write y for x^ it takes the form
.-. (y-9)(y-l) = 0,
y=d or 1.
Buty = x', .'. a;^=9 or 1,
.r= + 3 or ±1.
We see that this equation has four roots. This is what we might
expect, as it is an equation of the fourth degree in x.
Verify each of the four roots.
Ex. 2.— Solve (:^2_5^^)2^4(^2_5^)_12^0.
Here we consider x~ — 5x as the unknown, whose value should first
be found.
Let cc^ — 5a; = t/,
2/2 + 4?/- 12 = 0,
.-. (y+6)(y-2) = 0.
y
= ~
6
or
2.
x^—5x=—Q,
or
a;2-5a; = 2,
a;2-5a;+6 = 0.
2.
a;2_5a;_2 = 0.
(x-^)(x-2) = 0,
a; = 3 or
5+\/25 + 8
^- 2
5±V33
This equation has foiu- roots, two of which ai-e rational and the
other two irrational.
Verify the rational roots.
Ex. 3.— Solve {2x^-\-3x-l){2x^+3x-2)=56.
Let 2x" + 3x = y.
rp, u • 3 o -3±V'^^9
J he result is x = jr, —3, •
THE GENERAL QUADRATIC EQUATION 275
Ex. 4.-Solve -X-+.^+2.= 5-
- , a;2 + 2a; 3 1
^^' "3-=^' •■■ a^^ = 2/'
1 26
y 5
Complete Ex. 's 3 and 4 and verify the rational roots.
Ex. 5.— Solve .6-3-1=0.
Factoring, {x—l){x~ + x-\-\)=(),
x — 1 = 0 or a;^ + x + 1 = 0,
x=i or X = ^ •
We thus see that if one root of an equation of the third
degree, or a cubic equation, can he found by factoring, the
equation can he completeUj solved.
This equation might be written ,^^=1, and each of the three
roots when cubed must give unity, which shows that unity
has three cube roots. This is what we might have expected,
as we have aheady seen that unity has two square roots
+ 1 and —1.
EJXBRCISE 129
Solve and verify the rational roots :
1. a;*-5x2+4=0. 2. a;4-13a;2+36=0.
3. V + 12=317/2. 4. 8.c6-65a;=»+8=0.
5. (a;H5a;+6)(a;2-9a;+14):-0. «. ^+16_^ ^^ ^ 2.
V T T A ^ ) 25 ^a;2+16
42
7. (a;2-4a;+5)(x2-4x+2) = -2. 8. a;^ + x + 1
9. (.r2+x+l)2-4(a;*+a;+l)+3=0.
10. x-3-4a;2-4x+16=0.
11. 6('.r + -y-35('x+^')+50=0.
12. (l+a;+a;2)(x+x2) = 156.
T 2
276 ALGEBRA
13.* {x+l)(x+2)(x+S){x-\-4:)^120. (Multiply the first and last
factors and the second and third.)
14. x{x-\){x-2){x-3)=360.
15. Find the three cube roots of 8 by solving the equation x^— 8=0.
16. Find the four fourth roots of 16 by solving the equation
17. Solve .c^— 19a; +30=0 being given that 3 is one of the roots.
18. Solve 12x3—29x2+23^—6=0 (use the factor theorem).
19. It is evident that 4 is a root of the equation
.r(a;-l)(x-2) = 4. 3.2.
Find the other two roots.
20. Find the six roots of 8x^—211x^+2: =0.
21. Solve (x-~xY^-8{x^~x) + l2=0.
22. Solve x^ + — - + a; + - = 4. (Add to x^ + - the quantity
x^ X \ x^
required to make it the square of x H j
EXERCISE 130 (Review of Chapter XXI)
1. Explain the different methods of solving quadratic equations.
Illustrate them, by solving in full the equation 3x^ — 4x— 15 = 0, by
each method.
2. Solve 323x2 + 2x=l.
3. The difference of two numbers is 8 and the sum of their squares
is 104. Find the numbers.
4. If X = 2f 1+^, ), find x to two decimal places.
5. What is the price of meat per lb. if a reduction of 20%
in the price would mean that 5 lb. more than before can be bought
for $3 ?
6. Solve 10x2- 19x- 9 = 0.
7. The sides of a right-angled triangle are a, a— 10 and a+10.
What are the sides ?
THE GENERAL QUADRATIC EQUATION 277
8. Solve ^ + -^ = 4.
a; — 4 x—b
9. The sum of two numbers is 45 and tlie sum of their reciprocals
is -09. Find the numbers.
10. Solve 6375a;2- 10a;= 1.
11. The length of a rectangular field is o rods more tiian the width.
The area is 3^ o.cres. Find the sides.
\Qa-\-2n'^
12. What must be the values of n in order that ,^ — — ;r^ may
10« + 21«
equal | when a = nV '
13. The perimeter of a rectangle is 5G and the area is 192. Find the
diameter of the circle which passes through its angular points.
14. Solve •0075x2 + •75a;= 150.
15. By solving (x— 2)(a;— 3) = (a — 2)(a — 3), find a quantity which
can be substituted for a in (a — 2)(o.— 3) without changing its value.
16. Solve a;3- 2x2 -89^+90 = 0.
17. Two trains each run 330 miles. One of them, whose average
speed is 5 miles per hour greater than the other, takes \ an hour less
to travel the distance. Find their average speeds.
18. Solve f±j^ + J^^ = 2J.
19. Solve^^ + -^ = — .
20. I sell a horse for $96 and gain as much % as the horse
cost in dollars. What was the cost ?
21. Solve (.T2-3a;-5)2+8(x2-3x-5) + 7 = 0.
22. Divide 25 into two parts so that the sum of the fractions formed
by dividing each part by the other may be 4-25.
23. The sides of a rectangular field are a;+17 and x— 17. The
diagonal is 50. Find the area.
24. Solve a;2 + .r+ ,'~ =18.
x^->rx
25.* Solve (w2_.^2)a;2^2x(?n2 + n2) + m2-n2 = 0.
26. Solve (x-2)(.-c-l)(.T+2)(x+3) = 60.
27. Find all the roots of the equation x^= 125.
28. Since x^ — 8a;+12 = (x— 2)(a;— 6), for what values of x will the
expression x* — 8x+12 be equal to zero, and for what values will it
be negative ?
278 ALGEBRA
^^ c, 1 « i> a — b
29. Solve r = r-
x — a x — b a-\-b
.'50. 8olve adx — acx" = bcx~bd.
31. The area of a square is trebled by adding 10 inches to one
side and 12 inches to the other. Find the side of the square.
32. Solve a;2(a2-c2)-a;(a6 + 36c)-262 = o.
33. Solve (a;2-|-6x+8)2 + 3.T(a;2 + 6.T + 8) = 0.
34. A man bought a number of acres for S300. If he had paid
$5 more per acre, the number of acres would have been 2 less. Find
the number bought.
o. o , 1 111
35. Solve — --, =-+-+--.
x+a+o x a h
36. Solve — -. \- = ,- •
b a x — a x—b
37. OX and OY are two roads at right angles. A starts at noon
along OX at 3 miles per hour. B starts at 2 o'clock along O F at 4 miles
per hour. Find to the nearest minute when they will be 20 miles apart.
38. Solve a2a;2-2a3a; + a*-l = 0.
6c"
39. Solve oa;2— , =cx — bx^.
a-\-b
40. A gravel path 2 yards wide is made round a square field and
it is found that it takes up xV of the area of the field. Find the area
of the field in square yards.
41. Solve s = i?<+16<2 for <.
42. What positive integer is that, the sum of whose square and cube
is nine times the next higher integer ?
43. vSolve (a:2 + a;-2)2-4(a;2 + .r-2) + 3 = 0.
44. The side of a square is 34 inches. Find at what points in the
sides the vertices of an inscribed square must be placed so that it
may have an area of 676 square inches.
45 . Write the equation ax^ + bxy + c^/^ = 0 as a quadratic in - . What
X V
are the values of - and of - ?
y X
46. What positive integral value of x will make a;*+10x most
nearly equal to 1000 ?
CHAPTER XXII
SIMULTANEOUS QUADRATICS
193. Consider the problem : The sum of two numbers is 12
and the sum of their squares is 74. Find the numbers.
Let a; = one of the numbers,
12 — a; = the other,
.-. a;2 4-(12-a;)2 = 74.
Solve this equation and find a; =7 or 5.
If a; =7 or 5, then 12 — a; = 5 or 7
/. the numbers are 5 and 7.
Here we have used only one unknown. We might have
solved by using two unknowns.
Let X and y be the numbers,
.-. x + y=\2,
and a;2 + ?/- = 74.
How can we obtain from these two equations the original equation
in the preceding sohition ?
194. Type I.
Ex. 1.— Solve a;+3.y=10, (1)
x^-\-xy^=4:. (2)
From(l), a;=10-3t/ (3)
Substitute in (2), (10-32/)2 + y(10-3/y) = 4,
.-. 100-()0;y + 9(y2+10i/-32/2 = 4,
6^/2 -501/ + 96 = 0,
3.V*- 252/ + 48 = 0,
(.V-3)(3?y-16) = 0,
2/ = 3orJ/.
279
280 ALGEBRA
Substitute y= 3 in (3) and x==l.
!» y = ^ " ;> !» a;=— 6.
There are therefore two solutions,
x= 1, 2/ = 3 or a;= —6, y = 5^.
x= 1 or —6,
i/ = 3 or 5i.
Verify by showing that x=l, y = 3 satisfies both equations and also
x=-6, 2/ = 5i.
The pupil must note that x=l was obtained from ?/=3,
not from y=5^.
Therefore, x=l, y=5J is not a sohition, nor is x= — 6,
y=3. Verify this by substitution.
Equation (1) is a Unear equation, or an equation of the
first degree in x and y. Equation (2) is a quadratic equation,
or an equation of the second degree in x and y.
A system of equations of this type, that is, where one is
of the first degree and the other of the second degree, may
always be solved by the method of substitution, which does
not differ from the similar method employed in art. 107,
when both equations were of the first degree.
Ex. 2.— Solve 3x~y^5, (1)
x^+3xy=l5. (2)
From (I), 2/ = 3x-5, .-. a;2 + 3n;(3x-5)= 15,
.-. 10,r2-15a;-15 = 0,
2x2- 3a;- 3 = 0,
3±V3^ 3+5-745
4 4
2-186 or -686.
.-. y = 3x- 5= -Ai^'^^ =1-558 or - 7-058.
Here the roots are irrational and it is customary to leave them in
that form, unless the decimal form is asked for.
SIMULTANEOUS QUADRATICS 281
EXERCISE
181
Solve and verify
1-6
1. x+y^l.
2. .T— ?/ = 4,
3. x-2y=0.
xy=l2.
xy=m.
x^-y'^=21.
4. x-y^3.
5. x—y=6.
fi. 2.r+//=9,
x^+y^=m.
x^-y^ = 60.
a;2-i/2=15.
7* x+Sy=U,
S. 2x+3y=l2,
9. 3a:— 4?/=2,
x''+y=21.
a;2 + ?/2=13.
3a;2+2?/2=140.
10. x2+3a;!/+«/2+2.r=37, 11. 3x2— 2a;?/+5a;— y=17,
x—y=Z. 2x—Zy=\.
12. If a;— 3?/=2 and x'^—xy^2y"=&, find the values of x and ?/
to three decimal places.
13. The hypotenuse of a right-angled triangle is 25 and the
perimeter is 56. Find the sides.
14. A is 10 years older than B. Eight years ago the sum of the
squares of the numbers representing their ages was 148. Find their
ages.
15. The diagonal of a rectangle is 50. The difference of the sides
is 10. Find the area.
16. The area of a right-angled triangle is 96 and the difference
of the two sides about the right angle is 4. Find the hypotenuse.
17. Solve 3x4- 52/^:2, Zx^-my^-xy^2%=(i.
18. If each digit of a number be increased by 2, the product of these
increased digits wUl be the original number. When the digits are
interchanged the resulting number is thirteen times the tens digit of
the original number. Find the number.
19. The sum of the areas of two squares is 40 square inches. The
side of the smaller is 10 inches less than three times the side of the
larger. Find their sides to three decimals.
20. Solve - + 5-= 14, i/-l=a;.
y- y
195. When both equations are of the second degree in x
and y, they can not always be solved by elementary methods.
282
ALGEBRA
There are special cases in which they can be solved without
difficulty.
196. Type II.
Solve
Factoring (1),
{x-4ij){x-y) = 0,
x=4y or x=y.
We are now required to solve :
x'^-{-y'^-\-^x-
= 291
and
Substituting the value of x,
16,/2 + y2+122/ = 29,
17i/2+122/-29 = 0,
(2/-l)(172/ + 29) = 0,
y—\ or — ?§,
x = 4 or — TV•
-2/2+3x=29^
x = y f
y'+y'- + 3y = 29,
2y'' + 3y-29 = 0,
■3+\/24T
y
-3+\/241
(1)
(2)
Here there are four solutions ;
4 or —
X
1 or
-3+V241
4
-3±V24l
4
In this type the first equation contains only terms of the
second degree. When that is the case the left-hand member
may be factored and each of the resulting linear equations
may be combined with the second equation, thus giving two
cases of Type I.
EXERCISE 132
Solve and verify 1-5 :
1. x-—y"=0.
2. x~—4xy+3y^=0.
x-+xy+y-=36.
x^+y^=lO.
3. 3x^-2xy-y^=0,
4. x^+y^+2x=l2.
x+y+y^=32.
3x^+2xy=yK
SIMULTANEOUS QUADRATICS 283
5. 4x2+20.r;/+%2=0, fi.* •^'" + '^ = 14,
ir y
2.1//+ 1=0. .T- 1-7/2.
7. 6.r2-17xy/+12^-=0, 8. ,'1:2+2^^=5,
x'^—xy—y^\. Qx'^^Ay-^Wxy.
9. Find four solutions of the equations
(x--^)(.r-2)=0, U-+y/-6)(//+3)=0.
197. Type III. Homogeneous Equations.
Solve x^—xij=Q, (1)
y/2+3x-y=10. (2)
Multiply (1) by 5 and (2) by 3 and subtract, to eliminate the absolute
terms, and we get
,5x2- 14x7/- 3y 2 = 0. (3)
This equation (3) is of tlie same form as the first equation in
Type II. Grouping (3) with (1) we proceed as before.
Factoring (3), {x—Zy)(Z
a; + 2/) = 0,
x = Zy or —ly.
Substitute x—Zy in (1),
Substitute x
--i7/in(l).
.-. 9y2_37/2 = G,
izV
Hl2/2 = 6,
2/-^=l,
2/2 = 25,
2/=±l,
2/=±5,
.^=±3.
_•_
a;= + 1-
Hence the four solutions are :
x = 3, ^^ a;=-3, ^^
or or
y=\. 2/=-l.
x=-\,
7/ = .5.
or
a;=l.
Verify each of these four pairs of roots.
If we had grouped (3) with (2), the results would have been the same.
Show that this is true.
In this type, terms of the first degree were absent from
both equations. The expression on the left in each is homo-
geneous, that is, every term is of tlie same degree. For this
reason, this is called a homogeneous system.
284 ALGEBRA
The pupil should be on the look out for special methods of
obtaining from the given equations an equation of the first
degree. Here we might have done so by simply adding the
equations and taking the square root. Solve it by this
method.
EXERCISE 133
Solve and verify 1-9 :
1. 3x^-5!j"=28, 2. 2.i-2-3?/2=23, 3. x^-xy+y^=2\,
Sxy—4:y^=8. 2xy-3y^=3. 2xy—y^=15
4. 2x"—3xy^l4:, 5. x-+xy^66, 6. x~~xy=54:,
3y'—x^+l=0. x'^-y"=U. xy—y^=18.
7. x^+2xy=32, 8. 3x^-5xy+2y"=14:, 9. x^~-iy^=20,
2y^+xy=l6. 2x--5xy+3y^=6. xy=12.
10.* x^-3y^^4, 11. x^+xy+y^=7, 12. 32y^^-2xy+U,
x^+xy+y^-=2S. 3x^-l=xy. x^+'kij^=10.
13. 2x^-9xy+9y^=5, 14. x^+xy+tj^=7,
4:X^—10xy+Uy^=35. 2x^+3xy+4y^=24:.
15. 3x^~3xy-^2y^=2x, 2x^+3y^~-xtj=4:X.
16. Find, to two decimals, the real values of x and y which satisfy
x^—xy=20 and 3xy—y^=50.
17. When a number is multiplied by the digit on the left the product
is 105 ; when the sum of the digits is multiplied by the digit on the right
the product is 40. Find the number.
198. Special Methods.
Since {X'\-y)^—{x—yy'^-\-4:Xi/, it follows that if the values
of any two of the quantities .t -(-?/, ^—y and xy are given, the
remaining one can be found.
Ex. 1.— Solve x-\-y=U, (1)
xy=l8. (2)
Squaring (1), x'' + 2xy + y^=l2l.
From (2), 4xy = 72.
Subtracting, x'^ — 2xy-\-y' = 4:9,
:. x—y—±7.
SIMULTANEOUS QUADRATICS 285
If x-[-y=U, If
x + y=n.
and x—y = l. aiid
x-y=-l.
a; = 9, y = 2.
x = 2,y = 9.
Hence there are two solutions :
a; = 9 or 2,
2/ = 2 or 9.
Ex. 2.— Solve .T— ?/=ll, .r//=60.
Find {x-\-y)^ by adding 4a;^ to (x — y)' and complete the solution.
Ex. 3.— Solve .t3+^3^35, (1)
x-^ij=5. (2)
Dividing (1) by (2), x^-xy + y^ = 7, (3)
Squaring (2), a;2 + 2x2/ + 2/2 = 25.
Subtracting, 3xy=lS,
xy = 6. (4)
Subtracting (4) from (3), x^ — 2xy + y^=\,
x—y= + l.
Complete the solution as before.
Also solve by substituting x = 5~-y from (2) in (1).
Ex. 4.— Solve a:4+a;V+2/''=91> (1)
.»--+^2/+y'=13. (2)
x*-\-x^y^-^y* = {x^-{-xy-{-y^){x^ — xy+y^).
Dividing (1) by (2), x^-xy + y^==l. (3)
Subtracting (;5) from (2), 2xy = 6,
xy = 3. (4)
Adding (2) and (4), x- + 2xy + y''=H\
:. x + y=±4. (5)
Similarly from (3) and (4), x — y= + 2. (0)
(5) and (6) can be grouped in four ways, thus :
a;+w = 4, x + v= —4, a; + w = 4, x + y= —4,
\ a 'or •^ or ^ or ^
x — y = 2. x — 2/= — 2. X— ?/=— 2. a;— 2/ = 2.
From these four solutions are obtained :
x=3, -3, 1, -1, x=+3or+l,
y=\, -1, 3, -3. y=±lor+3.
286
ALGEBRA
Ex. 5.— Solve {x+y)^-5{x+ij)-Q--^0,
xy=8.
Factoring (1),
Now solve
{x + y-6)(x + y+\) = 0,
a;+2/ = 6 or — 1.
xy = 8. xy = 8.
(1)
(2)
EXERCISE 134
Solve, by finding x+y and
x-y.
and VI
erif y :
1.
xJf-y=8, 2. :i
:-y=
4,
3. x2+!/2=25,
xy=lb.
xy=
12.
x—y=\.
4.
a;2+2/'=61, 5. (:
v-yY
'=1,
6. x-—xy+ij^=51.
a;+?/=ll.
xy
=30.
x-y=8.
7.
x^+xy+y~=19,
x+y=5.
8.
x^-xy+y^=19,
a:+2/=13.
9.
5x'^ + xy+5y~=23,
x+y=l.
10.
a;?/ =40.
11.
x^—7xy+y'^= — 101,
xy=SO.
12.
2a;2+3a;i/+2//2=8,
x-?/= —6.
13.
X3_7/^=19,
x-y=l.
14.
x^+y^=l064:,
x+y=U.
15.
x-—xy^y^=39,
a;3+?/3=351.
16.="
' x'+x^y^+y*=2l,
x''+xy+y^=l.
17.
x*-\-x^y^+y*=l33.
x^—xy+y"=7.
18.
x^-xhj'-^y^ = n,
xy=2.
19.
(x+yr--3{x-hy)-2S--
=0, .X-
-y=i
20.
{x-yy~-l{x-y) + l2:
=0, xy=\2.
21.
x^y^-27xy +180^0, ;
«+Z/=
=8.
22. The perimeter of a rectangle is 34 inches and the diagonal is
13 inches. Find the sides.
SIMULTANEOUS QUADRATICS 287
23. The diagonal of a rectangle is 25 and the area is 300. Find the
sides.
24. The sum of two numbers is 12 and the sum of their squares
is 72-5. Find the numbers.
25. The product of two numbers is 270. If each number is decreased
by 3 the product will be 180. Find the numbers.
2G. The sum of two numbers is 10 and the sum of their reciprocals
is f*5. Find the numbers.
27. Solve (x-l)(y+2)=9, 2a-//= 15.
28. A and B are two squares. The area of A is 63 square inches
more than B, and the perimeter of 4 is 12 inches more than B.
Find the side of each.
29. Find two numbers whose product is 1 and the sum of whose
reciprocals is 2^\7.
30. Solve x^—%if=b%, x-2ij=2.
31. The sum of the two digits of a number is 1 of the number.
The sum of the squares of the digits is 4 less than the number. Find
the number.
32. The area of a rectangle is 1161 square yards, and its perimeter
is 140 yards. Find the dimensions.
33. Solve 1 + 1 = -3, - - ^ = -03.
X y X- 7/^
34. The sum of a number of two digits and the number formed by
reversing the digits is 121. The product of the digits is 28. Find the
number.
35. Find the sides of a right-angled triangle whose perimeter is
24 inches and whose area is 24 square inches.
36. Prove, algebraically, that if two rectangles have equal areas
and equal perimeters, they are equal in all respects.
37. Soh'e x'^-\-xy-[-y'^=l-15, x-—xy-\-y^=5-25.
38. What must be the dimensions of a rectangular field containing
7J acres, if the greatest distance from any point in its boundary to
any other point is 50 rods ?
288
ALGEBRA
39 . The sum of the radii of two circles is 8 inches and the sum of
their areas is | of the area of a circle whose radius is 9 inches. What
are their radii '!
40. What must be the length of a rectangular field that contains
a square rods and which can be enclosed by a fence b rods long.
199. Graphical Methods. What is the distance of the
point P(4, 3) from the origin 0 ?
Since
(9P2.
OP^--
OP --
-OM^+MP^,
:42+32=25,
^5.
If any point {xij) is the same
distance from the origin that P is,
then the point {x,y) must lie on
a circle whose radius is 5 and
whose centre is 0. But the
square of the distance of the point
{x,y) from the origin is x^-\-y^,
It is thus seen that the equation x'^'\-y^=25 represents a
circle ivhose radius is 5 and whose centre is the origin.
Similarly, x^-{-y^=lQ, x^'j-y^=lOO, x^-iry^=\8, represent circles
with the origin as centres and whose radii respectively are 4, 10, VlS.
It is seen that it is a simple matter to draw the graph of
the equation of the circle in the form x^-\-y^=r^. All we
require to do is to describe with the compasses a circle whose
centre is the origin and whose radius is r.
When the radius is a surd as in x^-[-y^=18, it is simpler
to find a pair of values of x and y which satisfy the equation.
Here .t=3, y=3 satisfies the equation, and the circle is then
described through the point (3, 3).
SIMULTANEOUS QUADRATICS
289
200. Graphical Solution of Simultaneous Equations
"Solve x^-\-y"=2o, (1)
x-y=\. (2)
(1) represents a circle whose radius is 5.
(2) represents a straight hne, two
points on which are (1, 0) and (0, — 1).
The graphs of (1) and (2) are shown in
the diagram.
The hne cuts the circle at the points
(4, 3) and (-3, -4).
.". the roots of the given equations are
a; = 4 or —3,
2/ = 3 or —4.
201. Equal and Imaginary Roots.
Solve, (1) a:'^+//'^=18, x-y=0.
(2) a;2+i/2=18, x+?/=6.
(3) a;2+y2^18, a;+?/=8.
y
-
^
""
■~~'
K
/
/
/
\
/
\
K
^
/
X
1
/
/
^
/
/
'^y
k
^
_,
^
-
^
■A
y
~
N
N
yu
n
\
J
/
N
s
''N
<y
/
X
^\«>
_^
^^"^
kI
jJn
<
/
y
Ny
\
/
'
N
\
X'l
/
N
s
X
/
?
^
\
^
/
J
\
V
/c
'>\
/
\
^
%
i. 1
y
/
1 r
^
The diagram shows that the
roots of (1) are
of (2) are
x*=3 or —3,
?/=3 or —3
a;=3 or 3,
y=Z or 3.
