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► 



flarbarli College Htftrars 




LIBRARY OF THE 



DEPARTMENT OF EDUCATION 



COLLECTION OF TEXT-BOOKS 

CONTRIBUTED BY THE PUBLISHERS 




TRANSFERRED 

TO 

HARVARD COLLEGE 
LIBRARY 



3 2044 097 014 583 





VIETA 



DESCARTES 





NEWTON 



GAUSS 



^ 



DURELL'S ALGEBRA 

TWO BOOK COURSE 
BOOK TWO 



BY 

FLETCHER DURELL, Ph.D. 

HEAD OF THB MATHEMATICAL DEPARTMENT IV THB 
LAWBENCEVILLE SCHOOL 




CHARLES E. MERRILL COlVt-fc». vlii'^ 

NEW YORK AND CHICAO^~>^ 



D pt. '!« •". 






JUL -6 1915 

TRANSFERRED TO 
HARVARD COLLEfiF,.,ff^jy^ 



URELUS MATHEMATICAL S: 

ARITHMETIC 

Two Book Sbsibs 

Eiementaiy Arithmetic, 40 cents 

Teachers' Edition, 60 cents 
Advanced Arithmetic, 64 cents 
Three Book Series 
Book One, 60 cents 
Book Two, 66 cents 
Book Three, 60 cents 

ALGEBRA 

Two Book Course 
Book One, $1.00 
Book Two, 96 cents 

INTRODUCTORT ALGEBRA, 60 CentS 

School Algebra, SLIO 

GEOMETRY 

Plane Geometry, 76 cents 
Solid Geometry, 76 cents 
Plaub and Solid Geometry, $L26 

trigonometry 

Plans Trigonometry and Tables, $1.26 
Plans and Spherical Trigonometry 

AND Tables, $1.40 
Plane and Spherical Trigonometry 

with Surveying and Tables, $1.60 
Logarithmic and Trigonometric 

Tables, 60 cents 



COPTRIOHT, 1911, 1915, 

By CHARLES E. MERRILL CO. 



PREFACE 

The present volume contains a Second Course in Al- 
gebra adapted to the latter part of the high school curric- 
ulum. The methods which are characteristic of the 
author's Algebra, Book One are here continued and 
developed. The chief aim is to simplify principles and 
give them interest^ by showing more plainly, if possible, 
than has been done heretofore, the practical or common- 
sense reason for each step or process. Each new process, 
for instance, is introduced by what may be termed the 
efficiency-inductive method. In the Exercises also there 
are special examples which cause the pupil to realize the 
efficiency meaning of processes from various points of view. 

As in the author's other mathematical texts, pivotal and 
permanently valuable number facts and laws from other 
branches of study are introduced in various ways. This 
gives a correlation of algebra with geography, history, and 
other subjects. A further correlation with physics and 
engineering is obtained by the use of some of the most 
important formulas in these branches, and also by famil- 
iarizing the pupil with their fundamental concepts and 
number facts. 

As in Book One, special attention has been paid to 
arranging written problems according to types, and to 
making each problem real and full of human interest. 
Similarly, the graphs of statistics and other graphic prob- 
lems are made to illustrate important efficiency principles. 

3 



4 PREFACE 

The self-activity of the pupil is aroused and developed 
by examples which require him to invent and solve prob- 
lems of special types, these problems being carefully varied 
and graded. 

Much oral or sight work is called for, and thorough mas- 
tery of principles is further assured by a series of spiral 
reviews similar to those characteristic of the author's other 
textbooks. 

Attention is called to the way in which the history of 
algebra is treated. The successive changes in algebraic 
symbols and processes which have been made in the past 
are shown as a series of improvements, and examples in 
various places in the book compel the pupil to realize 
these increases in efficiency. 



CONTENTS 

OHAPTBB PAOX 

I. Review Exercises in Fundamental Processes; 

Detached Coefficients 7 

n. Factoring; Factor Theorem; Highest Common 

Factor and Lowest Common Multiple . . 22 
HI. Review Exercises in Fractions; Fractional 

Equations 32 

IV. Simultaneous Equations 49 

V. Graphs 67 

VI. Involution and Evolution 81 

VII. Exponents 92 

VIII. Radicals 106 

IX. Imaginary Quantities 118 

X. Quadratic Equations of One Unknown Quantity 126 

XL Simultaneous Quadratic Equations . . . 146 

XII. Graphs of Quadratic and Higher Equations . 164 

Xin. General Properties of Quadratic Equations . 174 

XrV. Ratio and Proportion 181 

XV. The Progressions 197 

XVI. Binomial Theorem 221 

XVII. Inequalities; Variation 228 

XVIII. Logarithms .239 

XIX. History of Elementary Algebra .... 255 

APPENDIX 

Fundamental Laws of Algebra .... 266 

Cube Root 267 

General Review Exercise 270 

Material for Examples 278 

INDEX 287 

5 



BOOK TWO 

CHAPTER I 

REVIEW EXERCISES IN FUNDAMENTAL PROCESSES; 
DETACHED COEFFICIENTS 

EXERCISE 1 

In each of the following examples, state the order of 
operations before working the example. Wherever pos- 
sible, use cancellation. 

When a = 6, 6=3, c == 1, x = 6, find the numerical 
value of 

1. 8a-|-2(a:-0 ^ Bg^-y 

2. (3a-|-2)(a:-c) ^"2<? 