The roots of (2) are equal, as
the line a;+?/=6 touches the circle
at the point (3,3). We might say
that in this case the line meets
the circle at two points which happen to be coincident.
The diagram shows that the line a;+y=8 does not meet
the circle at all, and there are no real values of x and y which
will satisfy (3). The roots in this case are imaginary.
Solve these equations by the usual methods and see if the
results agree with the diagram.
U
290 ALGEBRA
EXERCISE 135
1 . On the same sheet draw the graphs of the circles whose equations
are x^+y^=4:, x^+y^=9, x^+7j^=l3, x'^+y^=34:.
2. Solve graphically x^-'ry^=13, x—y=\.
3. Find graphically the positive integral roots of x^+y'^~25
and 2x+3y=18 ; x^+y^=10 and 2x~y=5. Approximate to the
other roots.
4. The sum of two numbers is 8 and the sum of their squares
is 25. Show, graphically, that this is impossible. Is it impossible
if the sum of the numbers is 7 instead of 8 ?
EXERCISE 136 (Review of Chapter XXII)
1. Solve x+?/ = 28, a;2-y2 = 336.
2. Solve 5a;-2y=12, 25a;2-4y2 = 96.
3. The sum of two numbers is 10 and the sum of their squares
is 58. Find the numbers.
4. Solve 2a;-32/ = 4, :r2+(y- = 29.
5 . Solve Sx - 41/ = 4, 2x^ + 3xy=^56.
6. The sum of two numbers is 5 and the sum of their reciprocals
is |. Find the numbers.
7. Solve x^ + xy + 2y^-2x-7y + 5=^0, x+y^3.
8. Solve a;2 + a^- 62/2 = 0, x^ + 3xy-y^ = 36.
9. A field whose length is to its breadth as 3 to 2 contains 664
square rods more than one whose length is to its breadth as 2 to 1.
The difference of their perimeters is 60 rods. Find the dimensions of
each field.
10. Solve x'2+2a;2/=55, x?/ + 2y2 = 33.
11. Solve 2.t;2 + 3xy = 8, y--2xy = 20.
12. The area of a rectangle is 300 square feet. If the length is
decreased by 2 feet and the width by 3 feet, the area would be
216 square feet. Find the dimensions.
13. Solve x(a; + 2/) = 150, 2/(a;+2/) = 75.
14. Solve x{x — y)=l5, 2/(0; + ?/)= 14.
15. Sodding a lawn at 9 cents a square rod costs $108. If it had
been 10 yards longer and 6 yards wider the cost would have been
half as much again. Find the dimensions.
SIMULTANEOUS QUADRATICS 291
16. Solve a;3-2/3= 126, a;2 + a:y+2/2 = 21.
17. Solve x^ + Zxy-by^ + 2x'-y=\2, x+y = 7.
18,* If {x + yy-l{x + y)+l2 = 0 and xhj^-6xy + 8 = 0, find the
values oi x-\-y and cc^/, and thus solve these equations for x and ?/.
19. The product of two numbers is 28 and their difference is 5.
Find the sum of their squares, without finding the numbers.
20. Solve 8x^+1/^ = 280, 2x+y=lO.
21. Solve y = x+V2,x^+y^=l.
22. Find two positive integers whose sum multiplied by the greater
is 192 and whose difference multiplied by the less is 32.
23. Solve ^ + ^,= 10, ^ = 3.
x^ y^ xy
24. If 12^2 — 41a;!/ + 35i/2 = 0, find the values of -•
25. The product of two niunbers is (5 and the difference of their
squares is 5. Find the numbers.
26. Solve ^ + - = 6, a;-2/ = 4.
y- y
27. Solve (x + i/)(a; + 2.v) = 300, - + =^ = 3.
y ^
28. A regiment consisting of 1625 men is formed into two solid
squares, one of which has 15 more men on a side than the other. What
is the number on a side of each ?
29.
1 2
Solve - -j — :
x y
1 4
= 8, - + — = 40.
X- y-
« , 1 1
Solve :
X y
1 4,6 5
~ 12' a;2"'"2/2~ 12
30.
31. The difference of two numbers is 15 and half of their product
equals the cube of the less. Find the numbers.
32. Solve a;2-t-32/2 = 37, a;y= 10.
33. Solve a; + - = 4, w--=3.
34. Two men start to meet each other from towns which are 25
miles apart. One takes 18 minutes longer than the other to walk a
mile and they meet in 5 hours. How fast does each walk ?
u 2
292 ALGEBRA
. „. « , 1,1 1 1,9 1
35. bolve - + - = -, — 2 + ^ = Q*
36. Solve (a; +2/) 2- x- 2/ = 20, a;?/ = 6.
37. The difference of the cubes of two consecutive odd numbers
is 218. Find the numbers.
38. Solve a;*-cc22/2+162/'» = 28, x'^ + -ixy + '^y^=\4:.
39. Solve x2 + y=2/^ + a;=3.
40. Tlie diagonal of a rectangle is d, and the difference of the sides
is 5. What are the lengths of the sides ? Apply the formula thus
obtained to find the sides of a rectangle whose diagonal is 13 inches,
and one side is 7 inches longer than the other.
41. Solve 9a;2 + 2/^-21(3.r + t/)+ 128 = 0, xy = ^. (Make the first
equation a quadratic in 'ix-^y, by adding to Qx'-^y- what is necessary
to make a complete square.)
42. Solve x2 + 42/--18x-362/+ 112 = 0, xy = %.
43. Solve a;3 + 2/^=12(i, a;-*/ + x2/- = 30.
CHAPTER XXIII
INDICES
EXERCISE 137 (Oral)
1. What are the values of 3-, 2\ \*, P", O^ ?
2. Shnplify 3 X 22 ; SxlO^; 5x0^; 0-^^4.
3. When .t=10, what are the values of :
.1-3, 6.r2, 200-^.T, 500-f.T2, 6x^^x- 1
4. Give the values of (-1)2, (-1)3, (-l)*,(-l)-'\ (-!)"«.
5. What are the values of (-2)^, (-2)% (-2)« ?
G. Find the difference between 2^ and 3-, 2^ and 5'^.
7. What does x* mean ? How many factors are there in x"' <x^ ?
8. Express in the simplest form a^xa^xri*.
9. How many factors will remain when x'' is divided by x^ ? What
is the quotient ?
H>. What are the values of: x*^x^, .t^^-^.t^
x^ a^° Tir-' a*b^ r,
x^ o^" ttt a%-
11. What does (a^) 3 mean ? Read its value without the brackets.
12. State the value in the simplest form of :
(^2)2^ (?y3)2, (y3)3, {a^)\ (a2)'".
/ ^'\^
13. What does (oft)^ mean ? What does ( , ) mean ? Read
their values without brackets.
14. Express as powers of 10 : 100, 1000, 10,000, lOx 100, 10^ x 10^,
10=^^103.
298
294 ALGEBRA
15. Simplify (-l)2x (-1)»X (-1)* ; {-ay-x(-a)*x{-a).
16. What is the value of x if
10^=1000, 2^=16, 5^=125, 3^=81 ?
17. Express 32,794 in descending powers of 10.
202. Definitions of «*". As a^ is the product of three factors
each equal to a, so a'" is the product of m factors each equal
to a.
a'^=^a.a.a . . . to m factors.
Here it is understood that m is a positive integer.
203. The Index Laws. We have already seen that :
(1) a3xa^=-a3+^=--a'.
(2) a5-ha2=a5-2^a3
(3) (a2)3^^2X3^a6,
(4) (a6)4=.a454
Let us now express these statements in general form, using
letters to denote the indices.
(1) f«'«X «"=«'""*■".
(2) w"~n"=^a'
(3) {(i>»)»=a'
(4) {aby"=a"'b**K
a'"
ttn-n
iinth
These are called the index laws. The letters m and n
represent any positive integers, and in (2) m>n (m is greater
than n), to make the division possible. The laws, as stated
in the general form, may he proved as in particular cases.
INDICES 295
204. Law I. Law for Multiplication, a'" X«"=a'" + '^
By definition,
a'" = a . a . a . . . to m factors.
a" = a-. a . a . . . to n factors,
o™ X a" = {a . a . a ... to ni factors)(a . a . a . . . to n factors).
= a . a . a . . . to (m + n) factors,
= «"'+", by definition.
Also, a'»xa''xaP=a>"+"xaP,
205. Law II. Law for Division. «."'^a'»=re'«-»
a'" a . a . a . . . to ni factors
a" a . a . a . . . to n factors
=a . a . a ... to (m— m) factors, if ni>n,
= a™~".
Here the n factors in the denominator cancel with an equal number
in the numerator, leaving m — n factors in the numerator.
If, however, n>m, the n factors in the numerator cancel with an
eqiial number in the denominator, leaving n—m factors in the denomi-
nator.
when ?w>n, a'" -i- a" = «"*"",
and when n>m,, a'" -^ a" = — - —
a" "'
206. Law III. Law of Powers. {a'»y^=a''^'\
(a'")"=a'" . o'" . a'" ... to n factors,
= (a . a ... to m factors) (a . a ... to m factors) . . . the
brackets being repeated n times,
=a . a . a . . . to inn factors,
= a""'.
Also, {(o'")»;'' = (a""')''=a'""'\
207. Law IV. Power of a Product. {ab)'^=a»b'K
(ab)" = ab . ah . ah ... to n pairs of factors,
= {a . a . a . . . to n factors) {h . b . h . . . to n factors),
= o"6".
Also, (abc)" = (ab)" . c"=a"b"c'K
296 ALGEBRA
208. Law V. Power of a Quotient. , , , — ,
(f
7- • T . r • • . to n factors,
boo
a . a . a . . . to n factors
6 . 6 . 6 . . . to « factors'
a"
209. We have given five index laws. They are not all
independent. The second and third laws may easily be
deduced from the first.
(1) When m>n, a'" = a'"~"xa" by Law I.
.". a"^-Ha" = a'"~«, which is Law II.
(2) a"* X a'" = a"'+'" = a~"', by Law I.
Similarly, o'" X a™ X a"' = «'"+"«+"' = o^"',
and a"' . a'" . a'" ... to to factors = a'"+"'+"' • • • to n terms — c^mn^
(a'")" = a""', which is Law III.
For this reason the first law is frequently called the
fundamental index law.
EXERCISE 138 (1-18, Oral)
Simplify :
1. ary.a^xn^.
2.
X^X'X^ — X''.
:?.
(,,•2)4^.1-3.
4. (avy->y\
5.
(32)2.
«.
(33)2_^(3>);
7. {ab)^^a-b.
8.
5«^r)^.
9.
((-2r-)3.
6^
10. 6-5-
11.
(-1)'
12.
(ab)^
13. a;" X a;'' X a;". 14. a--^ . a'-' . w" ' >>. 15. x'" - " X .r^"' + ".
16. a;" + ''-f-.T«-''. 17. {a^b^c'^y'. 18. .x" + '' . .r'' + " . a;"^ + «.
19.* fn X (- j X (- j . 20. a;2« + ''xx2'' + <^xa;2« + ''-<'.
INDICES 297
-• ©"• ©■"• ©'"• -• ^^i^:x.--
23, • 24. a;" + ''. a:'' + ''..r'' + " — (;c" . .r''.x<=)2.
25 Express 4" as a power of 2 and 9® as a power of 3.
26. Divide 27^ by 9^ by expressing each as a power of 3.
2"x2"-ix22 9"x3» + s
27. Simplify ^^ and ^y^-n '
28. Solve
^2x~ i^^jc + s . 43-_2'-*"; 93j- + 3__27'*5 . 2-'' . 4-f . 8-^=16^^"^.
210, Fractional, Zero and Negative Indices. We have defined
a"* to mean the product of m factors each equal to a. This
definition requires m to be a positive whole number.
Thus, the definition will tell us what a" means, but will not
tell us what a- means, nor what a~^ means, nor what a°
means.
If we wish to use in algebra such quantities as a^, a~^, a",
it is necessary that we define their meanings.
Now it would be very inconvenient if we gave to these new
forms of indices such meanings that the index laws, already
established for positive integral indices, would not apply to
them. We will, therefore, give to fractional, zero and
negative indices such meanings as will make the index laws
vaUd for them as well as for positive integral indices.
211, Meaning of a Fractional Index.
Since x'" X;K"=a;'"+", then if we suppose that the same
law applies to fractional indices, it follows that
•T^ X X- = X^ + ^ = X^ = X.
Thus, x^ when multiplied by x^ gives the product x, or the
square of .r^ is x.
But we have already represented the quantity whose square
is X by Vx,
298 ALGEBRA
That this is a reasonable value to attach to x'^ might appear as
follows :
We know that x*= Vx^, x^= Vx*, x'^= \' x^, the index of the quantity
under the root sign in each case being half of the index in the preceding
case. If now we take half of the index on each side again, it would
seem but natural that x'^ should be equal to yfx.
Similarly, a;^ X a;^ X x'^ = a;* * ^ = a;.
a;^= Va; (the cube root of x).
Also, x^ = Vx (the fourth root of x),
and x"'= \^x (the ntli root of x), where «, is a positive integer.
Thus, 42 = Vi = 2, 125^ = f'l25 = 5, 32^= v'32 = 2.
By Law III, (x^^ = x^,
• x^=^X^.
Similarly,
(a;0 =xP,
.'. X — v'xi', where p and q are positive integers.
We thus see that if the same laws apply to fractional indices
as to positive integral indices we are led to the conclusion
that .T''=\/a:^, when 'p and q are positive integers, that is,
ichen the index is a fraction, the denominator of the fraction
indicates the root to he taken and the numerator the power.
p / \_\p I
By Law III, a;' = Uv = (x'')'' ,
p
.-. a;9=(v'x)P=^P.
p
So that x^ means that the p^^ power of the q^^ root is to
be taken, or the 7*'' root of the ^*'' power.
Thus, 8^=^82=^64=4,
or 8§ = (\'/8)2 = 22 = 4.
INDICES 299
It will be seen that it is simpler to take the root first and then the
power.
Thus, 32'° = ('v/32)3 = 23=8.
212. Meaning of a Zero Index.
By Law I, a" x a'" = a''+"' = a"*,
a'' = a"'-Ha"'= 1.
Therefore, if the same law appUes to zero indices as to
positive integral indices, we are led to the conclusion that
any quantity {zero excepted) to the index zero is equal to unity.
Thus, 3''=1, (5a;)''=l, (-2)o=l, (-ia6)o=l.
213. Meaning of a Negative Index.
By Law I, a-^xa^ = a-^+^ = a'^=l,
1
Similarly, a-PxaP = a-i'+P = a*^= I,
1
/. a-p = •
aP
We thus see that, any quantity to a negative index is equal
to unity divided by the same quantity to the corresponding positive
index.
Thus, 4- = l=j^.
27S (\y27)2 9
(a;t)-"'==x-« = i-
Since x^—x*-^x, x^=x^^x, x^ = x^^x, what would you naturally
expect x° to be equal to ? What would you expect x" ' to be equal to ?
214. Since a~'' = — and ai^ = , it follows that any
ai' a~P -^
factor may he removed from the numerator to the denominator
of a fraction, or vice versa, by changing the sign of its index.
300
Thus, g,2
ALGEBRA
2 . 62
^' b-" ' y" '
4ct»
4.r-%3 = -^
a;2
Ex.— Simplify ^/S^x^^le^; (y|^) '•
■^82 X \/l63 = 83 X 16+ = (2»)« X {24)i = 22 x 23=32.
/ 9a* \"' 9"*.a-« 16^ 64
Vl66--/ 16"? . 69 9%s69 27a«69
EXERCISE 139 (1-82, Oral)
What is the meaning of:
i 1
1. a-2. 2. x^.
4
5, x^. G. .T°.
9. x~*. 10. ?/-".
What is the vahie of :
3.
1
y-.
4.
al
7.
a-\
8.
a;-i
LI.
.«-*
12.
a;""^,
13. 92.
14.
16^.
15.
125^.
16.
10,0003^
17. 42.
18.
27^
19.
(i)^-
20.
5-2.
21. 10-3.
22.
4-i
23.
^ 1
8 -.
24.
(-6)«.
25. 8"i
20.
(«")--•
27.
(•25)i
28.
(•16)t
29. (J)-2.
30.
(-2)-*.
31.
.'^n . 4".
32.
2^. 2" ®.
Write with positive ind
ices:
33. a2ft~3.
34.
2«-3.
35.
a-2
36.
a6-3
37. -^^ "^
38.
1
x~^y
.39.
2a;- 1
By-^
40.
4-3. a;3
3-2.2/-3
Write without a
denominator :
41 2^^.
42.
4a3
43.
3x
44.
5
* • ^2
a2(c+rf)-2
INDICES 301
Simplify :
_ 3 1
45.* 16 i. 4B. -. 47. (-04)^2. 48. (-027)'^.
32"^
49. 2oi-5. 50. (— 8)~-i 51. f/g^*. 52. I6I".
2-2_2-3 \9gy \l6
56. ^I^Z:^. 57. fi^y. 58. (l^y^
3-1-2-1 V243/ V 816V
59. Solve j;^=4, .ry=32, .r*=27, .r^2=3, .i;"^^=8.
215. Operations with Fractional and Negative Indices. The
following examples will illustrate how the index laws may be
apphed to the multiphcation, division, etc., of quantities
involving fractional and negative indices.
The work will usually be simj)lified if all expressions are
arranged m descendmg or ascending powers of some common
letter.
Thus, Q-\-x^^x~''^—x~'^-\-x'^ would be Avritten in descending
powers of x, thus :
x^-\-x^-\-b—x~'^-\-x~'~.
Ex. 1.— Multiply 2,r2 + 3-x^ by 'ix^-2-5x~^.
Ex. 2. — Di\dde a—b by a^—b^.
(1) (2)
3x^-2 - 5x~^ a~ah^
,ih^i
6x +9a;-- 3 a^lr'
_4x^_ 6+ 2a;" ^ 0^-0^6^
-10-15a;~2 + 5x-i 7~
■b
6a; +5a;-^-19-13x •-' + 5x-i aM-6
302 ALGEBRA
Ex. 3. — Find the square root of 9a;— 12a;i+10— 4a;~2+a;~^.
9x
6a;2-2
12a;^+ 4
1 _i -i
6a;- — 4+a; - 6 — 4a; - + x~^
Q—4x -+a;~^
Verify by squaring 3x'- — 2+a; ^ by the method of art. 93. Also
check by putting x=l.
EXERCISE 140
Multiply :
11 11
1* x- + '3, X-—2. 2. a;+a;2 + l, a;2 — 1.
3. a;2— a;+a;2 — 1, x^-\-l. 4. 3a;— 2a;2_j-5, x—2x^.
5. a*_i + 2«~ia2_^l— 2a-2. 6. (a-a2 + i)2.
7. a;+5a;S+6a;2, a;^— 1— a;~i. 8. (a;24-2)*.
11 11 1 „
9. x-{-x"y--\-y, x—x-y~-\-y. 10. {a- — \Y.
Divide and verify :
11. a + 5a 26 2 -j- 66 by a 2 _j_ 26 2.
12. a;3— a:2+a;— 2— 2x-2— 2a;-3 by a;2+2+2a;-2.
4 2 J2 4 2 1 1 _:i
13. x'^-\-x-^y^-{-y"' by x'''+x'^y'-^'-\-y'^.
14. 1— 5a;'— a; by 1— a;^+3x'^, as far as four terms.
Find the square root and verify :
15. a+6a2-i-9 and 25x-2— 10+a;-2
16. a2+4a2+6a+4a2 + i.
17. 4x^-20a;^+37a;— 30a;*+9a;3.
INDICES 303
18. 49-30a;3-24x""-^+25x3+16a:"i
1 1
19. Show that ^ ^ ,^ - (a^+P)-i.
5.-0 1 _i
20. Divide x-—x - by x~—x -.
21. Divide lOa^'"— 32a'"-27a2m^l4 by 2a'"— 7.
22. Simplify (x+a;^+l)-+(^-a;- + l)2.
23. SimpUfy (V'a+l)(v^-l)-(\/3a+ V2)(V3^-\/2).
24. Find the square root of x^— 4x\/a:+10a;— 12V'.r+9.
216. Contracted Methods. The following examples will
illustrate how contracted methods may be employed.
Ex. 1.— Multiply a;+a;i— 4 by x^x^^4:.
If x + x- be considered as a single term, the product
= (x+x^)2-42, [(a + 6)(a-6) = a2_{,2.j
= a;2 + 2a;* + a;-16.
Ex. 2.— Divide a+6 by a^+fe^.
Since a is the cube of a^ and 6 of b'-^, this is similar to dividing
x^-\-y^ by x + y.
Since {x^ + y^)^{x-\-y) = x''- — xy+y^,
so (a + 6)^(o* + 6^) = a^_aM + 6S.
Ex. 3. — What is the cube root of
This is evidently the cube of a binomial whose first term is 2x* and
last term is ~Zy^.
.'. the cube root is 2x^ — ^y^ , if the given expression is a perfect
cube. Check when x = t/= 1.
Using the method of art. 155, the cube root of more
complicated expressions may be found.
304 ALGEBRA
217. Factors with Fractional or Negative Indices. If we are
permitted to use fractional or negative indices, many ex-
pressions may be factored which Avere previously considered
algebraically prime.
Ex. 1. a — b may be written as the difference of two squares, tliu-s
1 1 i 1
a — 6==(a-+6'-^)(a"^ — 6-).
Ex. 2. ^x—\ — 2x~^ may be factored by cross multiplication.
The factors are {Zx'^'Ix -)(a;- — x -).
Ex. 3. x~ -\- xy -\- y'^ is an incomplete square.
It may be written (x-\-y)^ — {x^y'-)^.
11 II
/. the factors are {x-\-x-y'-' -\-y){x — x''y'' -\-y).
EIXEBCISB 141
Use contracted methods in the following :
1.* Multiply x^—2 by x^+2 ; a-—b'^ by a2-f62-
2. Multiply a2 _ 1 -j-a" 2 by a^-{- 1 +a" 5.
1
3. Find the square of x—x- — \ and of 2a—2—a~K
i 1
4. Write down l;he cube of a^-j-i and of 1— :f-.
1 i JL i
6. Multiply x-{-x~y~-\-y, x—X"y~-\-y, x^—xy-\~y'^.
2 1. L 2
G. Divide x-\-y by x''^—x'^y'^-\-y'K
7. Divide a+2ah^+b-c by a^+b^—c^.
8. Find three factors of x"—y^.
1 1
9. Find a common factor of a->ra^b"—2b, a—h.
i
a;+a;-— 6 a—b a"-\-ab-{-b^
10. Simplify — ^ » -3 j' zr
INDICES 305
11. What is the cube root of .f-— 6.f+12;f'-— 8,
and of a:3-3a;'^+6a:2-7a;2+6a:-3.T^ + l ?
12. What is the square root of 4,f~*+12a;""^-l-9.e~^
and of a;--+4a;+2-4a;-iH-.T-2?
EXERCISE 142 (Review of Chapter XXIII)
1. State the index laws.
2. Explain how meanings are assigned to such quantities as
a^, a», a- 3.
3.* When x= 16, ?/ = 9, find the values of :
(x-\-y)'',x -+y '', (x'^+y-)^.
4. Find the numerical values of :
8^, 9~^ v' 125^^ 32*, 16-1-5, .25--i, (-64)"^.
5. Show that ^Px ^^8^= 108.
6. Simphfy i2~^^{J^)^ and 8i2^^(jVrn^-
7. Find, to two decimals, the value of 10' when x = ^, |, ^, §.
8. Simplify 2^+ 10«-4*-(|r* + 0* + ( Vt^)-!.
9. Find, to three decimals, the value of (3^)"^ X V^27.
10. Simplify 16^ + 16^- 16"^- 16"^ and 32* -32* + 32"'^ + 32"*.
11. Simphfy 5^ x 5* x 5* x 16^'^ x 16^'' x 16'''.
12. Solve x'^ = S, 2^. 4^ = 64.
13. Simplify 4^ x 6"* x ^^3 and (8^ + 4^) x 16"^.
9«+i Q-ii 3 2" — 4 2"~i
14. Simplify ^^ X Y^^ and ^1^;--— —^ .
15. Reduce to lowest terms :
a+SVa+l5 3x^ + 5x^ + 2 a^+ab
a + lVa + \2 ^i^j ' ab-b^
306 ALGEBRA
3 ^ 13 11
16. Multiply x-y' — 2xy-{-4cX-y- by x^-j-zy^.
1 _l _1 i JL _i _i i
17. Multiply x'^y '■^-\-l-\-x -^y- by x'^y -'— 1 + x ^y^.
18. Divide x^ — y-^ by x^-\-x^y~^-\-y~^
2 _ 2 1 _i
and a^+l + a ^ by a^+l+a ^.
19. Divide a*">-b*"' by a"'-6'".
5 •■' i
20. The dividend is y- +22/2_3y— 2, the quotient is 2/'~2/ 1>
1
the remainder is 3y'^—l. Find the divisor.
21. Find the square root of (x^x~^)^ — 4(x — x^^).
22. What is the cube root of la^^ — ^a2^6^ + §a^62^-/y6»» ?
23. If x = a~+l and y = a-^+l, show that ^y^^~y = a^.
xy—x+y
24. Simphfy -008*, 1-728^ 2-251-5, -0625"^.
1 _x
25. If .-r + 2/ = o^ and x—y = a -, fuid the values of xy and x^+y^
in terras of a.
26. If 2a=2^ + 2-'- and 26 = 2^-2--', find a^-b^. '
27. Find the square root of (e*— e"^)2 + 4 and of
r, i .". -.1 15
a;'— 4x^2/^ + lOx^i/— 14x-2/-' + 13a;?/" — 6a; -2/'- + 2/^.
3 1 -4- --1- -1 -\ -8
28. Simplify — j" ^ — ZT' ■
c^ b ^ c
1
29. Factor x- — y, a; — 5x- — 6, a;— 1, 4a— 6^ and
x^-4:X+ 10- 12a;-i + 9a;-2.
30. If 10-3oio3^2, find the value of 10-«o2»'> and lOi'^o^is,
31. If -^'oio^^SO and 72-0593 = 55^ g^d the value of 7*"»6"
9ii+l (9h-1)'(+1
32. Simplify ^^"^ X ^^;^+3
INDICES 307
33. Solve 3-'+ 1 + 2* = 35, 3^+2.«+2 = 4L
34. Divide x-2(x^' — x'h + 2(x'^-x~h—x-'>- by x^ — x~K
35. Show that x'^ ■' = (xV x)' is satisfied by a; = 2|.
36. Find the square root of
i{2Vx)^-2x-' + x+'lx'+Vx-^- ^ .
x2
CHAPTER XXIV
SURDS AND SURD EQUATIONS
218. In Chapter XVIII. we have ah-eady dealt with
elementary quadratic surds. It was there shown by squaring
that y/axVh=Vah.
We might now deduce it from the index laws.
From Law IV (a6)" = a"6".
Letw=^, .". (a6)'-=a-6-',
Va6=\/axV6.
1 11
Similarly, (a6)"=a"6",
219. Orders of Surds. We have already defined a quadratic
surd as one in which the square root is to be taken. A cubic
surd is one in which the cube root is to be taken. When
higher roots are to be taken as the fourth, fifth . . . n^'^, they
are called surds of the fourth, fifth . . . w"' orders.
220. Changing the Order of a Surd. A surd of any order
may be expressed as an equivalent surd of any order which is
a multiple of the given order.
n
Thus, ^/x=x^ = x'^ = x^=x^''\
Similarly, x^ = x^'=x^=x^ ,
308
ST'RDS AND SURD EQUATIONS 309
221. To Compare Surds of Different Orders. Any two surds
may be rednced to surds of the same order and their vahies
compared.
Thus, to compare the vahies of \^2 and f'^3,
\ 2 = 2^ = 2*^= V'2'=V^8.
f 3 = 3^ = 3^= V^=\/9.
It is thus seen that \ 3 is greater than A 2.
222. Changes in the Form of Surds. Any mixed surd can
be expressed as an entire surd.
Thus, 2 \V5= ^23 X ^5= i^40.
av'6= v^a" X Vb= \^a^.
V m + n A ^ ' m -\- n
Conversely, \'''250 = ^^l25 x ^2 = 5-^2.
^/^ib^=^/W'x y/a = b^-\/a.
-^^8T=f/^^27x^3= -3^3.
EXERCISE 143
Express as mixed surds :
1. Vfl, VlOOa, VsP, VSo^fe, \/32^, VSGSa^P.
3. v'32, 'v/243; \/^^, \/6i, v'8.T2+16a:2/+8i/2
Express as entire surds :
4. 2\/3, 10\/2. 3\^a, 0^5, ahx^b, {a-b)Va^.
310 ALGEBRA
7.* Reduce ^^2, VS to surds of the same order. Also reduce
^2 and \/3 ; V2, ^3 and \/5.
8. Which is the greater:
3\/2or2\/3; sVGorlVS; \/5 or \KlO ; 1-26 or ^2; ^3 or V^5 ?
Reduce to like surds and simplify :
9. VS + VlS+VdS. 10. \/500+\/80-V'20.
11. 3\/32+5\/50-i\/l28. 12. -^T6-\^/i28+\^''250.
13. -^96-2^^=^+^324. 14. v'32+\/l62+\/i260.
15. \/75-3\/l2+5\/300+2V'48-7\/l47+3VI.
16. xVx-\-y-\-Vx^-{-x'^i/—V{xA-y)^—V{x^—y^){x—y).
17. Express as equivalent surds of a lower order :
i/9, \/l25, ^/x^, \/i6xh/% v^32.
18. If ^2=1-26 approximately, find the values of :
\'/r6, -^54, •^2060, ^KJ, ^^002, 1^6^.
19. Show that 2 x V2 x V2 x v^4=4.
223. Rationalizing a Surd Denominator. When the numerical
value of a fraction with a surd denominator is required,
the value is more easily obtained when the denominator is
rational (art. 165).
When the denominator contains only two terms, it may be
rationalized by multiplying by its conjugate (art. 164).
EXERCISE 144
Multiply :
1. 2VS, 3V5. 2. V2ax, V3ax. 3. Vx, Vxy.
4. ^4, -^5. 5. 6VI4, JV2I. 6. ^^, .n''^.
7. </a^b, ^a+b, i''a^l)^.
8. Vx+2, Vx—S, Vx—2, Vx+3. '
SURDS AND SURD EQUATIONS 311
9. \/2+V3-a/5, \/2+V3+\/5.
10. ^'a— 1, ^a— 2, ^a+3.
11. \/6-\/lT, \'6+\/II.
12. (\/r8+Vl2+V8)2.
What is the simplest quantity by which the following must be
multiplied, to produce rational products ? What is the product in
each case ?
13. 3V2. 14. 2V'5. 15. \/32. 16. V64.
17. V5T2. 18. V2. 19. V^^^. 20. \/^.
21. 3-\/2. 22. Va+b. 23. 3\/2-2\/3. 24. aVb-bVa.
Rationalize the denominator of :
26. Vh 27. Vrs.
29. (v/S+VS)-!. 30.
25.*
., I0V2
V5
28.
4+2\/2
2+2\/2
31.
a+6— c
V'a2_^62_ft
_. 32. .
Va+b+Vc Vx+y+Vx—y
33. Find, to three decimal places, the value of
22^(3\/2-a/7)(2\/2+\/7) and of (5J-\/7)^(3 + \/7).
34. If X = and w = -J^ , find the value of x~+y^.
2+V3 2-V3 ^
35. Simplify (2\/3-\/2)3-(\/3-\/2)3.
36. Simplify ^^^ + i±^ and ^±^ + ^t^^? .
3+\/5 4-\/5 I + VS 2+A/5
37. SimpUfv^^2V5_lo+6V5.
4-V'5 2+\/5
38. Show that 3 — V? is a root of the equation .r^— Sx^— 4a;+2=0.
312 ALGEBRA
39. The three dimensions of a room are equal. If the longest
diagonal from the ceiling to the floor is 18 feet, find the length of the
room to the nearest inch.
224. Surd Equations. When an equation contains a single
quadratic surd, and the equation is written with the surd
alone on one side, the surd may be removed by squaring
both sides of the equation (art. 166).
If the equation contains three terms, two or three of which
are surds, the operation of squaring must be performed twice.
Ex. 1.— Solve l-f Vx=Va;+25.
Squaring, l-\-2Vx-\-x=x-\-25,
:. 2\/^ = 24,
Vx=U,
x= 144.
Verification 1 + Vx= l + -\/l44=13.
V'x+25=VT69=13.
Solve by squaring in the form Va;= Vx + 25— 1 and in the form
l = Vx-\-25—Vx, and compare the three solutions.
Ex. 2.— Solve l — Vx=Vx-\-25.
Squaring, 1 — 2'v/x + .T=.r+25,
.-. -2\/x = 24,
a;=144.
Compare, line by line, this solution with Ex. 1. The answer is the
same to both, although the equations are different. We have verified
Ex. 1, and we know that x= 144 is the correct result.
Let us now verify Ex. 2.
Verification: 1— -v/i=l— •\/l44= — 11.
\/x+25-:VT69=13.
It is seen that our attempt at verification shows that .t= 144 is not
the correct root of the equation in Ex. 2.
If in verifying we could say that Vl44 is —12, the equation would
be satisfied. But this is not allowable, as the symbol V always
represents the positive square root (art. fi3).
SURDS AND SURD EQUATIONS 313
This may be explained as follows :
(1) The equation 1 — Vx=v.x+25 is impossible of solution,
as Va;-f-25 is a positive quantity, and therefore l — Vx must
be positive, that is, x must be less than 1. But it is evident
that no value of x which is less than 1 can satisfy the equation.
(2) If we square both sides of an equation, the resulting
equation is not necessarily equivalent to the given equation.
A simple example will show that this is the case.
Let .r = — 6.
Squaring, .". x'^—36.
Now the equation x" = 36 has two roots +6 and —6, and is,
therefore, not equivalent to the given equation.
This is similar to the case in which both sides of an equation
are multiplied by a factor containing the unknown.
3),
The equation a;^ — 5x+6 = 0, which has the roots 2 and 3, is not
eqioivalent to the given equation.
225. Extraneous Roots. Roots which are introduced into
an equation by squaring or multipljdng are called extraneous
roots.
Thus, x=6 in the first ecjuation and x=3 in the second are
extraneous.
Refer to Ex. 4, art. 145, where reference is made to the effect
of dividing both sides of an equation by a factor containing the
unknown.
We have already seen the necessity of verifying the results
in the solution of equations. In the case of surd equations
there is an added reason for verifying, for although there may
be no error in the work, the root which is found may not be a
root of the given equation.
Let
x = 2.
Multiply by x—3, .'
x{x-S) = 2(
. x^-5x+6 = 0.
314 ALGEBRA
EXERCISE 145
Solve and verify 1-15. Reject extraneous roots :
1. v2.r-5-3=0. 2. \/3a:-2=2V'a:-2.
3. 3xi=x^+4:. 4. ^5a:-7=2.
5. 2x^=3. 6. 2f'3x-25-f-3 = 7.
7. 2(a;-7)*'=(a:-14)5. 8. Vx+Vx+5=5.
9. Vx+45+V^=9. 10. 1 + ^^+2=^^.
11. V:r-t4+Va;+15=ll. 12. v/4x-2+3x-16 = 2a:+2.
X- 1_ 7\/x+10
Vx^ ' a/4x-2
^ x-1 ^^ lVx+\0 ^
13. Va; + 5=;7-==- 14. ,- ^ =4.
15. V^-3^Vg+l. 16.* Vx+4-V^4=^4.
Vx'+3 Vi-2
17. (12+a;)^+a;2=6. 18. (x+8)^-(x--f3)5=2.ri
,^ Vx+16 a/x+29 „„ 6\/x-ll_2\/x+l
19. -— = -— • -^u. — •
^/x+4: Vx+n 3Vx Vx+6
, ,- , 36
21. \/a;2-a2+62^x-a+6. 22. \/x+Va;-9= -/===•
23. Va;-15+Vx=-7==-. 24. V'a:-f3+\/x+8=2\/x.
Vx— 15
25. V5+4^-Vx-2Vfe+i. 26. ^a;3-6x2+llx-5=a;-2
1 1 OS Vx+6+V^_o
27. 5(70x+29)^'=9(14a:-15)^-. -8- ^^^^ZV^ZIe ~
29. _5x-l_^j^V5^ 30 V^4V^^^
SURDS AND SURD EQUATIONS
226. Surd Equations Reducing to Quadratics.
Ex. 1.— Solve :i-+\/:r+5=7.
315
Transposing,
Squaring,
Verification
Vx+5 = 7 — cc.
x + 5 =49-14.r+a;2
a;2-15x + 44 = 0,
(a;-4)(a;-ll) = 0,
x = 4: or 11.
When x = i,x+ Vx^5 = 4 + V9 = 7,
When x= 11, a;+V'x+5=ll+ VT6=15,
the correct root is re =4.
a;=ll is evidently a root of x— Vx + 5 = '7'
Ex. 2.— Solve V8x+ 1 — Va;+ 1 = VSx.
Transposing, .-. V8x+l = V3x+Vx+l.
Squaring, .-. 8x+l = 3x+2Vdx^ + Sx + x+l,
4:X = 2V3x^ + 3x,
2x= V3x^ + 3x,
4:x^ = 3x^ + 3x,
x2-3a;=0,
x=0 or 3.
Here we find on verifying that both roots satisfy the given equation.
Ex. 3.— Solve 2V2x-\-l = 3-SVx^.
Solve as in the preceding and the roots are x=4 or 364, neither of
which satisfies the equation.
Of what equation is x = 4 a root ? Of what equation is a; =364 a
root ?
Ex. 4.— Solve x^-3x-QVx^-SxS = -2.
If the surd is removed to one side, we get
a;2-3a;+2 = 6V'^^-3x-3.
If we now square both sides to remove the surd, we will obtain an
equation of the fourth degree which we cannot easily solve.
316 ALGEBRA
We may obtain the solution by changing tho unknown from x to
V.T^— 3x— 3, similar to the method employed in art. 192.
Let
\/.r2-3x-3 = ?/,
x'~Sx—S = y-,
x- — 3x = y- + '6.
in the original equation,
?y2 + 3-62/=-2,
,/2^6^-f5 = 0.
y = 5 or 1.
-3a;- 3 = 25,
or .T-- 3a;- 3=1,
-3x-28 = 0,
x^-3x-4: = 0.
7 or -4.
a; = 4 or —1.
We, therefore, have foiu* roots : 7,-4, 4, —1.
Verify each of these and show that they all satisfy the given equation.
Here both values of y were positive ; if either of them had been
negative it could at once be discarded as impossible.
Ex. 5.— Solve x-\-y—Vx^y^20, (1)
xy—2Vxy=l20. (2)
From (1), Vx+y = 5 or —4.
From (2), Vxy= 12 or - 10.
Here the negative values of the surds are discarded,
.-. Vx+y = 5, Vxy=l2.
x-\-y = 25, xy=l44.
Solving these, x = Q or 16, 2/= 16 or 9.
EXBRCISE3 146
Solve and verify 1-17. Reject extraneous roots :
1. a:-f-V'a;=20. 2. a;— \/a;=20.
3. V3x—5+Vx—2=3. 4. VSx-5-Vx-2=3.
5. 3a; +\/5x2+l 1+5=0. 6. 3.r+5=\/5a;2+ll.
7. —;^ = 5 — 2Vx. 8. 3a;-2V'7.T+4=15.
SURDS AND SURD EQUATIONS 317
\/x+16 \/4— x- T) .,
\/4-.i- \/a;+16 2 ^ ^
11. \/x+a+\/x+6=\/a-6. 12. V'2a;+5- V'a;-1=2.
13. 4(x--+a;+3p=3(2.x---+5.c-2)i
14. 3(.r+\/2-a;2)=z4(a;-\/2-a;2).
^' \/3a;2 + 4 + \/2x2 +1^7" ' V 2 V 2
17. 2\/a;+3\/^=12, 18.* u;(/-\/x7/=30,
3V'x+2\/^==13. x-+y=13.
19. a:+\/^+2/=28, 20. ;c+y+A/^+^=30,
x— V .i-;/+?/=12. X-— (z+a/x— 2/=12.
21. a;2+x?/+2/2==91, 22. x-2+.r2/+2/2=5i,
a;+V'x^+i/=13. x— \/a;«/+2/=l2-
23. a:2-3x-+6-\/a:2-3x-+6=2.
24. x^— .r— Vx^— a;— 6=36.
y A/o; 2
227. Square Root of a Binomial Surd.
(V'3 + V2)-=3+2+2v6 =5+2a/6.
(V5-V3)2=5 + 3-2Vl5 = 8-2\/l5.
(3-\/2)2=9 + 2-6V2 =11-6^2
{Va-Vb)^=a-\-b-2Vab.
The square of \/a-\-Vb is made up of a rational quantity
a-\-b, which is the sum of the quantities under the root signs,
and a surd qiiantity 2Vab, the ab being the product of the
quantities under the root signs.
318 ALGEBRA
The form of the square of va-l-vb will show us how we
can sometimes find, by inspection, the square root of a
binomial surd.
>/a^b—2Vab = Va—V0.
Ex. 1. — Find the square root of 7+2'v/l2.
Here we want two factors of 12, whose sum is 7. They are evidently
4 and 3.
7 + 2\/r2 = 4 + 3 + 2V473.
V'7 + 2V'l2=V'4+v'3 = 2+\/3.
Similarly, V7-2V'l2 = 2- V3.
Verify by squaring 2+ Vs and 2— VS.
Ex. 2. — Find the square root of 14— GVs
To put this into the form a + 6 — 2^06, first change 6a/5 into 2^45,
14-6V5=14-2V'45 = 9 + 5-2V'45,
.-. \/l4-6\/5=\/9- V5 = 3- V5.
EXERCISE 147 (1-7. Oral)
Find the square root and verify :
1. 5+2\/6. 2. 8-2\/l2. 3. 4-!-2\/3.
4. 6-2 Vs. 5. 10+2 v/24. 6. 15+2\/56.
7 8+21/7. 8. 7-4\/3. 9. 9+4\/5. •
10. x+y^2Vxy. 11. x+y—2Vxi/. 12. 2x+2Vx^—y^
13. 15-4\/i4. 14. IS-SVS. 15. 20+V'300.
1(>. IO+a/64. 17. 47 + a/360. 18. 57-18\/2.
19. a— 2\/a— 1. 20. 4:X -{- 2 V 4^x^—1. 21. rt— 6— 2\/a— 6— I
SURDS AND SURD EQUATIONS 319
Ex. 1. — Find the square root of 56— 24\/5.
56-24^/5 = 56-2^720.
Here we require two factors of 720 whose sum is 56. Wlien tlie
numbers are large, as here, it may be difficult to obtain the factors
by inspection.
When this is the case we may represent the factors by a and b and
find the values of a and b from the equations
a6=720,
a + b = 56.
Solve these equations by the method of art. 19-i or of art. 198 and
obtain
o=36 or 20,
6 = 20 or 36.
The required factors of 720 then are 36 and 20.
.-. 56-24\/5 = 56-2V'720 = 36+20-2\/36. 20
.-. Vse- 24V5= V36- \/20 = 6-2V'5.
Verify by squaring.
Ex. 2. — Find the square root of f + Vs.
9 /- 9 + 4^5 9+2\/20
4+^^ = — T— = 4
the square root is ^ or — — (- 1.
Ex. 3.— Find the square root of 2\/lO+6V2.
First take out the surd factor V2, and we get
2V\0+QV2= ^2(6 + 2^5),
the square root =V2(l+\/5).
EXERCISE 148
Find the square root and verify :
1. 94--42V5. 2. 38 + 12Vr6. S. 47-12\/T5.
4. 107-24\/l5. 5. 94+6\/245. 6. 10I-28\/T3.
7. 67+7^72. 8. 28-5^12. 9. xy+2yVxy^i^-.
320 ALGEBRA
10,* Find the value of l-^Vl6— 6V7 to 3 decimal places.
4 2\/3
11 . Find the value, to three decimals, of the sq uare root of ^ .
7+4V3
12. By first removing a simple surd factor, find the square roots of :
7\/2+4\/6, lO+eVs, 7\/3-12, 59\/2+60.
13. Show that V 17 + 12^2+^/17-12^2=- 6, (1) by taking the
square roots, (2) by squaring.
14. Simplify V 3+Vl2+V49+8V3.
,^- 4-2\/3 ,_
15. By changing 2— V 3 into x , find the square root of 2— V 3,
also of # + V2 and of V +^V7.
16. From the result of Ex. 1, show that 94— 42\/5 is a positive
quantity less than unity.
17. If .r2(14-6V'5)-=21-8\/5, find x to three decimals.
18. The sides of a right-angled triangle are Vs and 3+2^2.
Find the hypotenuse
228. Imaginary Surds. When we solve the equation x^=Q,
we obtain .t= ± 3, and we know that this is the correct result,
for the square of either +3 or —3 is 9.
If we solve x^=5, we say that the value of x is± V5, and
we can approximate to the values of the roots as closely as
we wish by finding the square root of 5 by the formal method.
If we are asked to solve x'^——9, we might say that
the solution is impossible, as there is no number whose
square is —9. This statement is correct, but we find it
convenient to say
if .t2=-9,
then rr = ± V^^.
SURDS AND SURD EQUATIONS 321
Such a quantity as V— 9 is called an imaginary quantity,
and must be distinguished from such quantities as 5, — |,
^V?, etc., which are called real quantities.
We may define an imaginary quantity as one whose square
is negative, or as the square root of a negative quantity.
We have already seen how imaginary quantities sometimes
appear in the solution of quadratic equations (art. 190).
We will assume that the fundamental laws of algebra,
which we have applied in using real numbers, apply also to
imaginary numbers.
Thus, V^^=V'9x V^ = 3V^^.
\/^25-f- \/^=4 = 5 V^ + 2 V^ = 7 V^.
These examples show that an imaginary quantity can
always be expressed as the product of a real quantit}'' and the
imaginary quantity V— 1. The quantity V— 1 is sometimes
called the imaginary unit.
229. Powers of the Imaginary Unit. Any even power of
V— 1 is real, and any odd power is imaginary.
Thiis, [V^l)^= - 1, by definition.
.-. (V^^)* = (-l)2= + l,
.-. (V^^)5 = (\/^^)*x\/^= + V^, etc.
230. Multiplication and Division of Imaginaries.
Ex.1. V^x V^=V2 . V^^x V3 . V^n^,
= \/2. v'3x(\/^)2=-V6.
Note that the product here is - V 0, not VG.
322 ALGEBRA
Ex. 2. 3\/^4x5V^^ = 6\/^^xl5V'^I,
= 90(\/^)-=-90.
EX.3. ^-y-H>l^.^«.v/9=3.
V-2 \/2xV-l V2
Ex. 4. (a; + 2/V^n)'' = x2 + 2/'(v/^T)2 + 2a;2/V^^,
= x2 — 2/2+2a;yV-l.
Ex. 5. {a + bV'^)(a-bV^^) = a^-b^(\/^^l)~=a^+b'-
3
Ex. 6. Rationalize the denominator of jz==^ •
1 — V — 2
Miiltiply both terms by 1 + V - 2 and we get
3(1+ V-2) _ _ 3(1+ v;^) _ 1^ v^.
(l_V-2){l + V-2) l-(-2)
EXERCISE! 149 (1-9, Oral)
1. Express as a multiple of V—~l : V— 4, V— 16, V— 81, V— a^
\/=^625, V=9^*, \/-(^"6)2.
2. What is the value of (V^l)\ {V^^)\ (V^^)\ (V-f)^ ?
Find the sum of :
3. V^, V^, V^. 4. V'^25, V-100, V^^iO;
5. 44- V^, 2-\/ — 16. 6. a+V-ft-, a-6\/^.
What is the product of :
7, V^, V^. 8. \/^25, V-IOO. 9. \/^^, V^^^.
Simplify :
10.* 3\/-3 + 2\/^75-4\/-12+5a/^^.
11. (3 + 5V^)(3-5\/^) + (5-3V'^)(5+3\/^).
12. (4-3\/^)H(2+6V^)2.
13. 2 + (1-a/^). 14. (-1 + V^3) + (-1-a/-3).
15. {a+bV~iy-+(a-bV^^y-.
IG. Show that -j(-l + V^)2=i(-l- V^).
SURDS AND SURD EQUATIONS 323
17. By finding the cube of hi — l + V—S), show that this quantity
is a cube root of unity, (art. 192, Ex. 5.)
18. Are 2± \/^, the roots of x^—4:X+7=0 ?
19. If a=2-\-SV^-i and b=2—3V,^, show that a+6, ab and
a"^+6^ are real quantities.
231. Impossible Problems. We obtained the imaginary
number V—9 in answer to the question, " What is the
number whose square is — 9 ? " As we have said, this is
arithmetically an impossible problem.
When we obtain an imaginary result in solving a problem we
may conclude that the problem is impossible.
Ex. 1. — The sum of a number and its reciprocal is 1|-.
Find the number.
Let a;=the number .". - = its reciprocal,
X
2x2 -3x -1-2 = 0,
3±\/9^n[6 3±V^1
.'. X = ;, = z •
4 4
Here the roots are imaginary, and we conclude that there is no
number which answers tlie condition of the problem.
In fact, it may be shown that the sum of a positive number and its
reciprocal is never less than 2.
Change H into 2h and solve the problem.