3. 3^2-1- 7a-6(a-6) g (x- 0(2 6- a) 

4. 3^.62-a^ ' 3(62-f.)-2a 

7. 7rr(5a — 4a;)— ca:^ 

8. a(3a;-a6)2-|-rr(a262-a^ 

9. 3rr(a;-a)(a?»-a62-|.2ac2) 

If a =s 4, 6 = f , c = 0, rr = 1, y = 9, find the value of 
10. ■\/2ax 11. Vo^ 12. 3 6V668? 



13. a:V7 2^2 - a6 16. 3a-2VIFT3^ 



14. 3 (?a; Va2 -|- b(fl 17. Zx^aby - bx-V^c^ 

15. a+V^ .18. (a-2 6)Va8T4^ 



7 



8 DURELL'S ALGEBRA: BOOK TWO 

Express in algebraic symbols Exs. 19-20. 

19. The square of the sum of a and h equals the sum of 
their squares increased by twice their product. 

20. Four times the quantity a square minus nine ( is 
less than the square of the quantity seven a plus h cube. 

21. If jr=s J 6A, find the value of K when 6 = 28 and 
A = 13. Also when J = | and A = |. Also when h = 1.84 
and A =.92. 

In geometry, what is the meaning of the formula K—\hK1 

22. If K^irm and 7r = 3.1416, find the value of K 
when R = 25. Also when R = ^. Also when R = .08. 

In geometry, what is the meaning of the formula K = irR^t 

23. The formula for the area of a circular ring is 
K='ir(^R?—7^^. In the shortest way, find the value of 
JK" when ^ = 13 and r = 12. Also when ^=.82 and 
r = .18. 

24. If c2 — ^2 ^. j2^ fijijj ^)^Q value of c when a = 3.3 and 
6 = 56. Also when a = 4 f t. and 6 = 4 f t. 7 in. 

What use is made of the formula c^= a^-\- b^ in geometry? 

25. If 8= ^fffi^ find the value of « when g = 32.16 and 
f=3.2. 

Give the meaning of the formula s = i gfi, 

26. A stone dropped from the top of a monument 
reaches the base of the monument in 4.5 seconds. How 
high is the monument ? 

27. If (7=f(^-32), find (7 when ^=113°. Also 
when F^ 1800°. 

Do you know the meaning of the formula used in this example ? 

28. Make up and work an example concerning the 
formula 8=ivt. 



FUNDAMENTAL PROCESSES 9 

29. Make up and work an example concerning V=s Iwd. 

30. Define the word term as used in algebra. How 
many terms are there in each of Exs. 1-5? 

31. Define binomial Trinomial, Polynomial. Write 
an example of each of these. 

32. Define power. Exponent. Write a power and by use 
of it show the difference between a power and an exponent. 

33. Write a polynomial of the fourth degree arranged 
according to the descending powers of x. 

34. Write a polynomial of the sixth degree containing 
the letters x and y and arranged according to the ascend- 
ing powers of y. 

35. Make up and work an example similar to Ex. 6. 
To Ex. 23. 

EXERCISE 2 

Add and check each result : 

1. 2. 3. 4. 5. 

-.5 -8a: -3a62 Ta^y^ -2.5aV 

27 6 a? -Saja -Zx^y^ -^1.1 a^a? 

-15 -3a: 12ai2 -9a?y2 .9^2^ 

-3 -5 a: -a62 a^y^ -^ .Z a^a? 

7 X -3ai2 2x^y^ h c?^ 

6. 7. ^ 9. 

K^-^y) -4(a-6) 8 Va-a: ^-nr^ 

-5(a: + y) _7(a-6) - 9 Vg - x -^Trra 

-2(a; + y) (« - *) -V a-a: - 1 TrrS 

7(a:4-y) 6(a-6) -2Va-a: \irr^ 

Reduce each of the following to its simplest form : 
10. a:— 2y-|-8a:-|-4y '\-x 4-5y — y — 5a:— 5y4-a: 
U. aa_ ^5 + 3 52 + 2a2+ 2 06 + 3 a2 - 06 ^h^^a^ - 2 62 



10 DURELL'S ALGEBRA: BOOK TWO 

12. Tj^ + J + Sr — 10jt>g'-h2jt> — 3r + 13jt>r-4jt>r + 5r 
^5pq+p^l6r + 21pr — pr + 7p+9 r-'Sq+2pq'-4:pr 

13. ^x^iy + z-^ly + S^z + ^x+ix-^z+ly-S^x 

14. Bab + .2ac+ .75 be- .02 oJ - .36 ac+. OOS bc+.6 ah 

15. Reduce 6 aabbb — 3 aabbb •— 7 aabbb + 8 aabbb to its 
simplest form. Compare the number of symbols in the 
two forms. 

Subtract and check each result: 



16. 

aa-.3aft + 26a 
3a2-5a6-7J2 


17. 

9(2: -y) 
-2(.:-y) 


18. 19. 

— 7 Va -X a + b 


20. 

(2a-3J> 


21. 

(a + b — c)x 
(b + c — d)x 


22. 

(3a-2J-(?)y 
(6a-26 + i)y 



Find the expression which must be added 

23. To 2 a - 3 6 to make 6 a - 7 6. 

24. To3a2-2a + l tomake7a8-2a2-3. 

25. To a?'» + 5ic* + 7 tomake 8a?*-3a;*-2. 

26. From the sum of 3a-}-56— 2(? and 7a— 26 + 5<? 
subtract their difference. 

27. From the sum of (5 a — 3 h)x and (2 a — 5 6)2; sub- 
tract their difference. 

If ^ = 3a?- 2 ic + 5, 5= 5 2?- 7, (7= 3 a? -2, find 

28. A + B-C 29. B-A-O 30. C-A-B 
31. If Rome was founded in the year — 763 and cap- 
tured by the Goths in the year + 410, find the number of 
years between these dates. 