Ex. 2.— For S30 I can buy x yards of cloth at $(10— x')
per yard. Find x.
The total cost in dollars =x(lO — x) = '.iO.
.-. a;2-10a;+30=0,
10+ V^^^
What conclusion do you draw ? Would it be impossible if for $30
we substitute $25 ? $20 ?
y2
324 ALGEBRA
SXERCISE 150
Solve and determine if these problems are possible :
1. A line which is 10 inches long is divided into two parts so that
the area of the rectangle contained by the parts is 40 square inches.
Find the lengths of the parts.
2. The length of a rectangle is twice its width. If the length be
increased 10 feet and the width decreased 1 foot, the area is doubled.
Find the dimensions. Solve also when the width is increased 1 foot.
3. A man has 20 miles to walk. If he walks at x miles per hour
it will take him 8— a; hours. At what rate does he walk ?
4. If it is possible that x{\2—x) — ZQ^a, and a is not negative,
what must the value of a be ?
BXBRCISB 151 (Review of Chapter XXIV)
1.* Multiply l + \/3-V2by \/6- v/2.
2. Multiply V3+ \'2 by 4- + ^ •
V3 V2
3. Find the square roots of : 14+ \/T80, 25 — 4V2l, 22+ V'420,
11-V72, 12-6V'3.
4. Find to two decimal places the values of :
1 / — 7+^3 3A/8+V20
Vf^Z V5 + 2V6, 2v'3 + V8' V5+V2 "
Solve and verify :
5. x+\/x^=\\. 6. V'4a; + 7+\/4x + 3 = 6.
7. Vx + Vx^l = ,^-^-- 8. V6x + 7-V'.r+2=V2\/x+T.
Va;— 4
35
9. Vx+s + Vx-ie =
Vx+S
, , 24
10. V19 + a;— V15 — a; = — .T^T-
Vi9+a;
SURDS AND SUED EQUATIONS 325
11. Multiply V'2+V'3+V5 by v/5+a/3-V'2 and
x + y+2Vx + y by x + y-2Vx + y.
12. When x=2+VTi, y=2- Vs, find the value of
2x+y x-2y _
x-y x+y
13. Expand and simplify {V2a + Vh+V2a-Vb)^. Check the
result by substituting a= 13, 6= 100.
14. Solve p-\-x—V2px-{-x^=q.
15. Find the product of 2 \/3 + 3\/2+ VSO and V2+V'3-a/5.
16. Find the continued product of ^
a;- 1+^2, x-l-V-I, x+l+Vs and x+l-VS.
17. Simphfy 7;^--— ;^ 7=-— +
Ve+Vs Vs+i ^3+^2
18. When x = a-\- \/a-—\, find the values of :
x + - ' a;2 + — , x- H — :.•
X x^ ic-*
19. Express in the simplest form :
V'27- \/8+ ^17 + I2V2- ^28-6-^/3
and \/ll + 6v'2+\/l9-4Vl2fV'5-V'2i.
on Q- r* V12 + 6V3 , fm+n /^T^n
20. Simplify r= and ^/ h \/ — ;
VS+l ^ m— /!. V m+n
21. If x=-l + 2V'^, find the value of a;" -12a;.
22. Find the square roots of 7-}- VlS and 2a-\- \/4a- — 4.
23. Solve 2x2 + 6x - 6 - Vx'^ + 3a; - 3 = 45.
24. Simplify (\/.^+ \/3+ v/2)2 + (\/5+ \/3- \/2)2
+ (\/5 - v/3+v/2)2-f-(\/5- V'3-\/2)2.
25. Solve 3x2 — 9x+ ll = 4\/x2 — 3a; + 5, giving the roots to two
decimal places.
Va-\-b + Va — b Va + 6 — y/a — b
26. Simphfy
Va+b—Va—b Va+b+Va— 6
326 ALGEBRA
1
Show that — 7^===^_- +
-\/l6+2V63 Vie- 2^63
28. Show that va+ Vb cannot be expressed in the form Vx+Vy
unless a'^ — b is a perfect square.
29. S,n.p..fy f?'-^')' + [1=^]
30. Simplify {3-2V2)^ + {S + 2V2f.
.31. Find the value of x^-\-x^+x+l when .^•=V3^-l.
CHAPTER XXV
THEORY OF QUADRATICS
232. Sum and Product of the Roots.
Solve these equations :
(1) a;2— lLr+10=0. The roots are 10, 1.
(2) 2x2- 3^_ 5^0. „ „ „ I, -1.
(3) 15.T2+26.r+ 8=0. ,, „ „ -|, —i.
In (1) the sum of the roots = 11, product = 10.
In (2) „ „ „ „ „ = I, „ =-|.
In (3) „ „ „ „ „ =-f|, „ = T%.
Examine the sum and the product in each case and state
how they compare with the coefficients in the given equations.
Every quadratic equation may be rediiced to the form
a.r2-(-6.T+c=0.
— 6+\/62— 4ac , —h—Vb^—^ac
2a 2a
For brevity represent these roots by m and n,
-b4-Vb^-^ac-b-Vb^-4:ac -26
2a 2a
b
a
(-6+V62-4ac)(-6-V62-4ac)
id mn-y ^2
(_6)2_(V62i:4ac)2 b^-b^-\-4ac 4ac
c
4a^ 4a^ 4a^
327
328 ALGEBRA
Comparing these results with the coefficients a, b, c in the
equation, we see that :
The sum of the roots of any quadratic equation, in the standard
form, is equal to the coefficient of x with its sign changed, divided
by the coefficient of x^, and the product of the roots is equal to the
absolute term, divided by the coefficient of x^.
coefficient of x
Sum of the roots
Product of the roots
coefficient of x^
absolute term
coefficient of x^
See if these two statements apply to the roots and co-
efficients of the three equations preceding.
The formulae for the sum and product of the roots furnish a
convenient means of verifying the roots.
Thus, I find the roots of 3x^ + x — 2 = 0 to be |, —1, but the sum of
I and — 1 is — J and the product is — |, which agree with the sum and
product given by the formulae. Therefore, these are the correct
roots.
Are the roots of Ux-- iar-60 = 0, V", -* ?
233. Reciprocal Roots. If the roots of ax'^-\-bx-\-c=0 are
reciprocals (hke § and :f ), their product is unity, and therefore
c ,
- =1 or c=a.
a
So that any quadratic equation, in which the coefficient of
x^ is equal to the absolute term, ivill have reciprocal roots.
Thus, the roots of 6a;^— 13a;+6 = 0 are reciprocals, since their product
is I or I. Verify this by finding the roots.
234. Roots equal in Magnitude but opposite in Sign. If the
roots of ax'^-\-bx-\-c=0 are equal in magnitude but opposite
in sign (like 3 and —3), their sum will be zero, therefore
= 0 or 6=0.
a
So that any quadratic equation in which the second term is
missing will have roots equal in magnitude but opposite in sign.
Thus, 20;" — 9 = 0 and ax^ — c = 0 have such roots.
Verify by finding the roots.
THEORY OF QUADRATICS 329
BXBRCISE; 152 (Oral)
State the sum and product of the roots of :
1. a:2— 7x+12 = 0. 2. x^-ox— 11=0.
3. x24-6x-fl=0. 4, 2a;^-10x+6=0.
5. 3x^—12x-l=0. 6. 4x2-17x+4=0.
7. ax^—bx—c=0. 8. o.r'^— (6+c)a;4-cf=0.
9. pa;2— 9=0. 10. ax'^+a=0.
11. 3x2— 4a;=6. 12. (a+b)x^-'X+a^--b-^=0.
13. Which of the preceding equations have reciprocal roots ?
Which have roots equal in magnitude but opposite in sign ?
14. Are 4 and 5 the roots of x^— 9x+20=0 ?
15. Are3+V'2, 3— \/2 the rootsof x2— 6.r-L7=0?
In which of the following are the correct roots given :
16. a;2— 7a-+10=0; 5, 2. 17. .i-+3.r— 28=0 ; 7, -4.
18. x2-13a;+36=0; 4, 9. 19. x2-12a;+27=0 ; 4, 8.
20. a;2— 4a;— 5=0 ; 5, 1. 21. 2x2— 5a;+2=0 ; 2, J.
22. In solving x2—2x— 1.599=0, one root is foimd to be 41. What
must the other be ?
23. How would you show that 1-12.5 and 2- 168 are the correct
roots of x2— 3-293x+2-439=0 ?
24. If the roots of Gx^— 10x-)-a=0 are reciprocals, what is the value
of a?
25. If the roots of mx^—{7n-—9)x-{-m^=0 are equal in magnitude
but opposite in sign, what is the value of to ? What would then be
the product of the roots ?
235. To form a Quadratic with given Roots. First Method.
In the equation x^-\-px-\-q=0, the sum of the roots is —p,
and the product is q. Since every quadratic equation may be
reduced to the form x^-\-px-\-q=0, by dividing by the co-
efficient of x^, any quadratic equation may be written thus :
x^—x (sum of roots) + (product of roots) =0.
330 ALGEBRA
If the roots are given, the equation can at once be written
down.
Thus, tne equation whose roots are 3 and 5 is a;* — .t(3-|-5) + 3 . 5 = 0,
or x2 — 8a;+15 = 0.
The equation whose roots are 2+ \/3 and 2— V 3 is
a;2-a;(2+\/3 + 2-V'3) + (2+V'3)(2-\/3) = 0, or a;2-4a;+l = 0.
The equation whose roots are a-\-b and a — b is x^ — 2ax + a^ — b^ = 0.
Second Method. The equation whose roots are p and q
is {x—p){x—q)=0.
The equation whose roots are 3 and 5 is {x—S){x — 5) = 0, or
a;2-8x+15 = 0.
The equation whose roots are j and — f is (x— |)(aj+i) = 0> or
(3a;-2)(4a;+3) = 0.
The equation whose roots are 2+^3, 2— V3 is
{x-2-V3){x-2+ \/3) = 0, or (a;-2)2-3 = 0 or x^-4:X+l = 0.
Either method is simple enough to apply, but the first is
probably easier when the given roots are not simple numbers.
The second method may be applied to form an equation
with any number of given roots.
Thus, the equation whose roots are 2,3, — 5 is
{x-2){x-S)(x + 5) = 0, or .T»-- 19a; +30 = 0.
EXERCISE 153 (1 16, Oral)
State, without simplifying, the equations whose roots are :
1. 3, 7. 2. 3, —7. 3. —3, 7. 4. —3, —7.
K 1 1 <! 1 1 7 1 1 8 2 3
9. a, a. 10. — rt, —6. 11. 3, 0. 12. 0, m.
13. 3,4,5. 14. 2,3,-1. 15. a, b, c. 16. rr, 6, 0.
Reduce to the simplest form the equations whose roots are :
17* m+n,m—n. 18. 2a—b,2a+b. 19. S + VS, 3— \/5
20. If, -21 21. -2, -4, 6. 22. |, ^, J.
23. Show that 1-25 and 4-64 are the correct roots of
100a;2-589.T+580=0.
THEORY OF QUADRATICS 331
24. Construct an equation in which the sum of the roots is 7 and the
difference of their squares is 14.
25. Form the equation whose roots are a and h where a^-\-h'^=25,
2G. Form the equations whose roots are
a-\-h a—b ,_
-S., --,; i(4±\/7).
a—b a-\-b -^ '
27. Find the sum and the product of the roots of :
(1) (a;-2)2=5.r-3. (2) {x-a){x-b)=ab.
(3) x*x-p)^p{x-q). (4) {x+ay'+{x+by'^{x+c)^
28. Solve .T*— 21.t-— 20.i-=0, being given that one root is 5.
29. If one root of x'^—12x-\-a=0 is double the other, find the roots
and the value of a.
30. If one root of x^-\-px+4:8=0 is three times the other, what are
the values oi pt
236. Functions of the Roots. When m and n are the roots
of ax^-\-hx-\rC=^,
h c
m-\-n = , mn = - .
a a
Here it will be seen, that the sum and the product of the
roots do not contain surd expressions, while the separate
roots do.
If we wish to find the sum of the squares of the roots, we
can do so in the following way :
m^-^-n'^ = {m-{-7i)'^—2mn,
a a
- ^J _ 2^ =: ^^-^^
a^ a a-
It can also be found by taking the square of each root
and adding the results Find it that way and see if you get
the same result.
332 ALGEBRA
Ex. 1. — When m and n are the roots of ax^-\-bx-\-c=^0,
1 1 7/}' Ti
find the vahies of 1 — , — | , m'^-\-n^, m—n.
m n n m
11 7n-\-n ft . c b
m n mn a ' a c
62_ 2c
m n _ 7n2-)-n* (7n-\-n)- — 2mn_a^ a b"—2ac
n m mn mn c ~ ac
,,,,,,„„,,, b^ 36c Sabc—b^
m^-\-n^ = {m-\-nY — 6mn{m-\- n) = ; + — „- = — , — >
or m^ + n^ = ('m-\-7i)(m,^ — mn~\-n^) = (m-{-7i)\(m-\-n)^ — Smn} = etc.
/ \2 , , \9 A b^ 4c b^~4ac
(m—n)^ = {m,-\-ny — 4'mn = — „ = 5 — »
Vb^'-iac
m — n = + .
a
The same two values of the last expression might have been found
by simple subtraction, the sign depending on the order in which the
roots were taken.
Ex. 2. — If m and n are the roots of x^-\-px-\-q=0, find the
equation whose roots are m^ and n^.
Here m-\-n= —p and mn = q.
The sviin of the roots of the required equation is
m2 + n2 = (m + n)-— '2mn=p~ — 2q.
The product of the roots = w-n^ = 7-.
.". the required equation is x~ — x(p--~2q)-\-q- = Q.
Ex. 3. — Find the equation whose roots are each greater by
2 than the roots of 6a:2— 13a;— 8=0.
Solve the given equation and the roots are f , — \.
.'. the roots of the required equation are V, %.
.'. the required equation is
x^-{\t-\-%)x + h* . 1=0, or 6.-r2-37.r + 42 = 0.
We might have solved the problem without finding the actual roots
of the given equation.
THEORY OF QUADRATICS 333
Let p and q be the roots of 6a;*— 13a; — 8 = 0.
Then P + <l = ^i and M=— I-
.*. the sum of the roots of the required equation is
p + 2 + fy + 2=p + 5 + 4 = -V- + 4 = -«/,
and the product
= (p + 2)(g+2)=M+2(p + g) + 4=-A + V + 4 = 7,
.'. the required equation is
.t2-=Vx + '7 = 0, or 6a;2- 37a; +42 = 0.
When would the second method be simpler than the first ?
Ex. 4. — Find the equation whose roots are the reciprocals
of the roots of mx'^-\-nx-\-k=^().
Let p and q be tlie roots of the given equation,
then P + <1= and pq= —'
m m
The roots of the reqviired equation are - and — •
Find the sum and product of and - in terms of m, n and k and
p q
complete the solution.
Compare the new equation with the given one and see if you could
not write down, mentally, the equation whose roots are the reciprocals
of the roots of any given equation.
237. The following method will be found useful in solving
such problems as the three preceding.
Ex. 1. — Find the equation whose roots are each greater
by 5 than the roots of 4^2— 5a;+7=0.
Let y be the unknown in the required equation.
Then y — x-^5 or x = y — o.
Substitute x = y — 5 in the given equation, and the required
equation is 4(y — o)^ — 5{y — 5)-\-7 = 0,
or l-y*-40i/+ 100-52/4-25 + 7 = 0,
or 42/2-4%+ 132 = 0.
334 ALGEBRA
Ex. 2. — Find the equation whose roots are the squares of
the roots of ax^~\-bx-\-c—0.
Let y be the unknown in the required equation.
Then y = x^ or x= + Vy,
.". the required equation is a{±Vyy-{-b{±Vy)+c = 0,
or ay->rC= +bVy,
or a^y^-'i-2acy-{-c^ = b^y,
or a^^/2 + ?/(2ac — 62) + c- = 0.
Solve Ex.'s 2, 3, 4 preceding, by this method.
BXERCISE 154
1.* If tn and n are the roots of X'—5x-\-'S—0, find the values of
I . I m 11 , , , o
— |- -, — I — , nr-\-'mn->rn-.
m n n m
2. Find the sum of the squares of the roots of
a;2-7.r+l=0 and of 3a;2-4a;+5=0.
3. If p and q are the roots of Sx'^-\-2x—6=0, find the values of
- + -, 2 + -T' p^-pq+q^-
q p p^ q^
4. Find the sum of the cubes of the roots of
2x-2— 3a;+4=0 and of x^— x+a=0.
5. Find the equation whose roots are double the roots of
a;^— 9a:+20=0, (1) by solving, (2) without solving.
6. Find the equations whose roots are each less by 3 than the
roots of (1) a;2-lla;+28=0, (2) x^-x-l=0.
7. Find the equations whose roots are the reciprocals of the roots
of (1) 2x2+a;-6-=0, (2) x^-px+q=0.
8. If m and n are the roots of 3,*'-— 2.r+5=0, find the equations
whose roots are :
(1) — and -, (2) - and — , (3) m- and n^.
in n n m
THEORY OF QUADRATICS 335
9. Find the sum of the squares and the sum of the cubes of the
roots of x'^^ax-\-h—0.
10. Find the equation whose roots are the squares of the roots
of x--\-px—q=0.
11. Find the equation whose roots are each greater by h than tlie
roots of ax'^-f-6a;+c=0.
12. Find the equation whose roots are the reciprocals of the roots
of x'^-\-x=\.
13. If m and n are the roots of x^—px-\-q—0, show that m-[-n
and mn are the roots of x^—x{p^q)^pq=0.
14. Form the equation whose roots are m and n, where
7n--|-w-=20, TO+W.= — 6.
15. If m and n are the roots of x'^-\-px-\-q=0, show that wi+2?i
and 2m +71 are the roots of x^-\-Zpx-\-2p^-\-q^:^Q.
16. If p and q are the roots of ax'^-\-hx-\-c=0, Hnd the vahie of
p*-\-p'^q'^-\-q^ in terms of a, b and c.
238. Character of the Roots of a Quadratic Equation.
Solve the equations :
(1) x^—6x-[- 9=0, the roots are 3, 3.
(2) 6a;2+ a;- 15=0, „ „ „ |, -|.
(3) 5x^+lx- 2=0,
(4) 2x^—Sx-\- 2=0,
)J 5>
-7±V89
10
3±V^
4
In (1), the roots are equal. VVe might say that there is
only one root, but we prefer to say that there are two roots,
which in this case happen to be eqiial.
In (3), the roots are irrational, but we can approximate
to their values by taking the square root of 89.
In (4), the roots are also irrational, but we can not even
approximate to their values. Here the roots are imaginary,
while in each of the others the roots are real.
336 ALGEBRA
These statements might be written thus :
In (1), the roots are equal, real and rational.
In (2), the roots are unequal, real and rational.
In (3), the roots are irrational and real.
In (4), the roots are irrational and imaginary.
If we examine the roots of the general quadratic equation
we will see the reason why, under particular conditions, there
is this difference in the character of the roots.
The roots of ax'^-^bx-\-c=0 are
— 6+\/62— 4ac , — 6— \/62— 4ac
and
2a 2a
From these roots we may conclude :
( 1 ) If the particular values of a, b, c are such that 6^— 4ac=0,
then the roots are equal, for each is evidently equal to — ^ •
In equation (1), «= 1, b= —6, c = 9.
.-. b2-4oc = 36-36 = 0.
(2) If b'^—4:ac is a perfect square, then its square root can
be found exactly and the roots are rational.
In equation (2), a=6, 6=1, c= — 15.
.-. 62- 4ac= 1 + 360 = 361 = 192.
(3) If b^—4ac is not a perfect square, but is positive, the
roots are real but irrational.
Find the value of 6- — 4cfc in equation (3).
(4) If b'^—4:ac is negative, the roots are imaginary.
Find the value of b^-iac in equation (4).
Hence, the roots of ax2+bx-|-c=0 are real and equal if
b^ — 4ac— 0, real and unequal if b^ — 4ac is positive, imaginary
if b^— 4ac is negative, real and rational if b^— 4ac is a perfect
square.
THEORY OF QUADRATICS 337
239. The Discriminant. We see then, that we can deter-
mine the character of the roots of a quadratic equation
without actually finding the roots. All we require to do is to
find the value of &-— 4ac.
This important quantity is called the discriminant of the
equation ax'^-\-hx-\-c=0.
Ex. 1. — Determine the character of the roots of :
(1) 3a:2+5a:-ll=0. (2) 12a:2-25.r+12=0.
(3) x^- X-+ 3=0. (4) 2,1-2- 16^+32=0.
The value of the discr
in(l)is 157,
in (2) is 49,
in (3) is -11,
in (4) is 0,
minant (6- — 4ac)
•. the roots are real and irrational,
■. the roots are real and rational,
". the roots are imaginary,
•. the roots are real and equal.
Ex. 2. — For what values of k will 4.t2— A;a;-{-4=0 have
equal roots ?
The roots will be equal if 6^ — 4ac = 0, :
that is, if A,-2-64 = 0 or if A-= ±8.
Substitute these values for k and see if the roots are equal.
Ex. 3. — Show that the roots are rational of
3mx2— a;(2m+3w)+2»i=0.
Here 6^ — 4ctc = (2m+3«)^ — 24mn,
= 4m^ — 1 2m>t + 9/1^ = ( 2^?i — 3ji)-.
Since b^—^ac is a square, the roots are rational.
Verify by finding the roots.
EXERCISES 165 (1-5. Oral)
1. What is the discriminant of x-"--f4x+4=0 ? What is tho
character of the roots ?
2. What is the nature of the roots of x^-\-^x-\-2—Q ?
3. What is peculiar about the roots if b^—4:ac=0 ?
4. What kind of roots have .r-— 5a; +7=0, :c- — 6.r 4-9=0,
a;2-.x--6=0, x^—4.x—Q=0 ?
5. If the discriminant is -25, what is the character of the roots ?
Z
338 ALGEBRA
Determine the character of the roots of :
6.* 2:r-+5a;+3=0. 7, 3x'—lx—5=0.
8. 4x2+7.T+15=0. 9. 9a:2— 12.r+4=0.
10. abx'-+x(a^+b'^)+ab=0. 11. x^—mx—\^0.
12. Show that x^-^-ax+b^O has real roots for all negative values
of i.
13. If 9x^-\-12x-\-k—0 has equal roots, find k.
14. If «a;-—10a;+ff=0 has equal roots, find a.
15. Show that the roots of x'^—x{l-\-k)-\-k=0 are rational for all
values of k.
16. If .T-+2a:(l+«)+«"=0 has equal roots, find a.
17. By solving the equation x" — 4:X-\-5=k, show that if x is real,
k cannot be less than 1.
18. Show that the roots of — | \ = 0 are real if
X x-\-a x^b
a^—ab-\-b^ is positive.
19. Eliminate y from the equations y=mx+c and?/^=4ax, and find
the value of c if the resulting equation in x has equal roots.
20. If 2mx^ -\-{5m-\-2)x-'r(im-'rl)=0 has equal roots, find thevalnos
ot m and verify.
240. Factors of a Quadratic Expression.
When m and 7i are tb.e roots of ax^-\-bx-{'C=0,
h c
a a
,\ ax^-\-hx-\-c=a(x"-\ — x -\ — )
\ a a/
=a{x^—{m-{'n)x-\-mn]
=a{x—'m){x — n).
So that, if m and ri are the roots of ax^-{'bx-{-c=0, the
factors of the quadratic expression
ax^-\- bx-\-c are a(x—m){x—7i).
THEORY OF QUADRATICS W.V.)
We can, therefore, find the factors of a trinomial like
ax'^-\-hx-\-c by solving the corresponding equation.
Ex. 1.— Factor 6.i--+.r-40.
Solving by formula, we find the roots of
6x2 + a;- 40 = 0 are |, — f.
.-. Ga;2+a;-40 = 6(x-fi)(a; + f)
= (2x-5)(3a;+8).
Ex. 2.— Factor 12.r2_47a;+40.
The roots of the corresponding equation are ^, f,
.-. 12a;2-47a; + 40=12(x-|)(,r-.:;)
= (4a;-5)(3a;-S).
241. Character of the Factors of a Trinomial. Since
a.r-+6.r+c=0 has equal roots when & "^—400=0, it follows that
ax^+bx+c has equal factors, or is a square, tvhen b^— 4ac=0.
Thus, in Sx^-SOx+To, 62 _4cjc = 900 -900 = 0.
.'. 3x^—30x+75 is a perfect square when the numerical factor 3
is removed.
// b^—4ac is a jJerfed square, the expression ax~'\-bx-\-c has
two rational factors, for under this condition the corresponding
equation has rational roots.
Thus, in 20x2-a;-12, 62-4ac = 961=312.
20x'^ — a;— 12 has rational factors. Find the factors.
242. Surd Factors of a Trinomial. When we say that a
trinomial can be factored, we usually mean that it can be
expressed as the product of rational factors.
As we have seen, this can always be done when h'^—4:ac
is a perfect s([uare.
When there are no rational factors we may use the pre-
ceding method to find surd factors.
Ex. — Find two surd factors of a:-— 6.r4-4.
If a;2-6a;+4 = 0, x = -^.^ "^ = 3± V5.
• x^-6x+i = (x-S-Vo){x-3-i-\/5).
Verify by multiplication.
Z2
340 ALGEBRA
EXERCISE 156
Factor, by trial if you can, otherwise by solving the corresponding
equations and verify :
1. 3a;2-17,r+10. 2. 20a;2 + 3x--108.
3. x2-2x--1763. 4. 1800a2-5a-l.
5. 299.t;- + 10x-l. 6. 221a;2-458aa;+221a2.
7. Show that 12.c-— 15x+4 has no rational factors.
8.* x^-\-4:X—'3 has no rational factors. Find two surd factors
of it.
9. If x^—8x-\-k is a perfect square, find k.
10. If ax'"—kx-'r9a is a perfect square, when the factor a is removed,
find L
11. Express .r-— 6.f— 11 as the product of two surd factors.
12. Factor 144.?;*— 337.c-.^''+144//*. When this expression is
equal to zero, find four values of the ratio of x to y.
13. If .r— 2 is a factor of 120.^3— 167.1--— o,;--|- 56, find the value of
a and find the other two factors.
14. By finding the square root of ax'^^bx-\-c, find the relation which
must connect a, b and c when this expression is a perfect square.
243. A Quadratic Equation cannot have more than two
Roots. We have .seen that the e(|uation ax'^-\-bx-\-c—'d has
two roots, and since this equation represents every quadratic,
it follows that every quadratic equation has two roots.
It cannot have more than two roots.
Let m and n be the roots of ax'^+hx-\-c=0.
Then ax^^hx^c=a{x—m){x—n) (art. 240)
«(.}•— rn)(.x—w)=0.
Since this product is zero, one factor must be zero. But
a is not zero, for the equation would }iot then be a quadratic.
Therefore, eitJier
•X'— m=0 or .T— 71=0.
THEORY OF QUADRATICS 341
But no values of x other than in and // will nuikc cither of
these quantities equal to zero.
m and ti are the only roots.
Since the quadratic equation aa:'-+6.T+c=0 has only two
roots, then the quadratic expression ax'^+bx + c can he resolved
into linear factors in only one way.
EJXBRCISB 157 (Review of Chapter XXV)
1. What is the sum and the pi-oduct of the roots of a.r' + 6a; + c = 0 ?
2. Under what condition are the roots of ax"-\-hx^c = {) reciprocals ?
When are they equal in magnitude but opposite in sign ?
3. When are the roots of oa;2-i-6x + c = 0, (1) equal, (2) real,
(3) imaginary, (4) rational ?
4.* Ifj)-r(/ = 4 and pq = o, find the values of
5. Find the sum and the product of the roots of
(3.T-2)(.T-3) = (.T-l)(.-r-o).
6. Find the equation whose roots are each one-lialf of the roots
of 4.r2-20.T^21 = 0.
7. Find the sum of the squares of the roots of S.r^— lla;+ 1 = 0.
8. Find the equation whose roots are twice as great as tlie roots
of 24x2-38x+15 = ().
9. For what value of k w ill .r-— ]Ox = k have equal roots ?
10. Find the equation whose roots are m and n wlian OT* + n^=74
and 7Hn = 35.
11. Factor 52o6.T2 + ,r 1 and 221a;2-8a;- 165.
12. Form the equation whose roots are w and n where m^-f-n' = 28
and 7M + n = 4.
13. Find the siun of the roots of {x — a)^ + (x — b)- = (x ~ c)-.
14. Construct the equation whose roots are the reciprocals of the
roots of 17a;- + 53x — 97 = 0.
15. Express x^-{-6z+7 as the product of two linear factors.
342 ALGEBRA
16. Construct the equation whose roots are each greater by 7 than
the roots of 2a;2+lla;-21 =0.
17. Find the equation whose roots are each three times the roots
oi ax^ + bx-\-c = 0.
18. If ?/i and ■», are the roots of ax'^-\-bx-\-c = 0, find the equation
whose roots are — and — •
n m
19. Show that {a-{-b + c)x'^—2x{a + b)-\-a-{-b — c = 0 has rational
roots. What are they ?
20. If one root of x"^ —px-^q = 0 is double of the other, show that
21. If m and n are the roots of x^-{-px-\-q = Q, show that p and q
are the roots of x--\-x{m-\-n — m)i) — inn{7n'\-n) = 0.
22. Show that the equation ym.x-\ — -j ='kax has equal roots for
ail values of m.
23. Find the values of k for which the equation
x^ + x{2^k) + k-ir^l = 0
has equal roots.
24. Since a;^ — Sx— 20 = (a;— 10)(a; + 2), for what values of x is the
expression a;^ — 8a;— 20 equal to zero ? For what values is it negative ?
For what values is it positive ?
25. Show that it is impossible to divide a line 6 inches in length
into two parts such that the area of the rectangle contained by them
may be 10 square inches.
26. For what values of k is 4x-2 — a;(/i;+8)4-A; + 5 a perfect square?
Verify your result.
27. Find the sum of the cubes of the roots of x~-{-m,x-Yn = 0.
28. Find the sum of the squares of the roots of
a;2^a;(l+a) + |(l+a + a-) = 0.
29. Find the sum and the product of the roots of
O' . b _ c
x — a X — b X ~ c
30. If the sum of the roots of ax^ — Qx-\- 12a = 0 equals their product,
find a and verify.
31. It is evident that x = a is one root of {x--c){x — b) = {a — c){a — b).
Find the other root.
THEORY OF QUADRATICS 343
32. If X--- ox— 3a and x^— lla; + 3a have a common factor, it must
be a factor of their difference. Make use of this to find the value of a
for which x~ — 5x—Za = 0 and a;^— llx+3a = 0 will have a common
root. Verify by finding the roots.
33. The absolute term in an equation of tlie form a;^+pa; + g = 0 is
misprinted 18 instead of 8. A student in consequence finds the roots
to be 3 and 6. What were the roots meant to be ?
34. If m and n are the roots of aa;2 + ^-'c + c = 0, show that m-\-n
and 1 — are the roots of acx^-\-bx{a-\-c)'\-b" = ().
m n
35. Two boys attempt to solve a quadratic equation. After
reducing it to the form x^-{-px-\-q = 0, one of them has a mistake only
in the absolute term and finds the roots to be 1 and 7. The other has
a mistake only in the coetficient of x, and finds the roots to be — 1
and —12. What were the correct roots ?
36. Express a;- + 26.T + c as the product of two linear factors in x.
CHAPTER XXVI
SUPPLEMENTARY CHAPTER
Additional Examples in Factoring
244. Product of two Trinomials. If we multiply
a— 2b— 3c by 2a—b-^c
the product may be written
The first three terms of the product, which do not contain
the letter c, are evidently the product of a— 2b and 2a— 6.
The last term, — Sc^, is the product of —3c and c.
If we wish to factor the product of two trinomials, we may
do so by the method of cross multiplication, which we used to
factor a trinomial.
Ex. 1.— Factor 2a^--5ab-^2b^-5ac^bc~3c^.
First factor 2a^ — 5ab + 2b^, and then choose such factors of —3c''
as will give the remaining terms in the product when the complete
multiplication is performed :
2o2-5a?) + 262_5ac + 6c-3c2
a -2b -3c
2a - b + c.
If the terms of the factors are written under the terms from which
they are obtained, it is not difficiilt to obtain by trial the factors of
an expression of this type.
Ex. 2.— Factor 4a2+362_i2c2-8a&-8ac.
Arrange the expression thus :
4a 2 - 8a& + 36 2 - 8ac - 1 2^^
2a - b + 2c
2a -36 - 6c
Show by multiplication that these factors are correct.
344
SUPPLEMENTARY CHAPTER 345
EXERCISE 158
Write, mentally, the products of :
1.
a— 26+c
a— b—c
2.
5.
8.
3x-^+l
Zx^y-2
a-6+4
2a- 6
3ra-2«, + l
2m— 3?i+4
3.
6.
9.
3a-46+ c
2a— 6— 2c
4.
3.r+ y-4t
2x-3«/+3
2a -36 -5c
2a +36
2x—oy-\- z
3x—2y—3z
7.
a- -2a +3
a2+3a-2
Factor and verify :
10. a--4a6+462-a+26-12.
11. 2x'^+xy-Qy^+2xz+Uyz—izK
12. 2a2+662-3c2+76c-5ca-7a6.
13. 2x^-lxy-22y^-5x+35y-3.
14. 6a2+a6-1262-2a+316- 20.
15. x^—xz—6z^—2xy-\-Gyz.
16.* Divide the product of Ga^- 5a6+62+lla— 46+3 and
a+6-2 by 3a2 + 2a6— 6^— 5a+36-2.
17. Reduce to lowest terms
4:p^+2lq'^—lSr^+33qr+&rp-31pq
4:p^—7pq+3q^--2pr-\-3qr-6r^
18. If 3x-\-2y—5z is a factor of
3x'^-\-axy—Qy- |-6,r2+c</2— lOs-,
what are the values of a, 6 and c ?
19. Write the expression x'^-\-xy—2y^—x-\-\0y—\2 in the form
a;2+a;(?/— 1)— (2?/2— 10(/+12). Solve the corresponding equation for
x and thus find the factors of the given expression.
20. Solve .r2-5ax + 6a-+7.T^17a + 12=0, (1) by factoring, (2) by
the general formula.
346 ALGEBRA
'21. Express, iii the factor form, the L.C.M. of
6«,- — 5ab-\-b'^-{-ac—c'^
and 6f/--|-a6 — 26^— ac+46c — 2c2.
245. Sum and Difference of Cubes. We have seen that
a^ + b^ = (a + h){a"-ab + b^), a^-b^ = (a-b)(a^ + ab + b"),
{a + b)^ = a^ + Sa-b + 3ab" + b^, (a-b)3 = a^--3a'^b + 3ab^-b^.
Similarly, {a + b)^ + c^ = {a + b + c){(a + b)^-c{a + b) + c^\,
and (a-6)3-c3 = (o-6-c)l(a-6)- + c(a-6)+c2}.
Ex. 1.— Factor a^^b^^c^—Sabc.
Add to a^-{-b^ svifficient to make the sum the cube of a-\-b, that is
add 3a26 + 3a62.
Then a^ + b^ + c^-3abc,
= a^ + b^ + Sa~b + Zab^ + c^-3aV)-Sab''"-3abc,
= {a + b)^ + c^-3ab{a + b + c),
= {a + b + c)\{a + b)^--c{j-\-b) + c--3ab\,
= {a-\-b-\-c){a"-\-b^-\-c^ — ab'-bc~ca).
The factors of this expression are important, and the pupil
should endeavour to retain them in memory.
The expression is the sum of the cubes of three quantities
diminished by three tim,es their product.
One factor is the sum of the three quantities, and the other is
the sum of their squares diminished by the sum of their products
taken two at a time.
We should recognize expressions which are of the same
form as this type expression.
Thus, a^ + fe^ — c* + 3a6c may be written in the form
a3 + 63_|_(_c)3-3o6(-c),
and it is now seen to be the sum of the cubes of a, b and — c, diminished
by three times their product.
The factors of n^-\-b^ — c^-\-Sabc may at once be written down from
the factors of the type form by merely substituting — c for c.
.-. a3 + 63-c3 + 3a6c = (a + 6-c)(a2 + 62 + c2-a6 + 6c + ca).
a3_53_c3_3„;,c = a» + (-6)' + (-c)S-3a(-6)(-c),
= {a-b-c)(a^+b'^+c^ + ab-bc+ca).
8x^ + 27y^-z^+lSxyz = {2x)^ + {3y)^ + {-zy-3(2x)(3y)i-z),
= (2x + 3y-z){4-x'^ + dy^ + z^-6xy + 3yz + 2zx).
SUPPLEMENTARY CHAPTER 347
Ex. 2.— Factor a-^+h^-\-\ — oah.
a3-i-63-|_i_3a6==rt3 4,ft3_i_i3_3„(f, . 1.
= {a + b+\){a-^b--\-\-ab — a~h).
Ex. 3. — Find one factor of
(•^•+2/)'+(^+2)'+(^+•^')'-3{.^;+y)(^+:;)(^+.T).
This is of the same form as a^ + 6* — c' — 3a6c, where a = x-{-y,b = y-\-z,
c=z-\-x.
One factor is x-\-y-ry-r^ + z-rX or 2{x-~y-\-z).
The other factor is lengthy, but is easily written down.
Ex. 4.— If a+6+c=0, show that a^+b^+c^=3abc.
This is equivalent to showing that a^-\-b^-\-c^ — 3abc = 0.
Now this quantity will be equal to zero, if one of its factors is zero.
But a + 6 + c is already seen to be a factor, and since it is given equal
to zero,
• „3_L?,3a-c3-3a6c = 0. ov a^ + b^^c^ = 3abc.
We have thus shown that if the sum of three quantities is
zero, the sum of their cubes is equal to three times their product.
Prove this also by substituting ~b — c for a.
Ex. 5.— Show that
(a-6)3+(6-c)3+(c-a)3=3(a-6)(&-c)(c-a).
Here the sum of a — h, b — c, c~a is zero, and therefore the
result follows at once from the preceding theorem.
Similarly, (a-L2fo-3c)» + (6 + 2c-3a)3 + (c + 2a-36)3
= 3(a + 26-3c)(6 + 2c-3a)(c + 2a-36),
since the sum of a + 26 — 3c, 6 + 2c — 3a, c-f 2a— 36 is zero.
SXBRCISE 159
Factor :
1. (a + 26)^-c3. 2. a^~{b-cf.
3. (a+6)3+8c3. 4. {a+bf+{c+df.
5. (.r-2/)3-(a-6)3. 6. {2x-y)^+(x-2tj)^.
7. (3a-6)3-(a-36)3. 8. 8(3a-6)3-27(2a-3fty'
9. a3— 6'+c3+3a6c. 10, 8x^+y^+z^-6xyz.
348 ALGEBRA
11. 0.3+6^^ l + 3r/6. 12. l+c-3-r/3 + 3crf.
13.* 8.t-'-/y-'-125z3-30a:y3. 14. {a+bf+c^+l-3c{a+b).
What is the product of :
15. a—b—c and a^~\'b--{-c''-\-ab-\-ac—hc.
16. 2x—y+Sz and 4:X^+y^+9z^+2xy—6.xz+3yz.
17. l—a—bandl+n-+b^-\-a+b—ab.
18. 2fl-36-4 and 4r/-- + 6a64-962_l26+8a+16.
What is the quotient of :
19. l-a^+b'^+Sab by l-a+6.
20. 21m^—n^—l—Qmn by 3m— %— 1.
21. a3+12563-l + l5ff6 by a2+2562+l— 5a6+a+56.
What is one factor of :
22. (4« + 36)3-(rt+26)3.
23. (.t2-3x+7)3+8.
24. (a2_3ff+2)3-(a2-5a+7)3.
25. (a+6)3+(c+rf)3-H-3(«+i)(c+(^).
26. Prove that the difference of the cubes of ia^-\-a+l and
2rt2— 2a4-3 is divisible by the product of 2a— 1 and a + 2.
27. Show that a^-\-b^+c^—3abc is equal to
i(rt+6+c){(a-6)2+(6-c)2+(c-a)2|.
28. Write down a quantity of the type o^ + b^-\-c^—3abc, of which
3x—2y-\-z is a factor. What is the other factor ?
29. If a+b—c=0, show that a^+b^+3abc=c^.
30. If x=y+z, show that x^=y^+z^+3xyz
31. If a + b+c=0, show that
(a + 2b)^ + (b ^ 2cP + (^ + 2a)3=3(« + 26)(6+2r)(c+f^,i).
32. Show that
(x-yr + {y-zf-^(z--xf-3{x-y){y-z){z-x)=0.
33. Show that {a + 3b-4:cf + (b+3c-4a)^+(c+3a-ib)^
= 3(a+36-4c)(6+3c-4fl.)(c-t-3a-46).
SUFPLEMEXTABY CHAPTER 349
34. If x=a — b, y=a+b, z=2a, show that x^+y^+3xyz=z^.
35. Find the Vcalue of a^-b'-^+c^+dabc when a=-32, 6 = -46, c=-14.
36. Reduce to lowest terms :
2a2_5a6^362+ac-3c2 {x-2yf+{2y-z)^+{z-x)^
37. Find two factors of the first degree of
{ax+by+azf+(bx+ay+bzf.
38. When x=b+c, y=c+a, z=a+b, prove that
^.sj_y3j^z^_^^yz=2{a^+b^+c^-3abc).
39. Prove that a^+b-+c-—ab—ac—bc is unaltered if a, b, c be each
increased, or each decreased by the same quantity.
40. Solve (,r-o)3 + (6-.r)3-i-(f,_6)3=o.
246. Grouping Terms. We have already seen (art. 91)
that we can frequently, obtain a factor of an expression by a
suitable arrangement of the terras.
The following examples will give further illustrations of
this method.
Ex. 1.— Factor a^b-c)'\-b'\c—a)^c-{a—b).
Arrange in descending powers of a, and the expression
= a-{b-c)-a(b^-c-) + bc(b-c),
= {b — c)(a^ — ab — ac-irbc),
= (6 — c) \a{a — 6) — c{a — b)\ ,
= (b — c){a — b)(a — c).
When the factors are written in cyclic order (art. 140),
aHb-c) + b'-(c-a)+cHa-b)=-{a-b)(b-c)(c-a).
This expression may also be factored by writing it in the equivalent
form (a2-62)(6-c)-(a-6)(62-c2).
In this form a — b and b — c are seen to be factors. Complete the
factoring by this method.
The expression a{b'^ — c");-b(c- a-) -'r c(a- ~ b-) differs only in sign
from a^b — c)+b"[c a)4-c-(a — 6),
.-. a(b^—c-) + b{c~-a^) \ c{a^-b^) = {a-b){b-c)(c-a).
Also, ab{a — 6) + bc(b — c) + ca{c — a) = — (a — 6 )(6 — c)(c — o).
350 ALGEBRA
Ex. 2.— Factor a^b—c) + b^{c—a)^c^{a—h).
The expression =a^{b — c) — a{b^ — c^) + bc(b- — C'),
= (b — c){a^ — ab- — abc — ac^ + b''c-{-bc-).
Now arrange the second factor in jDowers of b, and proceed as before
and obtain ~{a — b){b — c)(c — a)(a + b + c).
Factor also by using the second method of Ex. 1, writing the
expression in tlie form (a^ — b^)(b — c) — {a — b){b^ — c^).
What are the factors of a{b^-c^) + b{c^-a^) + c{a^-b^), and of
ab(a"'-b^) + bc(b^-c^) + ca(c--a') ?
Ex. 3.— Factor a^b + c)^b~{c-^a)+c-{a^b)+2abc.
Arrange in descending powers of a, and the expression
= a^{b + c)+a{b''' + 2bc + c^) + bc(b + c),
= {b+c){a^+ab + ac + bc),
= {b+c){a+b)(a + c) = {a + b)(b + c){c+a).
Ex. 4.— Factor (a--b^)x^-+{a"+b-):v+ab.
Expressions of this kind, when written in descending powers of x,
are easily factored by cross multiplication in the usual way.
{a^-b")x^ + {a^ + b^)x + ab
(a +b )x -\-a
(a —b )x -f-6
The factors are {a + b)x + a and {a — b)x-{-b.
EXERCISE 160
Factor and verify 1-8 :
1, acx^-]-x{ad-\-bc)+bd.
2. mpx^-\-xy{qm—2^n) — nqy'^.
i. x-{a^-b-)+iabx-iu--b-).
4. {p~-q-)!r + 2!j{p- + q-)+2J'-r-
5. x%a'{-b)+x{a+2b+c) + b+c.
G« x-{a^—a)+xi2a~—3a+2)+a-—2a.
7. a^b-\-c)+a{b~+3bc+c^)+bc{b-^c).
8. ab{a-\-b)+bc{b-}-c)-{-ca{c-\-a)-\-3abc.
SUPPLEMENTARY CHAPTER 351
9.* x^{ij-z) + if{z-x)^z^x-y).
10. xy{x—y)+yz{y—z)+zx{z-x).
11. x{y^—z-')+y{z^~-x'-)+z{x'~-y%
12. a{h^-c^) + h{c'^-a^)+c{a^-h^).
13. a2(64— c4)+62(c«-a*)+c2(a*-fc4).
14. Divide
a»(6-c)+63(c-fl,)+c='(rt-6) by a2(^_c)4_ft2(c_«)^c-(a-6).
Solve and verify :
15. ahx'^—x{ad-\-hc)-{-cd=Q.
16. {a-—b-)x^—4:abx—a-—b'^.
17. .r2(a-&) + a2(6_.r)+62(,i;_„)=0.
18. «6x-2— a;(a'-+62)_j_a2_^,2^0.
19. {a--ah)x^-ir{a-+h-)x=ah+h\
20. Find a common factor of
ahx--\-x{a-—2ah—h-)—u--{-h^ and a"x-—a^x—ab—b^.
247. The Factor Theorem. We have already seen that
any expression is divisible by x—a, if the expression vanishes
when we substitute a for x (art. 101).
Any expression whose value depends on the value of x is
called a function of x (art. 114).
Any function of x may be conveniently represented by the
symbol /(a;), which is read " function a:."
The factor theorem might be stated thus :
/(■r) is divisible by x—a if /'{a)=0.
Thus, if f{x) = x^-7x-+llx-2,
/(2) = 8-28+22-2 = 0
.". x* — 7x-+ llx — 2 is divisible by x—2.
If f{x) = x^-4x^a + 5xa^--\-\0a^,
then /(-a)=-a='-4a'-,5a3+10a3 = 0,
x^ — 4x'^a-\-5xa^ + l0a^ is divisible by x-{-a.
352 ALGEBRA
248. Factors of x"±a**. We have already seen that
x^ — a^ = {x — a){x-\-a),
x^ — a^ = (x — a){x^-\-xa + a-),
x*-a*={x-^-a^){x^ + a-) = (x-'a)(x+a)(x^+a^).
Here we see that x—a is a factor of each.
Is x — a a, factor of x^ — a^ ?
When we substitute a for x,
x^ — a^ = a^ — a-' — Q,
x — a is a factor of x^ — a^.
( 1 ) Is a; — a a factor of x" — a" ?
When we substitute a for x,
X" — a" = a« — a" = 0-
x" — a" is divisible by x — a.
(2) Is .r + a a factor of x" — a" ?
When we substitute —a for x,
a;"— a" = ( — a)"— a".
Now { — a)" — a" will be equal to zero only when { — a)"— a", and this
is true only when n is even,
x" — a" is divisible by a;4-« when n is even.
Thus, rc^ — a^, a;* — a*, a;^ — a®, etc., are divisible by x-{-a, but x^ — a^,
x^ — a^, etc., are not divisible by x + a.
(3) Is x + a a factor of x" + a" ?
Examine this, as in the preceding, and show it is a factor only when
n is odd.
(4) Is a; — a a factor of x"-\-a" ?
We thus conclude that, when n is a positive integer,
(1) x"— a" is always divisible by x—a.
(2) x°— a" is divisible 6?/ x+a when n is even.
(3) x"-Ka° is divisible by x+a when n is odd.
(4) x°+a" is never divisible by x—a.
249. Quotient on dividing sc"±a" by x±a.
(1) = x+a. =x^ + x^a-\-xa" + a''.
x—a x—a
=x--]-xa-{-a^. =x* + a;*a+a--«^+a;a-'4-a*.
SUPPLEMENTARY CHAPTER 353
Verify these results by division or multiplication.
Notice that the signs are all positive, and that the powers of x are
descending and those of a are ascending.
Similarly, =x^-j-x^a-\-x^a'-^xhi^-i-x^a*-\-xa^-{~u''',
and =x'^~'^4-x"~^a-{-x'^"^a^-\- . . . -}-xa"~'^-\-a"~'^.
x — a
(2) = x — a. =x^—x^a+xa^ — a^.
x-\-a x-\-a
Verify and note the peculiarity in the signs.
x-{-a "" "' x-\-a
(3) 5!±^'==a;2_a;a_,_as. ^^^'^-=x^-x^a+xht''-xa^ + a^.
x + a x-\-a
x' + a' x^"+^ + a^""^'-
Write down the value of , — and of ; •
x-\-a x-\-a
EXERCISE 161
1. If /(a;)=a;3-8.r2+19.r-12, find the values of
/(I), /(2), /(3), /(4), /(5).
What are the factors of .r^— 8.r-+19.T— 12 ?