FUNDAMENTAL PROCESSES 11 

32. Hydrogen melts at a temperature of — 256^ C. and 
ammonia at — 34° C. Find the difference between these 
temperatures. 

33. Make up and work an example similar to Ex. 15. 
To Ex. 27. To Ex. 31. 

1. Mnltiplioation by the Use of Detached Coeflkients. The 

process of multiplying one polynomial by another can often 
be much abbreviated, and the likelihood of error dimin- 
ished, by detaching the coefficients of the terms of the 
polynomial, performing the multiplication with respect to 
the coefficients alone, and then supplying the proper 
powers of the letters in the product obtained. Thus, 

Ex.1. Multiply 6a;8-6a?-4a;-3 by 6a?+5a?-4. 

Detaching coefficients, 6 — 5—4—3 = — 6 

6 + 5 -4 =_7 

36 - 30 - 24 - 18 

30 - 25 - 20 - 15 
-24 + 20 + 16 + 12 



36+ 0-73-18+ 1 + 12 =-42 
Annexing the powers of x, 36 x» + x* - 73 a:« - 18 x« + a: + 12 
Or 36x« - 73a:« - ISx^ + x + 12. Product. 

The check is obtained by letting x = 1. 

By performing the above multiplication in fall, the pupil will find 
that by the use of detached coefficients the work is shortened by at 
least one half, and that the likelihood of error is also diminished. 

Ex.2. Multiplya? + 3a2a;-2a8by a?-4aa?+3a8. 

1+0+3-2 =-2 

l-,4.f 0-1-3 = 

1+0+3-2 
«4-.0-12 + 8 

+ 3+0+9-6 



1.4+3-11 + 8 + 9-6 = 
Hence, x« - 4ax« + 3 a«x*- 11 a^x* + Sa^x^ + 9 ah: - 6a«. Product. 



12 DURELL'S ALGEBRA : BOOK TWO 

EXERCISE S 

Multiply by the use of detached coefl&cients and check 
each result : 

1. 3a; + lbya? + 2 2. 3a; + 4y by 4ic— 5y 

3. a?-3ic + lby 2a;-3 

4. 3a2-4aa; + jr2by 2a- 3aj 

5. 2y8-4y2+y.iby2y-5 

6. 4a? + X'-2hySa^-x-5 

7. 2a?+3a?t/''4:xt/^ + y^hy Sa^ + if^ 

8. 5a;8 + a;- 5by 3a:2- 2ic-4 

9. 4a8+2a26-68by 3a8-a62 + J8 

10. 3a;3_4j^y + 3^y2_2y8by a? + 3icy + 2ya 

11. 2a^-^ia?'^Sx^-'2x+lhyl + 2x-{'a^ 

12. 3a:2+6a; + a;^ + 15by l + a?-3ic 

13. 2ir8-a;-5 + 32:2bya;-5 + 22;8_3j^ 

14. la^ + ^ab + ^b^hj^a-^b 

15. |a;8«ja? + ia: + iby|a;8 + |^«|^ + ^ 

16. 1.2a? + 1.5a; + 6.4by 2.4x-3 

17. 1.8a2-3.2a+.48by 2.5a+.5 

18. 4 ic«+i - 3 x« + a^-i by 2? + 2 ic + 1 

19. 3a:'»-i + 2ic~-2-3a;»-3by2a?-3a; + 2 

20. What is meant by the degree of a term? Write a 
term of the 2d degree. Of the 3d degree. One of the 
5th degree and containing two literal factors. Also one 
of the 8th degree and containing three literal factors. 

21. What is meant by a homogeneous polynomial expreS" 
Bion ? In the above examples, point out three homogeneous 
polynomial expressions. 



FUNDAMENTAL PROCESSES 13 

22. Also point out an example in which both the mul- 
tiplicand and the multiplier are homogeneous expressions. 
Show that in this case the product is also homogeneous. 

23. State the rule for signs in multiplication and illus- 
trate it. 

24. State the rule for exponents in multiplication and 
illustrate it. 

Multiply and check each result : 

25. 5(a: + y)2-4(a; + y)+2by 8(2; + 3^) 

26. 6(a:-y)2+3(a:-y)-2by2(a;-y)~5 

27. a?+ t/^ + z^ + 2 xy --XZ— t/z hy x + y +z 

28. a^-{-h^ + e^'-ab — ac--bchya + b + c 

29. ofl + y^ — z^'- xz + xy^yzhy x — y — z 

30. In Ex. 10 perform the work without the use of 
detached coefficients. Then multiply by use of detached 
coefficients. Compare the amount of work in the two 
processes. 

31. Make up and work an example similar to Ex. 9. 
To Ex. 14. 

32. To Ex. 17. To Ex. 18. 

2. Division by the Use of Detached Coefficients. 

Ex. Dividea^- 2a?^y2_8a:2/«- 8/by a?+ 2xy-y\ 

1.4-0-,2 + 8-3 |l + 2-l =4-^2 
1+2-1 1-2+3 =2 

-2-1+8 
-2-4+2 

3 + 6-3 
3 + 6-3 



Hence, a:^ — 2 a:y + 3 y*. Quotient 



14 DURELL'S ALGEBRA: BOOK TWO 

SXERCISS 4 

Divide by the use of detached coefficients and check 
each result : 

1. 8a?*-2a^+a?-lby2a; + l 

2. 6a:8-19a:^ + 21a;y«-10y»by Saj-Sy 

3. 2a:6-5a?*-2a;8+9a:2_7^^3by 2a? + a^-8 

4. Sa^y-lOa^t/^ + ia^ + exf/^ + S^ by Sa^ --^xy^ 

5. afi - 5 a^y -{- 10 a^y^ - lOa^V + 5xy^ - y^ hy a? - 
3 xh/ + 3 xy^ — y^ 

6. a* + a^ja + J* by a2-a6+J^ 

7. 8ir8-a:»-20-5a;+a^-.7a^+17a? by 5 + a^- 
3a? 

8. 16a^ + 81y* + 36a?^by4a:2 + 9y2+6a;y 

9. -a:6+2aa^ + 8a2aj8-16a8a:2_i6a4>i;+32a« by a? 
-6aa?4-12a22:-8a8 

10. Ja;8_i7|^4.2by |a; + | 

11. ^-|a;8 + ^^_^^ + ^by2?-ia: + i 

12. .Iaa-.23a6 + .126aby .2a-.36 

13. 4.5a?-7.1a?-.4a; + .24by 1.8a^^-3.2a; + .48 

14. ar»+*-ic'»-i-62:*»-a_2ic + 4by ic-2 

15. ar^+i - rc2n - 2 ar»-i + 3 ic*»-a - 10 a^-8 by a;~+i - Saf 
+ 4 ar»-* - 5 a^-^ 

16. When a homogeneous expression is divided by 
another homogeneous expression is the quotient also homo- 
geneous? Illustrate by one of the above examples. 

17. State the rule for signs in division and illustrate it. 

18. State the rule for exponents in division and illus- 
trate it. 



FUNDAMENTAL PROCESSES 15 

Divide : 

19. 12(a: + yy - %(x + y)8 - 16{x + yY by 4(a: + y^ 

20. 9(a - 6y - 12(a - 6)8 + 15(a - 6) by 3(a - 6) 

21. a8 + 68 + a:^-f-3a62 + 3a26by a + 6 + a; 

22. a:^ _|_ y8 _ 2^ + 3 a% + 3 icy^ by >|; + y — « 

23. a^ ^ j8 + ^ _ 3 ^j^ by a + 6 + (? 

If a; = 0, y = 2, 2 = — 3, a = 1, find the value of 

3 



24. 


xyh 


25. 


xy^ + z 


26. 


Za^y 


27. 


Zc^x 


28. 


5 + a2x 


29. 


4 a% + 5 a^^ 



30. 



31. 



32. 



bx-Zc^z 



3agy + g8 
x-Vy 36. icz^ + 5 a* 



33. 


y^i 


34. 


y + 5x 


35. 


a 



37. In Ex. 5 perform the division without the use of 
detached coefficients. Now divide, using detached coeffi- 
cients. Compare the amount of work in the two processes. 

38. Make up and work an example similar to Ex. 5. 
To Ex. 12. 

39. To Ex. 14. To Ex. 24. To Ex. 33. 

EXERCISE 6 

Simplify and check each result : 

1. a-[2a + (a~l)] 

2. 2.7 4- {-.4 -(3.07 -.015) I 

3. 5-{-3 + [4-(2-l)]-3} 



5. a;- (re - y)- { - a; - [- (ic - y) - (a; + y) - a;] - 

6. a-2(a-3) 8. 3ic- 2(5rc2- 3ic- 2) 

7. (a-2)(a-8) 9. (3aj- 2)(5ar2-3ic-2) 



16 DURELL'S ALGEBRA: BOOK TWO 

10. 3ic + 5(2: + y)-y 12. (ic- 2)(a:-3)(ic-5) 

11. (3a: + 5)(a? + y)-y 13. a;- 2(ic- 3)(ic- 5) 

14. 56a-(6-a:X2i-3ic) 

15. 5-3(a~2)2-2(3-2a)(l-f-a) 

16. 3a2-[ic(a-a;)-a(a:-a)]-2a? 

17. 3(a - 6)2- 2 Ka + i)^- (a - 6)(a + 6) | + 2 ja 

18. 3[(a; + 2y)-56y]-5[(2:-3)y-hJic] 

-4(aj-y)(3a:-2y) 

19. In the shortest way, multiply the sum of (a? — 2 y)^ 
and (2a;-y)2by 3a:-2(a:-y). 

If a; = 3, y = 0, a = — 2, J = — 5, find the value of 

20. Hhf-^-SaQa—b') 21. 4a2 — aa;y(4a — J) 

22. 2(a2 + 6) - 6a;y + a^a; 

23. 3 a:y(a - 6)2- 5 y2(2 a - 3 6)2 + 6 a^(2 a -by 

24. 3 a;(a; - 2 a) - { X- (a; - 1) (a + 1) - (a + a;)2} + 5 aa; 

25. From 3 times the product of a:+ 5 and 3a;— 5 sub- 
tract 5 times the product of 1 — 2 a; and 2 a? — 3. 

26. From the sum of 2x + 5i/ and 3 y — 5 a; subtract 
x — l y. Verify the result when a: = 1 and y = 1. Also 
when a; = 2 and y = — 3. 

In each of the following expressions, insert the last 
three terms in a parenthesis preceded by a minus sign : 

27. a;8-5a^*-3a;+l 29. a + h — c — d 

28. a^ + bofi'-Zx + l 30. a—b-'C'-d 
Collect the coefficients of a;, y, z in 

31. 5a;-|-3y + 72 — aa;+6« — ca; — dy + a« 

32. -3y-|-7a; + 4«-aa;-26y-2aaj + 56y + 7« 

33. Who first used the letters a;, y, and z to represent 
unknown numbers in equations in algebra? (See p. 256.) 
Find out all you can about this man. 