2. If f{x)=x*--2x^~x"-\-2x, find the values of
/(2), /(I), /(O), /(-I), /(-2).
What are the factors oif{x) in this case ?
3. Prove that x^*— ^^^ is divisible by x—y and x-{-7/.
4. Prove that x^^— 1 is divisible bj^ .r— 1, x+1, x^-\-l, x^-\-l.
5. Prove that x'^+t/^^ is divisible by .r+.?/ and that .r^-|-32 is
divisible by x+2.
Write down the quotients in the following divisions :
.r3+y3 ^ «lr^'' 8 "''~^^^ 0 ^'-1
x-j-y ' a—b ' a-}-b ' x—l
^y^xM:32 ^^ x'-8l ^2 ^'°-«' 13 («+^)lzl.
x+2 ' x+S ' ' x"—a ' ' («+&)+! '
A A
354 ALGEBRA
14. State one factor of :
x^-b"", (i'-^h\ a;3-64, m3+— ,, {x+yf-l.
What is the product of :
16, «"+«-+«+ 1 and a— 1.
16. m*— m^+m^— m+1 and m+1.
17. a^+a'^b^+a^b^+h^ and a^-b^.
18. Prove that a;64-3.r*+4a;2+224 is divisible by X'+l.
19. Show that x-^y, x'^-\-y', x^+y^, x•*+?/^ x^+y'^ and x-^^+i/ia
are factors of x^^—y-^.
20. If a;— a is a factor of x^-]-px-\-q, find the relation between a, p
and q. •
21. If f{x)=m,x^-\-nx-{-r, find /(a) and show that f{x)—f{a) is
divisible by .r— a.
22. If .r— 1 is a factor of x''^—k\c"-'rlOkx—10, find the values of
k and verify.
23. Write down the quotient when
1 1
(1) x—a is divided by x^^—a^.
1 1
(2) x-\-a is divided by x^-{-a^.
1 1,
(3) x—a is divided by x"—a^.
1 X
(4) x'+rt is divided by a;"' +«■"•.
250. Symmetrical Expressions. An expression is said to be
symmetrical with respect to any two letters if it is unaltered
when those two letters are interchanged.
Thus, x-{-y and x^-{-y^ are symmetrical with respect to x and y,
but x^-\-xy is not symmetrical.
Similarly, a + 6 + c and ab-{-bc-\-ca are symmetrical with respect
to a and b, b and c, c and a, for if any two be interchanged the
expressions remain unaltered.
251. Cyclic Symmetry. An expression is said to be
symmetrical with respect to the letters a, h and c, if it is
unaltered when a is changed to h, b to e and c to a, that is,
when the letters are taken in cyclic order.
SUPPLEMENTARY CHAPTER 355
Thus, a^ + 6- + c' — ra6 — 6c — ca is symmetrical with respect to
a, b and c, for when the letters are changed in cyclic order tiie result is
b^-\-c^-{-a^ — bc — ca — ab,
wliich is equal to the. given expression.
The expression a' + 6* + c^ — 3a6cd is symmetrical with respect to
a, b and c, but not with respect to a, b, c and d.
The only expression of the first degree which is symmetrical
with respect to a, b and c is a-\-h-\-c or some multiple of it as
A'(a+6+c).
There are two expressions of the second degree, a^-\-b'^-\-c^
and ab-\-bc-{-ca, and the sum of any multiples of these, such as
A;(a2+62_|-c2)4-Z(a6+6c+ca),
which are symmetrical with respect to a, 6, c.
252. Symmetry applied to Factoring. The factor theorem
may be applied to the factoring of many symmetrical
expressions.
Ex. 1.— Factor a{b--c^)+b{c^—a-) + cia^-b^).
If we put a = h, the expression equals zero,
a —6 is a factor.
Since the expression is symmetrical and a — b is shown to be a factor,
it follows that b — c and c — a must be factors.
We have thus found three factors each of the first degree. But the
given expression is of the third degree, and, therefore, there cannot
be another literal factor. There may be a numerical factor.
Suppose A; is a numerical factor,
.-. a(62-c2) + 6(c2-o2) + c{a2_62) = A;(a-6)(6-c)(c-a).
Since this relation is true for all values of a, b, c,
let a=l, 6 = 2, c = 0,
then 1(4-0) + 2(0- l) + 0 = A;(l-2)(2-0)(()-l),
.-. 2 = 2Jfc, orfc=l,
.-. a(62-c2)J-6(c2-a2) + c(o2-62) = (a-6)(6-c)(c-a).
In finding the value of /.% any values of a, 6, c may be used provided
they do not make both sides of the identity vanish on substitution.
A A 2
356 ALGEBRA
Ex. 2. — Factor
(a+& + c)3+(a-6-c)3+(&-c-a)H(c-a-/>)3.
If we put 0 = 0, the expression vanishes,
.■. a must be a factor, and, therefore, h and c.
Complete the solution as before, and show that the expression equals
24a6c.
Ex. 3.— Factor a\h-c)^h\c-a)-]-c\a-h).
As in Ex. 1, show that a — b, b — c, c — a are factors.
Since the expression is of the fourth degree it must have another
factor of the first degree.
The remaining factor must be of the form A;(a + 6 + c).
.-. a^{b-c) + b^{c-a) + c^(a-b) = k{a'-h){b-c){c-a){a + b-]-c).
Substitute numerical \-akies for a, b and c and show that the factors aro
-{a-b){b-c){c-a){a + b + c).
Ex. 4. — Simplify
(a-6-2c)2+(6-c-2a)'-+(c-a-26)24-(a+fe+c)2.
This expression is symmetrical with respect to a, b and c and is
of the second degree.
In the simplified result there can be only two kinds of terms, squares
like a- and products like ab.
The coefficient of a- in the resvilt is 1 + 4+ 1 + 1 or 7,
one part of tlie result is 7(a- + ?;2 + c^).
The coefficient of ab is —2 — 4 + 4 + 2 = 0,
.". the complete result is l{a^-\-b^-\-c^).
Check by letting a = b = c=\.
Ex. 5. — Simplify
(a+&)(a+6-2c) + (6+c)(6+c-2a) + (c+a)(c + a-26).
The coefficient of a^ in the result is 1 + 1 or 2,
one part of the result is 2{a'^-{'b^^c^).
The coefficient of ah is 2-2-2 or -2,
the other part of the result is — 2(a6 l-6c + ca),
/. the complete result is 2(a* + 6- + c^ — a6 -6c — ca).
SUPPLEMENTARY CHAPTER 357
EXERCISE 162 (1-12. Oral)
With respect to what letters are these symmetrical :
1. a-\-b. 2. a+c— 6. 3, x--\-ij^-\-xy.
4. ah^hc^ca. 5. a^-j-ft^+c^— 3a6c.
G. x^-'-^y'+x-y. 7. Z{p'^+q"+r-)-2{pq-\-qr+rp).
8. What is the simplest expression of the first degree whicli is
symmetrical with respect to x and ?/ ? a, b and c ? a, b, c and rf 'i
9. W^hat expression similar to «--(-fc-+3a6 is symmetrical with
respect to a, b and c ?
10. SimpUfy
(a+6)2+(6+c)2+(c+r02 and {a-b)-+{b-c)-+{c-a)~.
11. If a -|- 6 is a factor of any expression, synuuetrical with respect
to a, b and c, what other factors must it have ?
12. When {a-irb)^-\-(b-{-c)^-\-{c-\-a)^ is simplified, the coefficient
of «■' is 2, of a-h is 3 and of abc is 0. What must the simpUfied
form be ?
Simplify :
13.* {a-b+c)-+{b-c+af+{c-(i + b)\
14. (a+b){a+b—c) + {b+c){h+c-a) + {c+a)ic+a—b).
15. (a;— ?/)(i)a;-|-jDi/— z) + (y— 2)(;)?/+pz— z)+(z— x)(ps+j9.-r— t/).
16. (ff-fe)3+(6-c)3-f(c-a)='.
Factor :
17. .t2(?/— z)+?/2(z— a;)-|-22(a;_2/j.
18. xy{x—y)+yz{y-z)+zx{z—x).
19. a2(5^c)_^j2(c^„)^c2(a+6) + 2a6c.
20. («,+6+c)3-(fr-f6-c)3-(6+c-a)3-(c+a-6)'-
21. (a;-2/)3 + (y-z)3 + (2_;,.)3.
22. a(6+c)2-{-6(c+a)2+c(a+fe)--4a6c.
23. ab{a^--b^-) + bc(b^-c^)+caic^-a^).
24. rt2(ft4-c*) + 62(c4_(j4)^c2(a4_64).
358 ALGEBRA
Simplify :
25.
^iy+z) , y{z+x) _^ z{x+y)
{x-y){z-x)^ {y-z)(x-y) ' {z-x){y-z)
26. ^^_---- + y- + -^— .
{x-y){x-z) {y-z)(y-x) {z-x){z-y)
ah he ca
{c-a){c-h) ' (a~h){a-c)^ (b-c){b-a)
28. ^ ""- + -^ +
29.
bc{a — b){c—a) ca(h—c){a^b) ab(c—a){b—c)
b^—ac c^—ba a^—cb
(a—b){b—c) (6— c)(c— a) (c~a){a—bj
bc{b-\-c) ca{c-\-a) ^ ab{a-\-b)
31.
32.
a— 6)(a— c) {b—c)(b—a) (c—a)(c—b)
x^ y^ z^
{x^){z-x)+(y-z)ix-y) + {z-x){y'^z) '
(a-6)3+(6-c)3+(c-a)3
a(62_c2)+6(c2_a2)+c(a.2-fo2) '
33. Simplify {a+b+c)^-{b+c)^-{c+af-(a + hf+a^+b^+c^
being given that a is a factor of it.
34. Show that a—b is a factor of
a"(b~c)+b"{c—a)+c"{a—b).
What may be inferred regarding otlier factoi-s ?
35. An expression is symmetrical in x, y and z and each term is of
two dimensions. When x=7j=^z=l, the expression equals ] 5, and when
x—1, y=2, z=3, it equals 64. Find the expression.
3G. Point out wherein it is obviously impossible for the following
statements to be true :
(1) (a-+b^+c^){a+b+c)=a'^+b''i+a%b+c)+b^c+a).
(2) n^+b'^+c-^—:iabc=^{a+b+c){a^+b-+c^-—3ab).
(3) {a—b){b—c){c—n) = ah--{-b'^c-\-ca^—a^- — bc~—ba".
SUPPLEMENTARY CHAPTER 359
253. Identities. We have already had many examples of
algebraic expressions which are identically equal, that is,
which are equal for all values of the letters involved.
Thus, {x + y){x — y) — x" — y~,
{x-\-y -\-z)'^=x'^ -\-y^ -{-z- -{-'2xy -\-2xz-\-2yz,
a^ + 6*4-c^ — 3a6c = (a + 6-|-c)(a- + 6- + 0- — a5 — 6c — ca).
Any of these may be shown to be identities by performing
the operations necessary to remove the brackets on one side,
when the result is the same as the other side.
Ex.— Show that (a+6 + c)=^
=a3+&3^c=^— 3a6c+3(a+6+c)(a6-|-&c+ca).
Here the cube of a-\-h-\-c may be fomid by multiiaHcation or by any
other method.
The brackets are then removed from the right and the terms
collected.
The two sides are now the same, which shows that the given statement
is an identity.
We might also hav^e changed the second side into the first by
factoring, thus :
(a3 + 63 + c3-3a6c) + 3(a + 6 + c)(a6 + 6c+ca),
= (a+6 + c){a2-|-62-fc2-a6-6c-ca) + 3(a + 6 + c)(a6 + 6c + ca),
= {o + 6 + c)(a2 + 62 + c2 + 2a6 + 2ac + 26c),
= (a + 6 + c)^, which jaroves the proposition,
254. When two expressions are to be shown equal, the
result may frequently be obtained by showing that their
difference is zero.
The difference may be zero,
(1) because all of the terms cancel, or
(2) because it has a factor which is equal to zero, identically,
or which is given equal to zero.
360 ALGEBRA
Ex. 1. — Prove
(a-6)3+(6-c)3+(c-a)3=3(a-6)(6-c)(c-a).
Here we may prove that
(a"6)3 + (6-c)=»+(c-a)3-3(a-6)(6-c)(c-a) = 0,
(1) by removing the brackets when all the terms cancel,
(2) by observing that (a — fe) + (6 — c) + (c — a) is a factor of the
expression and this factor is identically equal to zero (art. 245).
Ex. 2. — If a-\-b^c, show that a^-\-bc=b^-\-ca.
Here, as in the preceding, we may show that a^-\-bc — b^~ca = 0,
by showing that a-\-b — c is a factor of it and this factor is given equal
to zero, or by substituting c = a-\-b in each side or in the difference.
Solve this problem both waJ^s.
Ex. 3.— If a+6+c=0, show that
(a+6)(&+c)(c+a)+a&c=0.
For a-\-b substitute — c, for 6 + c substitute —a, and for c-\-a
substitute ~b and
(a + ft)(6 + c)(c + o) + a6c = (-c)(-a)(-6'+a6c = 0.
Ex. 4. — If 2s=^a-\-h-^c, prove that
s2+(s-a,)2+(5-6)2+(s-C)2 = a2 + 62^c2.
When the first side is simplified it
= 4s2_2s(a + 6 + c) + a2 + 62 + c2,
= 4s2-2s(2s) + a2 + 62 + c2,
= a2 + 'j" + c^, which was required.
Of course, this could have been proven by substituting the value of
s at once. It is visually easier, however, to substitute in the last step.
EXERCISE 163
Prove the following identities :
1. a(?>+c)2+6(c+a)2+c(a+6)2— 4a6c=(a+6)(6+c)(c+a).
2. (.T+^)HxHy*=2(.r2+.Ty+y2)2.
3. (a + ft)3+(a-6)=' + 6a(«+i)(f/-?;)=8a='.
SUPPLEMENTARY CHAPTER 361
4. 2{a^+b^+c^-3abc)={a+b+c)\{a-b)-+{h-cy- + (c-n)^l.
5. a{b-c)^+b{c-a)^+c{a-b)^={a-b){b-c){c-a)(a+b+c).
If a+6+c=0, show that :
6. (3a-26+4c)2-(2a-36+3c)2=0.
7. a^-\-b^—c^-]-2ab=0 and c^—ab—b'^—ac.
8. (a+6)(6+c)+(6+c)(c+a)+(c+a)(a+6)=a6+6c+ca.
9. aH6*+c*=2a262^262c2+2c2cf2.
10. (3rt-6)=' + (36-c)3+(3c-«)3 = 3(3a~6)(36-r)(3c-a).
11. a{b^-+bc+c^-)+b{c^+ca+a-)+c{a-+ab+b-)=0.
12. If a+6=l, prove that {a-—b-)'=a^^b^—ab.
13. If a;+y=22, prove that -^— + ^- = 2.
a; — 2 y — z
14. If a = yiZl, b = ^-^^, c = ^^:Z^, show that a+b+c+ahc=0.
X y z
._ „ 1 , 1 2 .,112
lo. If - H — ,, prove that --!-- = -.
a a — c a—b a b c
16. If .T + A = y, show that x^ + -, = y-~2 ; o;^ 4- — = ?/3 — 3)/ ;
X ' a;^ x^
^* + -, = y'- 4/y2 + 2.
X*
If 2s=a-i-6-|-c, show that :
17. s(.s— rf)+(-s— '')(-5— r)=&'^-
18. a(.s-a)+6(s-6)+c(.s-c)+2.s2=2(«6+6c+ccf.).
19. {.s-ay-+{s-b)^+{s-cy^+2{s-a){s-b)+2{s-b){s-c)
+ 2(s— c)(s— a)=s-
20. (2rt.s+ic)(2ft.s+m)(2c.s+a6) = (a+6)2(6+c)2(c-|-«)2.
5— a 5—6 s—c s s(s— «)(s— 6)(.s— c)
22. 16s{s—a){s-b){s-c) = 2b^c^2c^a^+2rJa--a*~b*—c*.
23. If 6 + - = 1. c + - = 1, prove a + - = 1 3^nd afic = —1.
c a 0
362 ALGEBRA
24.* If a + - = 3, find the value of a^ + 1 .
a o*
25. If a=x{b-{-c), b=y{c-\-a), c=z{a-\-b), show that
xy-{-yz-\-zx-\-2xyz— 1.
26. If x-\-y=a and xy=b-, find the values of .-r'4-^" and x^-\-y^
in terms of a and b.
27. Eliminate x and y from the equations x-{-y=a, xy=b^,
x^-'ry"=c'^.
28. Eliminate x and // from ;c+2/=ct, .t^+2/^=^^ a;^4-?/^— c^.
29. If a:=o+&— c, y^b^c—a, z=c-\-a — b, show that
x^-\-y^-{-z^—3xyz=4:{a^-{-b^-\-c^—3abc).
EXERCISE 164 (Review of Chapter XXVI)
1. Show that x^-\-y^-\-z^ — 3xyz is divisible hy x+y + z, and hence
show that (6-c)» + (c-a)» + (a-fe)3 = 3(a-6)(6-c)(c-a).
2. Prove that
/„.,„ „\/l 1 1\ fb . c\fc . a\fa , h
( o,
V- + ^^ + ^;)( ^ + 6^ + ^j-(e + 6JU + cJU + «; = '•
3. If a + 6 + c + d = 0, prove that
(a + 6)(a + c)(a + f/) = (6 + c)(6 + c/)(6 + o).
4. Prove that (a — 6)" + (6 — c)"4-{c — o)" is divisible by
(a-6)(;)-c)(c-o).
when n is an odd integer.
5. If n is a positive integer prove that 12"— 1 is divisible by 11,
232" n+1 by 24, 72"- I by 48.
6.* Write down a quantity of the same type as x^-\-y^-\-z^ — Zxyz
of which }fX-\-\y — \z is a factor.
7 . Show that a, a — x and a — 2x are factors of
{a-b){a-b-x){a-{-2b-2x)-{-b(b-x){Za-2b-2x).
8. Show that (cc + y)" — x" — y" is always divisible by xy{x-{-y),
when n is an odd integer.
9. If (y — a)(l— a) = (2/ — fc)(l — ?j)=.t, find x in terms of a and ft only.
SUPPLEMENTARY CHAPTER 363
10. If .'c + 2/ + 2 = 0, prove that
(1) x'^^xy + y- = y^ + yz-^z'^ = z'^-\-zx-{-x-.
(2) {x-\-y-zY^{y + z-xY^{z+x-ij)^-^24.xyz^0.
11. Simplify .-,„—t^7-—-.+ ~Ji^—7^7u-.^-„^ +
bc(a — b){a — c) ca{b — c){b — a) ab(c — a)(c — b)
12. Solve (.r-o)3 + (.r-6)3 + (a;-c)3 = 3(.'c-a)(.-c-fe)(.r-c).
1.3. Show that {a + b)^ — a^ — b^ = 5ab{a + b){a- + ab + h^).
14. If 2s = a+b-\-c, show that
(1) s(s — b)-\-{s — a)(s — c)=ac.
(2) s'^ + {s—a){s—b)-{-(s—b){s-c) + {s — c)(s~a) = ab + bc + ca
(3) (s-a)3 + (s-c)3 + 36(s-a)(.s-c) = 63.
15. Prove that a"(b^-c-) + b"{c~-a^) + c"{a'^-b^) is divisible by
(a — 6)(6 — c)(c — a) and find the quotient when n = 3.
16. Simplify —■ j-- : + rn —r r + — rr •
^ -^ a{a — b)(a — c) b(b — c)(b — a) c{c — a){c — 6)
17. If x = a~ — bc, y = b^ — ca, z = c^ — ab, prove that
ax + by-{-cz = {a + b + c)(x-\-y + z).
„ ^. ,.. a3(b - c) + b^c - g) + c»(a. - b)
18. Snnphfy — -y ,„ , , ,.. , , — rrj- •
19. If ab-\-bc-{-ca = 0, show that
(1) (a + 6 + c)2 = o2 + 62 + c2.
(2) {a + b + c)^ = a^ + b^ + c^-3abc.
(3) (a + 6 + c)* = a4 + 64-l-c<-4a6c(a + 6 + c).
20. Show that cc"+i — a;" — a;+ 1 is divisible by (a;— 1)-, when n is a
positive integer.
21. Write down the quotient on dividing
a;* — a* by .r — o, .-r^-f 1 by x~-\- 1, a^ — 32 by a — 2.
22 . Factor a;^ - 1 — 3(x^ - 1 ) + 4(a;2 - 1 ).
23. Simplify r^, r + two similar fractions.
^ -^ (a — b)(a — c)
24. Show that x{y^ — z^)-{-y{z^ — x^)-'fz{x^ — y^) is not altered when
X is changed to x + a, y to y + a, z to z + a.
25. If .'c^ = a;+l, show that x-' = 5x-\-3.
364 ALGEBRA
26. Find two linear factors of
{ax+by + {bx + cy + {cx + ay~3(ax + b}{bx + c)(cx + a).
27. If x^ + y^ = z^, show that {x^ + y^-z^)^ + 27x^y^z^ = 0.
28. If a+b+c=0, prove that
a3 + 63 + c3 + 3(o + fe)(6 + c)(c + a) = 0.
29. A homogeneous expression of two dimensions is symmetrical
in X, y, z. Its vahie is 42 when x = y = z=2 and is 16 when x=l, y=2,
z = 0. Find it.
30. Eliminate x and y from x-\-y — a, xy — b, x^-\-y- = c.
31. li x-\-y = ^ &nd x^ -\-y'^ = b, find the values of a;^ + 2/^ and x* + ?/*.
32. If a + 6 + c=10anda6 + 6c+ca = 31,find the values of a^ + b^+c*
and a^ + fc^+c^ — 3a6c.
.«. *-•'• • *-*•"
ANSWERS
TO
HIGH SCHOOL ALGEBRA
ANSWERS
No answers are given to elementary examples, oral examples or
examples which ma_y be verified or checked without difficulty. In each
exercise the number of the first example to which the answer is given
is marked with a star.
Page 8
15. 108, 38, 10, 32, 60. 16. 3, 14, 39, 0. 17. 9, 29, 18. 19. 2.
21. 25. 22. 44,7. 23. 154,616.
Page 10
9. 47. 10. 70. 12. 10a;+10. 13. xft.E. 14. 15a;.
16. 2a3+2a2-j-3a. 17. 11 ox.
Page 12
24. 37. 25. 17. 26. 34 27. 1 28. 1. 29. h.
Page 14
30. ('- + —^ hours. 31. (5x+20?/-7z) cents. 32. ^±?^ cents.
33. 5a;+102/+502. 34. 1234, 4019. 35. -3, -02, 2, 15, 05, -03.
36. 1 + - . 37. 20. 20. 38. 24. 39. 2k
X y
367
368
7. 7a+6b—15c.
10. 4a+46+4c.
13. 8a-4&.
16. 2x+2y—z.
ALGEBRA
Page 38
8. 10.r2- l4,*+9.
11. 4.a-8b+3c-5d.
14. 6a2+862-bc2.
9. lOa-76.
12. 8a,-— (j^+52.
15. 3a+36+3c+3d.
18. 0. 19. 15X+5//.
23. 2a-^6-ic,
Page 40
20. Abe. 21. lOa^+ab.
22. 4z/2.
13. -3a;^. 14. 0.
18. 0. 19. X.
Page 41
15. 4p-. 16. 10m— 3n.
17. 6.V-4Z.
Page 44
15. 3a-26. 16. 2«+5c. 17. -3x^. 19. 4a2_4a_l5.
20. 2b. 21. 13r— 2;. 22. 36— 5c— 2a. 23. x^+Qx~5.
24. 2a2+a-12. 25. a+b+c. 26. 2a;-3.
27. 10a;3+2a;2+8a;+2.
Page 46
11. 3x+2^. 12. — 2rt — 36. 13. a+b. 14. 6— a.
15. 3a+6-3c. 16. 3x~-3. 17. 7. 18. 11
22. G, 4, 4, 6, 10. 23. 4rt+46-15c, 4a-46+4c-4rf, y, 0.
ANSWERS 369
Page 47
1. a. 2. 4.r-f6. 3. 5n. 4. 5?. 5. 46,8a.
6. 2a+26+6c. 7. -.r+y-52. 8. 3a-26-2c. 9. 14, G.
10. -7. 11. y-x. 12. 31. 13. «-2c, 2c-a.
14. 9m-2«. 15. 4.r-9. 16. a;- 12. 17. 26-4c.
18. 56— 5a, a+36— 4c, 7a— 6— 6c. 19. ■2x. 20. 3x-6.
21. a-lb+hc. 22. 52-3a.-. 25. 20. 26. l+2x. 27. 7.
28. 7n+4a;— 2m. 29. 3-a-6-c. 30. 5c-36.