FUNDAMENTAL PROCESSES 17 

34. Give some of the other symbols that were used to 
represent unknown numbers before the use of the last 
three letters of the alphabet was suggested. 

35. Can you point out any advantages in the use of 
X, y, and 2 instead of the other symbols once used for the 
same purpose? 

36. Find out, if you can, whether any other symbols 
than the last letters of the alphabet are now used to repre- 
sent an unknown number in an equation? How many 
different symbols can be used for this purpose ? 

37. Who invented the parenthesis sign and when ? 

38. Make up and work examples similar to such of the 
above examples as the teacher may indicate. 

Abbreviatb^) Multiplication and Division 

3. Type Forms of Abbreviated Mnltiplioation. The labor 
involved in the multiplication of algebraic expressions 
may frequently be much diminished by the use of one or 
more of the following relations : 

I. (a+ft)2 = a2 + 2aft+6?. 

Ex. (4 x2 + 3 y)a = lQx* + 24 x^y + 9 y^. Product. 

II. (a-ft)2=a2_2aft+»3. 

Ex. 1. (3 a?"-i - 2 a:y»+i)a = 9x^-^ - 12 a*y"+i + 4 xhj^+^. Product. 

Ex.2. (x + y-2zXx-y-\-2z) = Ix +(y - 2z)'ilx -(y -2z)} 

= x^-(y-2zy 
= x^-ly^-iyz + ^z^) 
= x^-y^ -\-4:yz- iz^. Product. 

III. (a+b+c + df:=d^-\-lf^+(?+d^ + 2ab + 2ac 

+ 2ad+2bc + 2bd-\-2cd. 

Ex. (2 X - y - S zy = i x^ + y^ + 9 z^ - i xy -12 xz + Q yz. 

Product. 



18 DURELL'S ALGEBRA : BOOK TWO 

IV. (jc+a)(jc+&)=r»+(a+ft)jc + aft. 

Ex. (x + 3 y) (a: - 2 y ) = x* + a:y - 6 y^. Product. 

V. (ar+ft)(cjc + d)=acx3 + (aif+6c)jc+M. 

Ex. (3 a: + 5)(2 a: - 7) = 6 a:« - 11 ar - 35. ProducL 
£XBRCIS£ 6 

Write at sight the value of each of the following and 
cheek each result : 

1. (3aj-5)a 6. (82?- 5)(4a?^ 7) 

2. (3a: + 5)2 7. (3a;8_ 5^,^)2 

3. (3a; + 6)(8a;-5) 8. (a + J -h 0(« + * - 

4. (3a: + 4)(2a:-5) 9. (l-2a;+r2)2 

5. (a: + 5a)(a:— 2a) 10. (a + 6 — (?)(a— 6 + c?) 
11. (a2+aa; + x2)(a2-aa; + a;2^ 

12. (a-2a; + 3y)a 14. (a'^-iJ + 3 a^J")^ 

13. (3a;»-5)(32:« + 6) 15. (l-2x-\'Sa?-2fi)^ 

16. [(a-6) + (a; + y)][(a-6)-(x + y)] 

17. (4ira + 2a; + l)(4ar^-2ic + l) 

18. (32^-5y)(4a;2_y) 

19. (|a; + J)2 21. (.3a-.5fr)a 

20. (|a-5 6)(Ja + 5 6) 22. (1.2 a; -.05)2 

23. Sliow that (2 ic — 3 y)2 has the same value as 
(3y-2a;)a. Why is this? 

24. Make up and work an example similar to Ex. 19. 
To Ex. 23. 

Simplify the following, using methods of abbreviated 
multiplication as far as possible : 

25. (a: + y)a+(a:~y)a 27. (x + 2 y)2 + (a? - 2 y)2 
2a (a: + y)a-(a:-y)a 28. (82:- y)2- (3a: + y)2 



FUNDAMENTAL PROCESSES 19 

29. (82:--l)2-(3ic+2)(3a;-2) 

30. 4(a - 3 6)2- 5(2 a + 6)2+ 3(2 a + S 6)(2 a - 3 6) 

31. (5a-36)(2a-ft)-3(a-5 6)2+(a-56)(a+4 6) 

32. (a + ft + <?)(a + 6 - <?)- 3(a - c)2- (6 + c^ 

33. Show thata2 = (a + 6)(a- J) + J2. By use of this 
relation, find the value of (7J)2 in a short way. (Let 
a = 7J, 6 = J, etc.) 

34. In like manner, find the value of (6J)2. Of (9J)2. 
Of (12^)2. 

35. How many of the above examples can you work at 
sight? 

4. Type Forms of Abbreviated Division. The labor in- 
volved in the division of algebraic expressions may 
frequently be much lessened by use of one of the follow- 
ing relations: 

L ^^ = a-6. 
a+ft 

Ex. "'r l^^"" " ^T. = «^ - 4(x - y)\ QuotienL 

IL ^tlZ^^a+b. 
a — b 

Ex. ^-rJj^=a^^3x». Quotient. 
IIL ^±^==d2„^J+ft2. 

a + b 

Ex. §^L±l! = 4a:2-2ary2 + y4. QtioiiVn^. 
2a: + y^ 

IV. ^^ = a2 + a6 + 6?. 

Ex. Q3:^-(g + ^0' = 4 a:4 + 2 arVc + rf) + (c + «?)«. Quotient. 



20 DURELL'S ALGEBRA: BOOK TWO 

a+b 

Ex. ?|^i±J^=16x*-8x«y + 4x2^2 -2ary« + y*. Q^otierU. 