Page 52
19. 1, 4, 5, -3, -1, -8, -9, 7. 20, 3, 16, 35.
21. a^ -a\ -8, -1, 1, 81, 32. 22. 29, 81. 23. 24.
25. 90. 26. 6. 27. 30. 28. 23. 29. -20. 30. -50.
31. -100.
Page 53
13. 8a+76+9c. 14. x-iij. 15. 3m. 16. 9a-b.
17. 4a+|6. 18. 6x^+8x. 19. a^ 20. a:3-9a;24-i0a:.
21. -4a6. 27. 6x2-15.r. 28. la^-5a.
29. 2x1/, 3a;2+x//+3//2, x-+5xij+ij'K
Page 56
19. 2a:2+4a:-4. 20. 5a2_8a-22. 23. 214. 24. ia--9b^.
25. 2x2+2//2, 4x7/. 26. a2+a6+462. 27. 14a;+30.
28. x^-6x-l. 29. 2.r-10. 31. 12.r24-12. 32. 3x2+ 10.
33. Ion. 34. 3.t2-l-12«-+14.
B B
370
ALGEBRA
Page 60
18. 1.
19. x2-3a:+2.
20. 2^.
23. a^
24. a;+13.
21. 5. 22. a-h.
Page 60
2. 9, 16, -12, 25, -7, 27, -64, 91. 3. 1, -1, 1, 16, -27.
4. in-. 5. 8«2_9a. 6. ^a^--^}f-. 8. 30a+406.
9. 12,T'-+12. 10. 4.r2+12.r!/-9;/2. 13. 13m2+13n2_24»m.
16. 4.r2. 18. 5a2_3a6_462. 20. 3a2-12a.+14.
21. 6.r2-2xy-6y2. 22. 4-a. 23.8,19. 24. .t*-16.
25. 8a2_9a-l, 6-lOa, 3a-4. 26. a'^-b^-, 2, a^-b^.
27. 2062- 56c. 28. 0.
Page 81
22. 7, -2. 23. -8. 24. 5, -2. 25. 5, 6. 26. i, 2.
27. 6, 1.
Page 83
21. 4,5. 22. -3,-3. 23. 4,9. 24. 5,3. 25. 12,12.
26. 19, 3. 27. 15, -56.
Page 92
28. 2(1. 29. 1. 30. .r + 1. 31. 3.i-— 8. 32. x+5, a+b.
33. 2(x-2)(a:-3). 34. 3(a+4)(a-3). 35. x{x-7){x-l).
36. ±5, ±1.
Page 93
17. 7, -11. 18. 10, -4. 19. 5. 20. 10 in. 21. 7 in.
22. 14 in. 23. 3i in.
ANSWERS
371
17. 2x2+2.
21. 5m'- — \0mn.
24. .r--4. 25. 18fl-lo
28. 36x. 29. 9a2-8ai+96-'.
Page 95
18. 2a-+2b-. 19. 5a;2
22. 5x-+24:xi/-5y-.
26. 3x--4a-^+G^
31. 8.
-5.
20. iab
23.
■Sx-+I2x+U
2.
27. 16X-34
Page 98
25. 5a2-5. 26. 3«--86-. 27. 0. 28. Idq^—ijiq.
29. x*-a*. 30. jr.. 31. 3(.i:+^)(.r-7/). 32. 5{x+2)ix-2).
33. a(a+l)(a— 1). 34. m{x-a){x+a). 35. 5(1 + 3p)( 1-3^3).
36. (a:+2/)(a;-2/)(a;2+2/2). 37. ^^ji^r){B-r).
38. (a;+l)(a;-l)(a+6). 40. a'^-2ab-3b'-. 41. 2.t, 7.
42. 4, ±8.
Page 102
22. .T(a:+l)(a;-l), 3(.-c-2)(a-+2), a(a-l)(a.-2). 23. 8,-2; 2,-1.
24. 2. 26. 2(.r-2)(a;+2)(a;2+4), (ffl+2)(ffl-2)(a+3)(a-3),
2w(m + 3)(m-3), {x+u){x-y)(a+b){a-b). 27. 2b^+2c-+4.
28. 3x2-52/2. 30. 12a6-3862. 31, 43,23,17,13.
32. 3x+6?/.
Page 104
13. a + 2b. 14. r/-6. 15. m-n. 16. x+j/.
18. a-2. 19. .r-3. 20. y-\. 21. a+b.
23. 2(3a+26). 24. a(rt-l). 25. 2, 3.
17. TO+2.
22. x-5.
22.
27.
x+1
x—y
x+2y'
23.
y
2/-1'
28. a+1.
Page 106
24
29
x-2
x^3'
a2+l
25.
TO + 3
30. x2+].
26.
a+46
BB 2
372
ALGEBRA
Page 107
,0.1
11. 4
12. ^'^. 13. "^^^
ay x-\-2
14. "-^
a— 5
15. 1.
Page 108
10. «>+!). 11. 3a;(a;+2). 12. ab{b+c). 13. 2(x-2-l).
14. .r(.r+/y)-. 15. {x+l){x-l){x-2). 16. ab{a-b).
17. («+6)(a-?>)-. 18. a;(x-l)(a;+l). 19. 4x(x-l)(a;+l).
20. (^_l)(^+l)(,y_2).
Page 109
22. ^^±^. 23. ^^^. 24. ^±^. 25. ^"il^. 26. 0.
6 8 12 a;2-//2
27.^. 28.--^^. 29.2^-1^-. 30. ^^+^^
G(x+2) ab{a-b) (a+l)(a+2)((7-f 3)
2
31.
(a-l)(a+2)
Page 111
2. .r-2,60(a;-2). 3. x+y,xy{x-^!/). 4. x-2,(a;-2){x+4)(.T-5).
5 "±^ _^ ^" ^ 6 4 7 1 8 — ■^— 2(a^) a+6
a ' x—y 4b' c' ' ' ' ^' ' x—S' x+3' 3(x-2)' a—b '
9.-^,- 10. ^f^,^. 11.^,1. 12.0.
13.^. 14. l,?^i:^. 15.0. 16. A. 17. '^^
46c 15 xz ■-"■ a2-62
18. -'^. 19. 1. 20. («+6)-. 21. 3. 22. +4.
AN8WEBS .373
Page 116
18.
Ix+G. 19.
a+b. 20. 'S—x—ij.
21. 3x+3//+3z
22.
22x-+24:X-U.
23. 10a;2-oa;+15.
24. x2-i-8a;-12
29.
a. 30. 2.
31. 4r,^.
Page 119
17. 3a;2-12x+ll. 18. 0. 19. 2ad-\-2bc. 20. 0.
21. 12a;2+12. 22. l-x^. 23. a;4-10a;3+35a;2_50a;+24.
24. a;*-10.r2+9. 25. «6_i. 26. 13. 27. 0.
28. 0. 29. l + 'ix+Gx^+lOx"^. 30. 2a:3+9x2+3a;-l.
31. 28x^+x^y-33x^y^+3lxh,3+20xij*-l2y^. 33. 2-a;+4a;2-2a;3.
34. 195. 35. 51. 36. abx*+x^b^-ac)+adx^+x{bd-c^)+dc.
37. p^x^-\-x(pr — q^)-\-q>', x\a^ — a)-\-x'^{a^-ra — 1) — 1.
38. 2ahj^-2bhj+2bc 39. px^'+x{p^+3p+3).
Page 123
30. x2+2a:+l. 31. —x^+9x^-]. 32. a'^+a+l, a^—a-\-l.
33. -2x^/. 34. 2a. 35. 0. 36. a-2. 37. a24.3a-2.
38. a;2+a;?/+2/^. 39. 2ax. 40. x+c. 41. x-{-p — l.
42. ax—b~c 43. oy+a+l.
Page 125
1.4. 2. -.5. 3. 22. 4. ?/-!-5. 5. 2,7/3. g. -2.v3.
7.1 + ^- 8.1 + -^,. 9.2--^. 10. .5a; -3+-?-.
x+l a-6 0+6 x+2
11. l+a;+x2^.a.3_ 12. l-a;+a;2-x3. 13. l+2a;+2x2+2x3.
14. l+2a+3a2+a3. 15. n-3. 16. 6. 17. a;-4, 7.
374 ALGEBBA
Page 126
15. a-^-^b^^c^-'iahr. 16. a^-'2m. 17. .i-'-^-4x+8. 19. 21.
20. llx2-7,r-8. 22. a;2(c-a)+.r(fZ-6) + (/-c). 23. 7.
26. 5//*. 27. a;+6j/-22. 28. 2(a6+6c+ca).
29. bx'^—bcx^^x{ac—a+h)-hc. 30. 35. 31. .r*— 4.'c2+12x.
32. 3.f-8. 33. 3+. 34. 9. 35. 6. 36. p-+p-2.
37. -52. 38. 4. 40. x''+2a;3+3a;2+2.r+l. 41. 9.
45. x'^+x%b+p)^x-{q+bp+c)+x{bq+pc)+cq. 46. .3a + 26— c.
47. x^+x^+X"{a-a-)-^x{\ — -2a)~\. 48. a^^2a%c+A.ab''-c'^+%b\^.
49. a;3— a;2(a+6+c)+a;(a6 + 6c+ca)— a6c. 50. 6— c.
51. 3.t5— 10.r* + 3a;3— 14.r2 — 7.r. 52. .r24-Z/'+-'*^.'/— 2.^;— 42/+4.
Page 129
21. 3(.r-2)(a;2+l). a(.c-l)(y-l). 22. a+b, x-l.
23. (2.i--.y)(5,r-33), «6(a+c)(a-3i). 24. a:-3.
25. (6;r-7y)(8a + 56). 26. (.r+//)(.x+;y+4), («-6)(2a-26-l).
Page 131
34. 14.r2+19vy2. 35. 3«2+362 + 3c2. 36. 2.T*+6.r2+2.
37. 3a2+3?j2_|-3c2_2a6-2ac-26f. 38. 8(x2— z2--a;?/+?/2).
45. 3(.T+1)2, fl(a+26)2. 46. (a+6+2c)2, (a+6_c-rf)2. 48. 3.
49. 14. 50. {x^-^y-){a^+b^+C'). 51. {ax-\-hyY^{ay-hxf.
52. 0.
Page 134
11. a--b^"-c^~+2bc. 12. 4.1-2+ 12.ri!/+9i/2- 25.
13. /)2_4^2_9^2.[_i2^r. 14. l+.T2+.r*.
15. a^-b'^-\-c^--d^-2ac-2bd. 16. a2^4ft2_c2_4^2_4(ji_^4c^,
44. 2(.r+2)(:r-2), a(a.+ l)(a-l). {a-x){a+x)(a'^+x'^).
45. .5(a-64-2c)(a-6-2c), (.T-36)(.r-6)(.r-56).
ANSWERS 375
Page 134 (continued)
46. {b+c){b—c){a-{-d){a-d), {ni-b+c){a+b—c){a—b+c){a-b—c).
47. {ax-{-c-{-by){ax^c—bi/), (to— ?i+3»i?i)(m— ?i— Sm?!).
48. (a;+l)(a;-l)(3a;-2), x{x-l){x-'3){x+3).
49. 2a-—2ab+2bc—2c^. 50. {x-\-y){x—i/){x+y+a){x+ij—a).
51. 2a2— 6a+l, 12xz~24:i/z, 24a+9a2-G«3, 20.i-2,(/2-40.rV
52. a*4-6*+c4-2a262_262c2_2c2a52. 53. (x-y){y-z)(z-x).
54. (a-6)(c-a).
Page 137
13. 2(.r2+2a;+2)(a;2-2.r+2), x{x^+x+l){x^—x+l).
14. {a-b){a+b){3a-b)i3a+b).
15. (a;2-a;+l)(a;2+a;+l)(a;*-a;-+l).
16. (a+6+c)(a+6-c)(a-6+c)(a-6-c). 17. (aHBKSa^+l ).
Page 139
31. 3(a;+8)(a;-9). 32. 2(a+l)(a + 3). 33. a;(3a;-l)(2a;-l).
34. {x-\-l){x-l){x+2){x-2). 35. a(a-l)(a+l)(a-3)(o+3).
36. (a + l)(a-l)(3a + l)(3a-l). 37. {x+l)(x+3){x-l){x+5).
38. ix-2){x-l){x+l)(x-l0). 40. x--5.r+6. 41. 4x-2-16a;+15.
42. ±1, ±11, ±19, ±41. 43. 33a2-38a6-862.
44. ix+y){x+4:y+l). 45. (3a+26)(a-6+2). 46. x^-\-l.
Page 141
22. 2(a-2)(a2+2a+4). 23. 'My+3)(y^--3y+9).
24. a(a+l)(a2-a+l). 25. ?>((/ + 6)(a2_aft+62)_
26. {a^+b^)(a*-a^b^+b*). 27. (a;-L?/+a)(.r24-2x2/+?/2_aa;-ai/+a2).
28. a;(x2-6a;+12). 29. (2a-6)(a2-a6+62). 30. 2a(a2+362).
31. x+y. 33. a2(a4-6a26c+1262c2), 2/2z(3a:-2/2)(9a;2+3a;2/2+2/2z2).
34. (a-6)(a+6)(a2+62)(a2_a6+62)(a2_^c^ft^62)((j4_a262^54),
35. (a;+l)(a;-2). 36. 2.
376 ALGEBRA
Page 143
7. {x-l)i2x^-9x-4:). 8. (.r-l)(.r+l)(x-2).
9. ix-l)(x-2){x+3). 10. (x-2)(.r-3)(.r+5).
11. (a-l)(a-2)(a+4). 12. {a+bfia-2b). 16. -12.
17. (a-6)(a+26)(a+36). 20.8,-4. 21.2,3.
Page 147
4. 2a;2+2a2-262. 5. 4:X^. 6. 4a2+462+4c2.
7. 14a2+1462+14c2+14a6— lOac— 226c. 8. 19997.
9. 14,860,000. 10.-5. 11. 2a+197. 30. (T-2)(4a;-9).
31. 3a(2a-6)(4rt2-f2a6+62). 32. 8{a+c)(c-a-b).
33. (3a--4)(4.r+5). 34. 4(3o-5)(9a24-15a+25).
35. {x+y){x-y+l). 36. {x-3)(x+3)ix^- + 2).
37. (a:-^)(a:2_a;2/+2/'). 38. {x+Uy)(x-12>j).
39. (a-6+c)(a-6-c). 40. (.r+2/)3. 41. {x-3b){ax-2).
42. (a+26)(rt-26-3). 43. {2x-y){2x+y+a).
44. (a+6)(a+fe+c). 45. (a-6)(a-6-l).
46. {x—y){T^+xy+y^+x+yi-l). 47. a6(a+6)(a— 6)2.
48. (2a+56)(2a-56+l). 49. 96(4a2+2o6+62).
50. {x^+^xy-y^){x^--4:xy-y^). S^. (a^-~b^+a-3){a'^-b^-a+3).
52. {x-l)(x^-lOx-3). 53. (a-l)(3a2-2a-10).
54. (a;+l )(.-»;- I)(c+l)(c2-c+l).
55. (a+l)(a~l){a+2)(a^+l){a-—2a+4:). 60. (o— 6)(6— c)(c— a).
61. (.r— 2)(2a;+3)(3a;-2).
62. (x--l)(a;-2)(a;-3)(a;-4), {x+l)(x^3){x-2)(x-G).
63. 1, 5, —6 ; 0, 1, 6, -7. 65. x^—c^-. 66. «2-62+c2+2oc.
67. -4, 5. 68. {x-a){2x+a+b).
69. {ab+cdf—{ac-'rb(jy-, {ab—cd)--{ac—bdf.
ANSWERS 377
Page 153
16. 30. 17. 13. 18. 1,3,10. 19. 29. 20. 1,3,5.
21. IG.
Page 155
6. 45. 7. 4. 8. 1.
Page 169
6. About 2 h. 35 m. after A started ; 31 m. from Toronto.
7. (a) At 10.55, 2 m. from C towards D. (h) 22 m., 17 m. (r) 11.10.
Page 172
5. 1.3. 6. A square, 16. 7. Right-angled, 4. 8. M.
9. 54. 10. IB; .5, 6^, 8. 11. 6.
Page 177
3. .5, 10, 13. 4. 6|. 5. 13. 6. 7|, 4. 7. .30, 30.
8-. (1, -7), (-3, -17), (5, 3). 9. 112i. 10. 24. 17. 24.
18.(4,4). 25. (3, 2), 90°. 26. $1200, 12th. ^
Page 181
I. 4xt/H, 24ax^y*z^. 2. a;— »/, a-7y(.r-— //'). 3. a-\-h,h(a~-h){a'{-h)^.
4. x-3, (x-3)(.T-4)(.T+3)(a:4-5). 5. o + 5, (a + 5)(a+3)(rt-7)(rt-2).
6. S{x-2), 3{x+\){x+2){x-2r. 7. x-y, y{x-y){x+z).
8. m— 2, im^n^{m-\-2){m—2)-{m^-\-2m-\-'i).
9. 2(a2+a6+62), 6a(a3_63). iq. cr-Lfe-c, a(a4-6-c)(a+6 + c).
II. a-\-h-\-c, {a-\-h-\-c){a—b—c){b—c—a){c—a—b).
378 ALGEBRA
Page 181 {continued)
12. x^-xii^>f, {x^y){x^-yxhf-^^y%
13. 3,i;-2, (3.f-2)(a;+3)(d;-3)(2a;-3).
14. 5x-l, (5.c-l)2(5.r+l)(2a+3c). 15. a:-3, .r(.r-3)(a;-2)(.T2+5).
16. u—v, {u—v){u^v){u^-{-v"){u'^-\-uv-\-v'^).
17. a;2_8^ (.c---8)(a;+2)(a;+3). 19. -a.
20. a;2_3_^^_|_2y2^ a;2+a;//— B.y^.
Page 182
1. a;-l,(:c-l)(;c-2)(^2-5.f+3). 2. a-\,(a-\){a-5){a^-\Sa-\).
3. x-2, (a;-2)(a;2+4)(2a;2-3x-6).
4. a-1, (a-l)(a2+i)(3a2+(7,+6).
5. a;-l,a;(x-l)(.r+4)(a:2+a;-6). 6. {x-2){x-+5x+\){x--2x-\).
7. ^^^^ , ^— . 8. .r3-6a:2+lla;-6, x3-9.c2+26a;-24.
a2+2a6-1562 2x+4
Page 186
1. x-S. 2. (a-3)(re-4). 3. 2(3a;2+2:c+2). 4. 2x-9.
5. 262-6-5. 6. 3a;-7//. 7. a-2. 8. x-S.
9. 3a2(a-l). 10. x-1 11. {x-'i){x + \){x\-2){x^-x+\).
12. (x+l)(.j;+2)(x+3)(a;+4). 13. (2.c+3)(3.r-4)(.i;2+3a;-l).
14. {x-\){x-2){x-'i){x-^). 15. (5a;2-l)2(4a;2+l)(5.r2+a;+l).
16. 3. 17. 35. 18. .r2+5x-14. 19. 11.
Page 187
1. a;-ll, (.r-9)(a;-I0)(;c-ll)(.r-13).
2. a;-3, (x--3)(:f-12)(a:2-2;(x2+3x+9
3. a-b, (a-6)2(a+6)(n2 4.^6+62).
4. a:+3, .T(x + 3)(a;+2)(.T-4)(a;-5).
5. (2a+l)(a-3), (2CT + l)(a-3)(rr + 3)(2a-l).
ANSWERS 379
Page 187 {continued)
6. x-b, {x-a){x-b){^-c). 7. x-\,{x-\){x-2){x-Z){x+2){x+Z).
8. {.i--l)(x+3), (a;-l)(a:+3)(.r2+a;+4)(a;2-6x-4).
9. (a^3)(2a+l), {a-2){2a+l){a+2f{a+Zf.
10. (x--//)-, (a--2/)2(a--22/)2(a;+2?/)2.
11. x-—x}j+y-,{x-—xy+y^f{x-+xif+y% 13. 3.
14. x*-a;2a2+a*. 17. 1, 3. 18. x2-3a;+2, x^-Qx+5.
2x+3
Page 191
7. «^+l 8.^. 9. ^-^ • 10. ^+^
a2+a+l ^--2 4a2+3a_6 2a;3(a;+l)
11 ^1 12 --^-'-^ 13 "'-3 . 14 ^-3
■ x+1 ■ 12x2-7x--4 ■ a«— 2a3+2a— 5 2a;-l
Page 192
1. ^^!±^\ 2. ^^ 3. ^^y 4.
a2-fe2 .r2_y2 x2-2/2 a2— 7a+10
_ 2ah g 2a^ ^ 2xj/ g^ 3.r g ^
a^—b' ' I— a* ' x-—y" ' {x+y)(2x—y)
10. 3 ^^ 3x^-5xy-2y\ ^^ 5
(x+4)(x+5)(a;-f7) x^-y^ x^-5x+6
13.^. 14.-^. 15. ?^2^. 16.0. 17. ,1 .
X— // a^—b^ x—y x^—\
18. ^ 19. - . 20. 0. 21. 2.
2a-36 (a:-l)(a;-2)(a;-3)
a
2(a+l)* a;2_^2' ^43^
_S — 29 ^^'
22. 0. 23. 1. 24. :.-^. 25. ^^ *^
26.-^^. 27. ^. 28. -„^„. 29.^. 30.2.
380 ALGEBRA
Page 195
1
x—a
(ix{a-\-x)
5
26 + 3a
ah{a—h)
g
a;2
a;2— 2/2
1/|
a2+c2
ac(a— 6)
IT
a;+?/
2. ;r - „, 3. ^^. 4.
2a: . -32/
2a— 36 a;— 2 a;^— Qy^
« x'^-\-ax _ a;2— a;4-2 _ 1
6. ~ • 7. — „ — — 8.
a(x—a) x^—\ x—2
10. 0. 11. ^ • 12. — ^-. 13. ^.
6— 3a 3a+2x x"^—!
15. "^ — , . 16
(c— a)(c— 6) (a;— a)(a;— 6)
/ ^2 ■'3- r-4-2- "^9. 0. 20. 0. 21. 0.
22.1. 23_ ^ n-ff ^^ -^^-^,^^., _ 24.-1. 25.0.
{^-y){y-z){z-x)
26. <i. 27. ^„— ,„^ . 28 ^^^
(a+6)(a2+62) (a;2-9)(a;2-25)
16a:
29. . 30.
(a+l)(a+2)(a+3)(a+4) (x2_l)(a:2-9)
Page 198
1.1. 2.^^. 3.^-=^. 4. ^^±^'. 5. -i?* . 6. -i^.
a*— 62 a;+4 a a^—b~ a;*+j/2
1 4. a- ft
-. 8. a«+-. 9. 1. 10. 1. 11. ^.
a^ a* a;-3
12. ^^"-^^ 13. "+*-'. (4. ? + ^+l. 15. ^'-1 + ^'
ax o— 6+c ij '^ j/2 a;2
ANSWERS 381
Page 198 (continued)
20. ^~ . 21 . '^^ . 22. 1. 23. 1- . 24. ^(^+^6)
a2-a;2 a^ a-8 a-66
Page 200
1. _^. 2.1?^. 3.-^. 4.—. 5.1. 6. --
106c 56 a^—b^ xy x b
7.^^. 8.^. 9.1. 10. ^^+i-. 11.^'-
x^+y^ a^ xy{xy—\)
■»2. ^!- 13. ?. 14. a+6. 15. a+6. 16. -?g^^. 17.
t3
Page 201
1. J^. Z.""^^. 3. ^^' • 4.0. 5.-?-. 6.^.
a«-6* 6c a;2-9i/2 \-3» x
7 1. 8. l±^^ 9 ^H^. 10. ^+^ 11. "+^
2 2a; 6+c x2+4a;+3 a^—ab-^h^
12.^^!+^'. 13. 2ia+^. 14. i±^. 15.0. 16.1.
a-\-b a^-{-ax+x^ 2
17. ^?ZlL^ . 18. J_. 20. x. 21. 1. 22. 1.
(2a;+3)(.3x-2) x-1
25. a;+x2. 26. 'J. 27. -1. 32. -+1^ 33. , -J'' ,-•
6 a+6 (a+6)(a2+62)
34. 0. 35. 'ix^-lOx — 3g ^ 3^ xy^
(x-l)(a;-2)(.T-3)(a;-4) x+y
382 ALGEBRA
Page 210
-_ cd—ab . —a ^_ , ._ ab
13. — , ,• 4. • 15. a—b. 1,6. •
a+b—c—d ra+o a+b~c
17 ^. 18 ""+"^+^^ 19 ??. 20 — ^^ 21 "^
■ a+6* ■ a+b ' ' 2 ' ' 2 ' ' b'
22. ^''. 23. a-b. 2^.'^^^. ;»5. ^, "^ .
a^ 2 ■)-\-c b-\-c
__ an-\-b am—b __ oA __ Safe --3a- __ mn{a-\-b)
m-\-n m-\-n a—b o + 3 ?mi — m — n
on ab—cd „. 25 2s— In 2s— an
a-^-b—c—d a+l n n
__ , , sr—s4-a s—a „_ 2s— qt^ 2s—2at
32. s-sr+rl, ^- , -. 33. ^-f-, — .
s—l 2t t^
Page 213
13. a, -b. 14. ^i:^, ^1^. 15. 2a. -b. 16. a,b.
a^—b^ a^—b^
17. -!-^ ^-S --!-^ ^^- 18. b+a,b—a. 19. ^a, lo.
fe.^Ci— 6jC2 a^c.i-a.TPi
22. c, 0, a. 23. 4a-36.