VI. ?llI^ = a* + fl86 + d26^ + a68 + 6*. 
a — ft 

BXERCIS£ 7 
Write at sight the quotient for each of the following : 
^ gg - y^ g g^ + y^ ^^ a^-4(x + yy 

2. ^'-3/' 7. 4:r2-.ya ^^ 8(a; + y)»-g« 

a — y ■ 2a;-f-y '. 2(a; + y)— a^ 

3. ^^1^1 8. 8^-/ 13. *^I^' 

a — y 2a; — y 62_y8 

4 «' + y' 9. 32a:5^y6 ^^^ 27:g8-8(a + 6)8 

a + y ' 2 a; — y ' 3 a:— 2(a4-6) 

16. In Ex. 12, by multiplying remove the parenthesis 
in both dividend and divisor, and then divide by long 
division. The work involved in this long division is 
about how many times that in the abbreviated division 
process ? 

Write a binomial divisor and the corresponding quo- 
tient for each of the following : 

17 ^-8i8 _ 20. 27a:8-8y8 ^ 

18. ^Zliy'= 21. :lill^' = 

19. l^ii^= 22. ^^:^^ 



FUNDAMENTAL PROCESSES 21 

23. * = 27. 5 — = 

25. = 29. !— 2— = 

2g .008 + .V» ^ ^ .00S^-.021y> ^ 

7 I 1*7 7 J»7 

31. Write the quotient for ^ , . Also for ^ "", . 

32. Al8ofor(aii+i^0-*-(«+O- For(a"- 6") -»■(«- *)• 

For what values of n will 

33. a» - 6« be divisible bya-6? By« + 6? 

34. a" + b"" be divisible bya-6? By a + b? 

35. Make up and work an example similar to Ex. 12. 
To Ex. 20. 

36. How many of the above examples can you work at 
sight ? 



CHAPTER II 

FACTORING; FACTOR THEOREM; HIGHEST COMMON 
FACTOR AND LOWEST COMMON MULTIPLE 

5. The First Three Cases in Factoring are 

Case I. A Polynomial having a Common Factor in all 
its Terms. 

Ex. 6x2y - 93*y^ - 18ax* - 12bx^y 

= 3 x2(2 y - 3 a;y« - 6 ax2 - 4 by). Factors. 

Case II. A Trinomial that is a Perfect Square. 
Ex. 16 x« - 24 xy + 9 y« = (4 X - 3 yy. Factors. 

Case III. The Difference of Two Perfect Squares. 
Ex. 16x* - 9y« = (4x« + 3y8)(4x2- 3y»). Factors. 

BXERCISS 8 

Factor and check : 

1. a;8_3a;ay + 9a:y2 7. 9^52+4^5 + ^62 

2. 2?-Qx^y+9xf 8. ^2^2-16^^2 

3. 2?^9xy^ 9. 7(a~J)a-14(6-a) 

4. a;6 + 2a^ + aj3 10. (3a:-2y)8-16y+24aj 

5. a:8-ic 11. .09a?-.12a:y + .04y2 

6. 4a**-8a2»» + a*'* 12. ^a^-t^ic 

13. (2:-3y)2-J^ 

14. a? + y2+22 + 2icy+ 22»+2y2 

15. 5(a + ft)(a:-y)-10(a + 6)a(3^-a;) 

16. ^a?-^y2 18. 15a2-60 

17. 9(a:-h3^)2--4(2a;-3y)a 19. 6(ic-y)-a: + y 



FACTORING 23 

20. (ic + y)2 - 2 ox - 2 ay + a^ 

21. . 0004a?-. 01y3 22. |a? + 4a:y + Yy^ 

23. a(a; — y) — 62; + % + cx — cy 

24. (a + 6)2-9(a + J-l)2 25. -4icya + a;a + 4y* 

26. (a + 6)a-6(a2-.ja) + 9(^_j)2 

27. ax^^-a 28. a?i2-6a^ + 9 29. 4a?-9(aj-y)2 
Compute in the shortest way the numerical value of 
30. 975x863-975x861 31. 4762-8242 

32. 975 X 863 + 975 x 413 - 975 x 1272 

33. 17 X 23 X 152 + 14 X 23 X 152- 11 X 23 X 152 

34. 7r^-7rr2 when 7r=s^, i2=32,r=24 

35. 89.732-85.272 

36. Define /a(?for« of an algebraic expression. 

37. Define prime factor. 

38. State and illustrate some of the advantages of being 
able to resolve an algebraic expression into factors. 

39. Make up and work an example similar to Ex. 30. 
To Ex. 32. 

6. Cases IV-VI in Factoring are as follows : 
Case IV. A Trinomial of the Form x^ + lx + e. 
Ex. x* - hxy -14 y2 = (a: - 7y)(x + 2y). Factors. 
Case V. A Trinomial of the Form a^ -\-bx + c. 
Ex. 6 a;2 + 11 xy _ 10 y2 = (2 a: + byX^ x - 2 y). Factors, 

Case VI. A Sum or Difference of Two Like Odd 
Powers. 

Ex. 1. 8 a:« + y« = (2 a: + y*)(4 x^ - 2 xy^ + y*). Factors. 
Ex.2. 27(a + by-l 

= [3(a + 6) - 1] [9(a + by + 3(a + 6) + 1] 

= (3a + 35 -l)(9aa + 18a6 + 96« + 3a + 36 + 1). Factors. 
Ex. 3. a:* + 32 y« 

= (x + 2 yXx* - 2 x»y + 4 j;y - 8 xy^ + 16 y*). Factors. 



24 DURELL'S ALGEBRA : BOOK TWO 

EXERCISE 9 

Factor and check : 



1. a8-4a*+3a 


16. 


Sa^-x 


2. 2a?-6a; + 2 


17. 


2a;8 + a%-10a;y» 


3. 8ic8-27 


18. 


06 + 27 a* 


4. a^-.13a^ + 12a? 


19. 


3f^-<ip + q^x+pq 


5. a^ + y9 


20. 


a6-32a* 


6. 3a^-lla?+62? 


21. 


12o2-7a6-12J» 


7. ya-(a+% + aJ 


22. 


8a» + 27(a; + y)8 


8. a»-a:» 


23. 


I + jV^ 


9. 15a2-7a-2 


24. 


9 a* - 13 «» + 4 


10. y^^ay — hy + ab 


25. 


27 + a» 


11. (a + 6)8 + (2: + y)8 


26. 


3?-9abx + 8aV>^ 


12. 8a«'»~27 6»« 


27. 


ai«-l- 


13. (a;+y)2-6(a:+y)+8 


28. 


afi + 6afiy + 9 3*2/^ 


14. 2aa+.la-.06 


• 
29. 


1-a* 


15. 3a? + 4a:-4 


30. 


a?-9(a + 6)« 


31. 3(a-6)2 + 7(a 


-6)a:-6ar» 


32. .008a;8-^y8 




33. Make up and work an example under each of the 
cases reviewed thus far in this chapter. 


7. Other Cases in Factoring. 






Case VII. A Polynomial whose Terms may be grouped 
80 as to be Divisible by a Binomial Divisor. 


Ex. ab + ax — be — cx = (ab'^ i 


ax)- 


(6c + ci) 



= a(b + x) - c(6 + x) 

= (a — c) (6 + ar). Factors, 



FACTORING 25 

Special Cases under Case IV. 

A. A Polynomial whose Terms may he grouped so as to 
form the Difference of Two Squares. 

Ex. a;2 + y2 _ aa_ 2 a:y = a;2 - 2 ary + y2 _ ^2 

= (^ - yy- «^ 

— {x — y -\- a){x — y — a). Factors. 

B. The Difference of Two Perfect Squares obtained by 
the Addition and Subtraction of a Square. 

Ex. a* + a^a:^ + ar* = a* + 2 a^x^+a^ - a^x^ 

s= (a2 + ar» + ax) (a^ + z^ - ax). Factors. 

SXERCISS 10 

Factor and check : 

1. px + py^qx — qy 6. 16 a^ — 9 a^y^ + y^ 

2. a^ + p^-^2px-^ 7. a^-.a^-b^+2ab 

3. a^ + jp^a^ + jt?* 8. x^-a^ + bx-ab 
^. a^ + ay — ab — by 9. 4 aJ* + 1 

5. a2-J2 + 26y-ya 10. 4a?* + 1 -4 a?- 9^^ 

11. a2-2a;y + 62_a^^2a6-y2 

12. 2(aa-62)-(a-6) 13. 2x^-4:x-%7^ + 2 

14. a* + a2y2^y4 

15. 2a:a-9aj+9+(2a;-3)2 



16. 


a - 6 - a8 + 68 




21. 


68 + 6*+l 


17. 


a8-7a-6 




22. 


a2 + a-6a-J 


18. 


64a4 + l 




23. 


4a:2_9y2.i_6y 


19. 


l + 3a2 + 4a* 




24. 


4a?-9y2+4a;-6y 


20. 


3(a8 + 8)-9(a 


+ 2) 


25. 


a;i2_6a^ + 9 



26. 3(a-6)(a + 6)2 + (a+6)8 

27. Make up and work an example under each of the 
cases in factoring studied thus far. 



26 DURELL'S ALGEBRA: BOOK TWO 

8. Examples Preliminary to the Factor Theorem. 

Ex. 1. Factor a8 + 3 o^ - 4. 

By separatlDg 3 a* into 2 a* + a^, we obtain 

a« + 3 a^ - 4 =a« + 2 a« + a« - 4 

= a2(a + 2) + (a + 2)(a-2) 

= (a+2)(aa + a-2) 

= (a + 2)(a + 2) (a - 1). Factors. 

Ex. 2. Factora?* + 2a^-13a;2_i4^^24. 
By repeated splitting of terms, we obtain 

a4 + 2x«-13xa-14x+24 

= ar* - x»- 3a;» - 13 x* - Ux+ 24 

= x*-a:« + 3x«-3xa-10xa-14x + 24 

= a:*-a:« + 3x«-3x«-10x2-10x-24x+24 

= x«(x - 1)+ 3xa(x - 1)- 10x(x - 1)- 24(x - 1) 

= (x - l)(x« + 3 x2 - lOx - 24) 

Hence, x — 1 is a factor of the original expression. 

This result might have been obtained in a shorter way. For, as 
this last expression reduces to zero when x = 1, we might test the first 
expression to see if it is divisible by x — 1, by substituting 1 for x and 
noting whether the expression reduces to zero. 

This last test may be further abbreviated to merely noting whether 
the algebraic sum of the coefficients of the terms is zero. 

By splitting the terms of x* + 3 x^ — 10 x — 24 in like manner, 
we obtain as a final result 

X* + 2x« - 13x2 - 14 x+ 24 =(x - l)(x + 2)(x + 3)(x + 4). Factors. 

Ex. 3. Determine by inspection whether 7^ + 7? — Qofi 
— 4 a? + 8 is divisible by a; — 1. 

Summing the coefficients, we have 

l + l-.6-4 + 8 = 0; 

hence, x — 1 is a factor of the given expression. 

In like manner, if an expression is divisible by x + If the sum of 
the coefficients of the even terms must equal the sum of the coeffi- 
cients of the odd terms. 