Page 214
25. ''"'-'-^«'. 26. afec. 27. 5,5. 28. 21^i , 27,"',.
a— 36
29. 2, -i. 30. J, 2. 31. -3. 32. $543, $457.
33. ^. 36. $16400, $13600. 37. 1540, 880, 616.
38. ^^"""1+^, 2«Z:^"^. 39. 2aH26^. 40. 35.
2n n^ — n
41. $2100, .$560. 42. 182040.
ANSWEBS 383
Page 219
18. a2+3a + l. 19. 6. 20. -4. 21. (.f4-l)(.T+2)(.r+3).
22. l-x-2a;2, 2-3a;-|a;2.
Paste 221
19. {x+!/)~-2{x+ij) + l. 20. ;i-2-3ox+«2. 21. a^+fts,
22. sHft' 23. .r2 + 2 + i. 24. ~6. 26. 13.
Page 223
21. 2a3+6rt62, 6o26+2fe3. 23. 2a3+2634-6a26+6a62+6ac- + 66c2
27. 27. 28. 242.
Page 225
17. a:2+a;^i. 18. l-2x-+3.r2. 19. •:^_i+?.
3 .r
20. 3a2-4a+l. 21. l-x^. 22. 4c 23. x-l.
24. a-3. 25. .r-2.
Page 226
1, 3x^—4:xy+2y\ 2. x^+2x^-:ix + l. 3. a;6+3a;4-2:r2+2.
4. ix2— |x+l. 5. 5x^—2ax—3a-. 6. 2a;2+3a + 7.
7. (a;+2)(a;+3)(x+4). 8. {x+l){x-5)i2x-3). 9. 2x2-5x-+2-- ■
X
10. 3— 5a;. 11. 2x2-a;+l. 12. a.
13. l-a:-|x2, l_|a-ia2, 2+|.r-J x^. 15. 7a,-2-2x-|.
18. 8a3. 21. 0. -82/3. 24. 2.c3-3a;2+a;-2. 25. 16.
27. 6x-4. 28. 7x2-2x+l.
384 ALGEBRA
Page 230
32. vT3. 34. V2I. 35. 2\/2 4V2. 36. 5V2
37. 4a/2, 12 V 2.
Page 232
9. 10\/;3. , 10. 7\/2. 11. 5\/5. 12. -3\/7. 13. 7\/2.
14. 8v/ll. 15. 7a/5. 16. -41/2. 17. 8-66. 18. 794.
19. 11-62. 20. 5-20. 21. -141. 22. 25 46.
23. ±6-083. 24. ±3-873. 25. ±6-782. 26. ±6-481.
27. ±9-592. 28. ±13-711. 29. 7-483.
Page 234
13. 24\/3. 14. UVI. 15. 5+2v'6. 16. 21-W35.
17. 30 + 12\/6. 18. a+b+2Vab. 19. 2+3^2.
20. 12 + \/6. 21. 6+1/10. 22. 6a+(3b-l3Vab.
23. 6 + 2v/l5. 24. 4v'6-4. 25. a+b-6-V~a+b. 26. 1.
27. 6 + 2\/3+2\/2+2V6. 28. 16+41/10-21/15-41/6
29. 2a + 2i/a2^2. 30. 13j,--5y-12i/x2-"^. 31. 1.
32. 12-4\/2. 33. 6\/6. 34. 70. 35. 30-5\/6.
36. V8+1/7. 37. 42, 43. 38. 46. 39. 9\/2.
40. 30-92.
Page 236
13. 14+'8l/3. 14. 6v/2+4i/3. 15. 5+2i/6. 16."^-^--
17. 1/15. 18. ^'^~^. 19. -577. 20. 3-536. 21. -817.
AN;SWERS 385
Page 236 {continued)
22. -318. 23. 1-491. 24. 084. 25. 1-225. 26. -894.
27. -072. 28. 2-12. 29. -82. 30. 1-39. 31. 3-15.
32. 11-71. 33. ±2-73. 34. 1-008. 35. \ 2.
Page 238
1. 12a/2. 2. 12V5. 3. lOVS. 4. 62. 5. 191.
6. -4V2. 7. 22-12v/2. 8. 12-4V6-2\/3+4V2. 9. ^.
10. 9-4V5. 11. 1. 12. 2\/l3+2\/2. 13. 74+11V6.
14. |. 15. 1-732. 16. \/l2+\/T0. 17. 1, 2.
18. 2\/2, iVe, iV30, iVli, 1{W2-2VS).
25. ±8-661, ±7-937, ±9-899, 1-291, -518.
26. -817, -447, 414, -757, -337. 27. 25^3. 28. 2^2.
29. ^^~ , 4V6. 30. 2-02. 31. 30. 32. 5. 33. 4-83.
Page 241
1. a;2+«-132=0, 2. x2-a;-156=0. 3. a;2-49=0.
4. a;2+6a;-112=0. 5. 5a;2-6x- 440=0. 6. a;2+6a;-9400=0.
7. x-2^19a;+88=0.
Page 242
1. 6, -1, -22. 2. 6, -25, 21. 3. 8, 19, -15. 4. 12, -11, 2.
5. 1, -10, 9. 6. 2, -.5, 2. 7. 1, 4, -32. 8. 5, -27, 28.
9. 2, -19, 44. 10. 2, -5, -3. 11. 0, 2, 7. 12. 0, 1, -1.
C C
386 ALGEBRA
Page 248
1. 4-236, --236. 2. 7-828, 2172. 3. 1-646, -3-846.
4. 1-916, -9-916. 5. -232, -3-232. 6. -851, -2-351.
7. S±Vn. 8. -4±3\/3. 9. |±V2. 10. l±|V4i.
5±V157 ^2. -i±^V22.
6
Page 251
20. 1-618. 21. 2i, -1. 22. 5, 12. 23. 6, -2.
24. 14, 6 or 16, 4. 25. .r=l or 2, 2/=2 or 1. 26. 540.
27. 3-236, -1-236. 28. 20c. 29. 60, 90. 30. 1-449, --949.
31. 20. 32. 3 m. per hr. 33. 8. 34. «'+«^ "'""^
-6' a+b
35. 20, 30. 36. x=2 or ^. 37. 4.
Page 255
18. 2-54, 1-0936. 19. 8 : 27. 20. -192, 1-302.
21. 3937 : 6.336. 22. 4:5, 11 : 27, a+3 : a+5.
23. 11 : 15, 13 : 18, 2 : 3, 3 : 5. 31. "^^^ ■ 32. i±^ .
c—d l+4a
33. 4 : 5. 34. "'^ "^ ■ 35. 10. 39. 206m : an
b+c b+c —
Page 259
29. 11-55. 30. 10^, 4i. 31. AE=6^, DE=lh 32. 240.
33.9-899. 34.2:3. 35. 2, '^-^ ^?^, - ? ■
a — c m—n p
ANSWERS -387
Page 259 (continued)
36. ;■; or H, o or -1. 37. - = ^ = - .
5 3 -8
38. AC=20, AE^o, DE=i. 39. 147 ft. 40. 3 or J.
43. 5:4:2. 44. :i or -i 45. G^ 5A ; ^"^ . ^ .
46. 2 : 3. 47. 17^, 25, 30. 48. 110 : 15 : 17.
Page 262
6. 6 ; 74 ; IIJ^,, 5, 4^^^ : 5, 2|, H. 8. 2, 8. 9. 3, 6, 12.
Page 264
4. f|. 5. -17*. 6. §i. 14. ?^ 15. V.
17. ~^ . 18. 7 : 16.
Page 266
17. "^ . 18."^. 19.2,.^. 20. |. 21. 3, §. 22.5.
23. f = ^ = ^. 24. iV 25. :n/-l. 26. i.
3 4 5 Zi/
32. ±10, ±,5, +5. 38. 41;-;. 39. 1,3,4.
Page 270
1. 3a, —a. 2. b, —56. 3. 3m±mV'6. 4. — 2p±p\/5.
5. cr±\/^2ii^. 6. -fttVftMlc. 7. -1± A + 1.
CC 2
388 ALGEBRA
Page 271
1. 1, §. 2. 1, 1. 3. jjj, -oV 4. i±wn.
5. tV, -VV- 6. :L3±iV89. 7. ^, -^. 8. ^^l. _j^.
9. 2m, -36. 10. 7, V. 11- i -5- 12. l^, -^.
19. ,''^± JoVm. 20. ■i±l\/51. 21. ;], -^. 22. |±2\/5.
32. l±^V2. 33. tVe. 35. l±2v'6. 38. 2414, --414.
39. 3. 43. 6-18, 3-82. 46. 2-786 or --120.
53. 703, 8-78, 8, 2-29, 642, impossible.
Page 276
13. 1, -6, -|±A\/^39. 14. 6, -3, flW^TT.
15. 2, -1±V^. 16. ±2, ±2V^. 17. 3, 2, -5.
18. 1, I, |. 19. -i±|A/=23.
20. 3, h, -i±iV^, -i±|V^. 21. 2, 3, -1, -2.
22. 1, 1, -i±iV5.
Page 277
25. ^^-— ^. ^~^. 26. -4,3, -J.±i\/^^T5. 27. 5, -#±5^/^.
TO+TO m—n
29. a+6±A/a2q^&+P. 30. ^, -- • 31. 15. 32. — , — .
c fl a— c a-'rC
33. -1, -2, -4, -8. 34. 12. 35. -a, -b.
36. a+h, 0, "^+^". 37. 5.10 p.m. 38. a±^. 39. ^, I^'^.
rt->rb a a-\-b a+b
ANSWERS 389
Page 277 (continued)
40. 6076 nearly. 41. -^-^"'+%. 42. li.
32
43. -J;±iv2T, -i±iVl3 44. 10 in. from a corner. 46.27.
Page 281
7. (5, 2), (-^, V)- 8. (3, 2), (^., |g). 9. (6, 4), (-?f, -Jf).
10. (4, 1), (-1, --/). 11. (2. 1), (-5, -V).
12. (2-525, -175), (-2-275, -1-425). 19. 4-196, 4-732.
20. (-2, -1), (-^, 1).
Page 283
6. (2, 1), (19±|1V_5 -7^5)_
7. (4, .3), (-1,-3), (3,2), (-1, -I).
8. (1, 2), (-5, -10), (-3±^89 -9±3v^\
\ 4 16 /
9. (2, 4), (3, 3), (2, -3), (-3, -3).
Page 284
/ 23 n \
'^- (-^-*H*V!1'*^> 13.(±4,±l).(±l.3^/,4,±5^/^).
14. (±1, ±2), (±^^, +^). 15. (0. 0), (1, 1). i^„ ^).
16. (±6-32, ±3-16). 17. 35.
390 ALGEBRA
Page 286
16. (±2, ±1), (±1, ±2). 17. (±3, ±2), (±2, ±3).
18. (±2, ±1), (±1, ±2), (±V^1, +2\/~), (±2V^, tV-I).
19. (5, 2), i-h, -I). 20. (6, 2), (-2, -6), (|±W57)(-|±i\/57).
21 . (5, 3), (3, 5), (6, 2), (2, 6). 39. 7-32, -68. 40. ^b+^Vb^-lQa.
Page 291
18. (2, 2), (2, 1), (1, 2), (2±\/2, 2 + \/2), {^^±W^, ?,+W-l).
19. 81. 21. ^-^-. ^J. 23. [a, 3J, ^3, 6j. 24. §. |.
27. (±10, ±5), (±5\/2, ±5^2).
36. (3, 2), (2, 3), (-2±\/^, -2 + \/^).
39. (-1, 2), (2, -1), (-i±^\/l3, -i±i\/T3).
^Q_ ^+V2^'-g'^ _s+V2^2-s2_ ^^_ ^^^ ^^^ ^2, 2), (1, 12), (f, 6).
42. (4, 2), (2, 4), (8, 1). 43. (5, 1), (1, 5).
Page 296
he
19. — . 20. 3^4'' + 4''. 21. 1. 22. 1. 23. 0". 24. 1.
25. 22", 312. 26. 3. 27. 2, 9 28. 2, 7, 3, 2.
Page 301
45. I 46. 8. 47. 625. 48. ll^. 49. 125. 50. J.
51. ,V 52. 32. 53. 4. 54. 5. 55. ^, 56. }^.
57. #-. 58. ^a^y". 59. 16, 8, 81, 1, \.
ANSWERS 391
Page 302
1. x+x^—6. 2. xi—1 3, x'—l 4. 3x2— 8a;'-+9.T— 10x5
5. a— l+4a"^— 4a-i. 6. a^~2a^ + 3a—2a^ + l.
7. x++4x— 11x2— 6x3^. 8. x2+8x'^'+24x+32x2 + 16.
9. x2+x?/+r/2. 10. ai—3a+3a^—l. 20. x^+x+l+x^^+x-^.
21. 5a2'«+4a'"-2. 22. 2x2+6x+2. 23. l-2a.
24. X— 2^x4-3.
Page 304
1. x-4, a3_i3_ 2. a+i+a-\
3. x2— 2x?— x+2x5 + l, 4a2-8a+4a-i+a-2
4. a^+3a+Za^ + l, 1— 3x2 + 3x— x'^. 5. x^+xV+y*-
6. x^+i/l 7. a^+fti+ci 8. (x+2/)(a----2/-)(a:^+A
i 1 r2_LS 2 1 1
i-^-b--^. 10. ^If^t^, a3+a^&3-
x3-3
11. x^-2, x-x^4-l. 12. 2x-2+3x-V x+2-x-i.
9. a2_62. -10. ^iXf, a3+a^&3+6^- a-V^+6.
x3-3
Page 305
3. 5, v„ 49. 4. 4, .;,, 25, 4, ,V 8, ,V 6. 5,i„ 2.
7. 31G2, 1-778, 1-333, 5-62. 8. 4. 9. 1-732.
10. 9f, -lig. 11. 100. 12. 4, 2. 13. ^, H.
14. f, f. 15. ^^^, 3x^+2, -^-^— . 16. x2t/*+8xV
Va+4 a^6-62
392 ALGEBRA
Page 305 (continued)
17. a;^?/-i + l+.r~^?y. 18. x^—y~^, a^— l+a~^.
19. a^'"-^a^"'h"'^a"%-"'-\-h^"'. 20. 2/+27/- + 1. 21. x—2—x-\
2!2. ha'-—%bK 24. -0016, 1-44, 3-375, 8. 25. Va -1\ Va + l
26. 1. 27. e'-+e-^ x^— 2a;?/2-j-3a;ii2/— 2/2. 28. 06c.
30. 4, 32. 31. 2750. 32. J. 33. 2, 3.
34. a;3+2a;^+l+2x"^+a;"t 36. 2a;+a;^—a;~l.
Page 310
7. ^4, ->^27; v^Ie, V'27; ^64, ^ST, ?L25.
8. 3V2, 5\/6, Vs: 1-26, ^5. 9. 12\/2. 10. 12\/5.
11. 33\/2. 12. 3^2. 13. 7\'T2. 14. 10\/2. 15. 9\/3.
16. 0. 17. \/3, a/5, .t\/^, 2/^4^, \/2.
18. 2-52, 3-78, 12-6, -63, -126, 1-26.
Page 311
25. 3\/rO. 26. IVS. 27. fVB. 28. V2.
29. 1(2a/2-V3^. 30. \/rt2+62+6. , 31. \/^6-Vc.
1 ,,
32. -(x-Vx^-y^). 33. 2-517, 1-354. 34. 194.
35. 27(\/.3-\/2). 36. ^Z(7-V5), 2\/5. 37. ^^(18-34^5)
39. 10 ft. 5 in.
4
ANSWERS 393
Page 314
16. No root. 17. 4. 18. 5^. 19. 100. 20. 9.
21. -a. 22. 25. 23. ()4. 24. Xo root. 25. fc^ •
2«— 6
26. 3. 27. 10. 28. 10. 29. V. 30. 4^
C2 + 1
Page 317
18. (4, 9), (9, 4). 19. (4, 10), (16, 4). 20.(17,8).
21. (9, 1), (1,9). 22. (2, i), (|, 2). 23.2,1. 24. 7, -G.
25. (2, 8), (8, 2).
Page 320
10. 2-823. 11. -196.
12. 2^(2+\/3), 5T(V.5+1), 3+(2-\/3), 2T(5\/2+3). 14. l+VS.
15. ^r\ ^^t\ ^^t^- 17. 2-309. 18. 24-3V2.
a/2 V2 VS
Page 322
10. 25\/^. 11. 68. 12. -25. 13. 1+V^.
14. ~^~^~^. 15. 2a2_262.
2
Page 324
•1. 2+2\/2-2\/3. 2. 2 + ,';\/6. 4. 1-98, 3-15, 1-39, 3-55.
5. 9. 6. Hg. 7. 7!. 8. 7, -1. 9. 20. 10. 13.
11. 6+2\/l5, a;24-2.r?/+j/2-4x-4?/. 12. ;\/3.
13. 4a+2\/4a2-6 14. A(M2_2pfl_^o2\ 15, 12.
2?
394 ALGEBRA
Page 324 (continued)
16. x*-lx^+2x+2. 17. 0. 18. 2a, Aa^-2, Sa^-%a.
,~ 2m
19. 4, 7 20. v/3, ,__=-• 21. 5.
22. y^ , \/a+\+\/a-\. 23. 4, -7. 24. 40.
25. 2-62, -38. 26. ^Va^^^^. 29. 1. 30. 10\/2.
b
31. 16+9\/3.
Page 330
17. x^—2mx+m^—n^=0. 18. x^— 4aa;+4a2— 62=o.
19. a:2-6x+6=0. 20. 16a;2+8a;-63=0.
21. a;3-28x-48=0. 22. 24a;3-26a;2+9a;-l=0.
24. 4x2 -28.r+ 45=0. 25. x2-7x+12=0.
26. a;2(a2-62)-2.T(a2-j-62)4-a2_52^0, 4x2-16x+9=0.
27. 9, 7; a+6, 0; 2;?, pq; 2c-2a-2b, a^+b^-c\
28. 0, 5, -1, -4. 29. 4, 8, 30. ±16.
Page 334
I. If, 6^, 22. 2. 47, -II. 3. -2g, 1^, 6J.
4. _5g, i_3a. 5. x2-18x+80=0.
6. x2-5x+4=0, x24-5x+5=0. 7. 6x2— x— 2=0, 5.^2—^:^+1=0.
8. 5x2-2x+3=0. 15x2 + 26x+15=0, 9x2+26x4-25=0.
9. a'~-2b, 'iab-a^ 10. x^-x{p^+2q)+q^=0.
II. ax'--x{2ah-b)+ah^-bh+c=0. 12. x2-4x-4=0.
14. x2+6a:+8=0. 16. i(62-a«)(62_3ac).
AN8WEBS 395
Page 338
6. Rational. 7. Real and irrational. 8. Imaginary.
9. Real and equal. 10. Rational. 11. Real. 13. 4.
14. ±5. 16. -L 19. -• 20. 2, -j.
m
Page 340
8. {x+2 + Vl){x+2-Vl). 9. 16. 10. ±6ff.
11. (.c-3+2\/5)(x-3-2\/5).
12. (3.r-4i/)(3:c+42/)(4x-3 )(4.t + 3//) ; ±|, ±|.
13. 174; (8a:+7)(15a:-4). 14. 62=4ac.
Page 341
4. 6, ^, |, 4, -14. 5. B, L 6. 16.x-2-40a;+21=0. 7. 12*.
8. 6x2- 19a;+ 15=0. 9. _25. 10. a;2± 12a; +35=0.
11. (72x+l)(73.T-l), (13.T+ll)(17a;-15). 12. a;2-4a;+3=0.
13. 2a+26— 2c. 14. 97a;2— 53a;- 17=0.
15. {x+3+V2){x-\-3-V2). 16. 2a;2-17a;=0.
17. a.T2-f36a;+9c=0. 18. acx^—x(b^—2ac)+ac=0.
19. 1, ^±^-^. 23. ±12. 26. ±4. 27. Zm,n-m^ 28. a.
a-\-b-\-c
29. ^"L, _^_ . 30. i 31. c+6-a. 32. a=8or0.
a+b—c a+b—c
33. 8, 1. 35. 6, 2 36. (x+b+Vb^^}{x+b-\/b^^).
396 ALGEBRA
Page 345
16. 2«-ft+3. 17. y-7g+3r^ ^g^ -7,1,19.
p—q+r
19. (.T-?/+3)(.T + 2/y-4). 20. 2a-3, 3a-4.
21 . (2ffl-?>+c)(3a-6-c)(3fl+2fe-2c.
Page 348
13. {2x-y-5z){Ax'^+y^+25z''-+2xy+\0xz-5ijz).
14. (a+6+c+l)(a2+62_^c2+2a6— ac— 6c— a— fe-c+1).
15. a^—b^—c^—Zahc. 16. 8a;3—?/3+ 2723+18x2/2.
17. l-f(3_63_3g,j_ 18. 8a3_2763-64-72o6.
19. l+a2_|_^2_^p(_ft_^(j5_ 20. ^m'+n^-\-\ + Zmn+Zm—n.
21. a+56-1. 22. 3ffl+6. 23. a;2-3.r+9. 24. 2a-5.
25. a+6+c+d-l. 28. 27a:='-8?/H3^+18.ryz. 35. 0.
36. «!±^M:^6n:.ac-6c^ x+2^^ ■ 37. (,+,+,)(«+,).
2a— 36+ 3c 2
40. a, 6.
Page 351
9. (.r— ?/)(?/- z)(a:— 2). 10. (x—y)(y—z){x—z).
11. (.r-;v)(?y-2)(2-aO- 12. (ff-6)(6-c)(c-a)(o+6+c).
13. (f/-6)(6-r)(c-a)(fl + 6)(6+c)(c+rt). 14. a+6+c.
15. ^, 'I. 16. ^^^ ^-^- 17. a, 6. 18. ^^ ""^ •
a b a—b b-\-a a h
19. t ^. 20. ax~a-b.
a b—a
ANSWEMS 397
Page 353
10. a,-«-2:i;3+4a;2-8.t-+16. 11. .i^-3x-+9x-21.
12. x^+x^a+x*a'^+X'a^+a'^. 13. (a^bf-{a+b)-+a+b—h
14. x—b, u+b, x—4:,m+-,x+y—l. 15. a^-l. 16. /w^+1.
m
17. a8-68. 20. a2^a?)+g=0. 22. 1,9.
3 1 1 2 2 1 _1_ 3 4 :•. 1 2 2 13 4
4 :i X 2 2 1 a 4
x-^ —x-^a-^ -\-x^a^ —x^a'' -\-a" .
Page 357
13. 3(a2+62_|_c2)_2(a6+6c+ca). 14. 2(a2+6Hc-). 15. 0.
16. —Z{a%—ab'^+b'h—bc-+c^a—ca-). 17. {x—y){y~z){x—z).
18. (x-«/)(2/-2)(x'-z). 19. (a+6)(6+c)(c+a). 20. 24a6c.
21. Z{x-y){tj-z){z-x). 22. (a + 6)(6+c)(c+a).
23. — (a-6)(6-c)(c-a)(a+6+c).
24. («-6)(6-c)(c-a.)(a+6)(6+c)(c+a). 25. 1. 26. 1.
27. 1. 28. -4-- 29. 0. 30. a+6+c. 31. -{x^y+z).
abc
32. 3. 33. 6a6c 35. 3(x2+.(y2-fz2)-f-2(a;2/+jy2+za;)-
Page 362
24. 18. 26. a2-262 a^^Sab-. 27. a2=c24-262.
28. aH2c=»=3a62
398 ALGEBRA
Page 362
6. W+^,y^-,hz^+l^yz. 9. x^{a-\){\-h). 11. ^^.
12. Ua+b+c). 15. -(a6+6c+ca,). 16. "^ ■ 18. -'(a+ft+c).
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22. (a;-l)(x3-2.r2+2.r+2). 23. -a-b-c. 26. (a;+l)(a+6+c).
29. 3(a;2+2/2+22) + |(^2/+^2+2x-). 30. a2=26+c. 31. 9,17.
32. 38, 70.
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