FACTOR THEOREM 27 

9. Factor Theorem. If any rational integral expressian 
containing x becomes equal to zero when a is substituted for 
jr, then x — ais a factor of the given expression. 

For, let E stand for any rational integral algebraic expression. 
If E is divided by x — a till a remainder is obtained in which x 
does not occur, denote the quotient by Q and the remainder by R. 
Then 

E= Q(x-a)+/2 
Let X = a 

Then = Q(0) + R (since JS: = when x = a) 

.-. 72 = 
Hence, E = Q(x — a), or x — a is a factor of E 

Ex. Factor a^ - 12 a; + 16. 

By trial we find that x« - 12 x + 16 = when x = 2 

.*. X — 2 is a factor of x* — 12 x + 16 
By division x« - 12 x + 16 = (x - 2)(x« + 2x - 8) 

= (x - 2) (x - 2) (x + 4). Factors. 

Note that the only numbers which need to be tried as values 
of X are the factors of the last term of the given expression. 
This follows from the fact that the last term of the divi- 
dend must be divisible by the last term of the divisor. 

SZSRCISS 11 

1. Determine whether a^ — 5a? + 7a? — 3 is divisible 
by a? — 1. Is a; — 1 a factor of the given expression ? 

2. Determine whether a; — 1 is a factor of a^-i- ofl 
— 8 a? + 8. Also wliether a; — 2 is a factor. 

3. Isa?-1 a factor of a?*-4a^-7a;2+22a? + 24? Is 
a?-2afactor? a;-3? a;+l? a; + 2? 

By use of the factor theorem, find the factors of 

4. ar2-9 6. afi-7x + 6 

5. afl + x^''2x 7. a^-6a?+lla;-6 

8. ar*-10a:8 + 35a?^-50x + 24 



28 DURELL'S ALGEBRA : BOOK TWO 

9. a^ + lla^+41a: + 61a; + 30 

10. a^^Uafi + lla^-Uix + nO 

11. 2a^-3a?-5a? + 6 13. S^r^-x^^ oQa;- 12 

12. 8a^-7a?-2a; + 8 14. a8-6a2 + 25 

15. y*-28y2 + 33y-90 

16. Prove that a:* — y* is always divisible by a; — y. 

17. Prove that a:* + y» is always divisible by a; + y when 
n is odd. 

18. Show that (1 — a?)* is a factor of 1 — a; — a;» + af^K 
Without actual division, show that 

19. 6<?(6 — <7) + ca(<?— a)+a6(a— 6) is divisible by J— <?. 

SuG. In the given expression, change each c into b and show that 
the result reduces to zero. 

20. a;(y + 2-a;)2 + y(2 + a:-y)* + 2;(a; + y-0* 

+ (y + 25 — x)(z + a: — y)(a; + y — 2) is divisible by x. 
SuG. In the given expression, let x = 0. 

21. a\b — (?) + fy^Cc — a) + c2(a — J) is divisible by 6 — {?. 
Also by c — a. Also by a — 6. 

22. a* + 6^ + c* — 8 aJ(? is divisible hj a + b + c. 

23. Make up and work an example similar to Ex. 3. 
To Ex. 6. To Ex. 8. 

SXERCISS IS 
General Review op Factoring 

Factor and check : 

1. a:«-5xa + 6x 6. 6zy«- 5ar»y -4a:« 

2. a:«-4a% 7. b^+b^x^+x* 

3. 2 a:« - 8 aa:^ + 8 a*r e. ay + by - ax -bx 

4. x"' - X 9. x» - X 

5. x* + 2y-l-y« 10. a" + xw 



FACTORING 29 

11. a^-z" 22. (6a-ft-|-3)2-(6a-|-ft-2)a 

12. a;8-13a: + 12 23. a:(x - a) - y(y - a) 

13. 10a!»x-10x 24. a« + 2a2 + 2a + l 

14. 5(ar-l)«-x« + l 25. l-Ux^ + ar* 

15. 72-a:--a:2 26. 21x* - 17x« - 30x« 

16. 8x«-(a + 6)« 27. aH^ - 4x« + 5r» - 20 

17. (a + 6)a - 4(a + 6)+ 3 28. a« - a^ - 1 + a 
la a'' + 128 29. x»~9x + 8 

19. (x - 6)2 - a2 30. 3aa - 27 + a^x - 9x 

20. 4(a - 6)2 - 9(x + y)4 31. (3x2y-3xy2)a 

21. l-i>" 32. 8(a + 6)«-(x-y)« 

33. (a2+62)4_15Q4J4 

34. (a + ft)2(x-y)+(a+6)(a:-y)a 

35. (a + 6)2 + 4(a + 6)(x-y)+4(x-y)2 

36. x2- 6ax- 962- 18 a6 

37. (x2 - 5x)2 - 2(x2 - 5 x) - 24 

38. (3x2~3y2)8 

39. (a2 - 62)8 + (2 a + 3) (a - 6)* 

40. 4j» + 4an+j»2_ 4^3 _ ^24.4 

41. 2(a8-8)+7a2-17a + 6 

42. a« - 4a* - 16a2 + 64 46. a262 - a2 - 62 + 1 

43. x» + 2 x2 - 6 X - 6 47. x*- 2 (02-1-^2)3.2^ (a^+h^^ 

44. 25a* + 34a262 + 496* 48. x«» - y^ 

45. x* + x«-7x2-x + 6 49. x^ - 2 x*^y^ + y*^ 

50. a2 + 62 + c2 + 2a6 - 2ac - 26c 

51. a» + 6» + a26 + a62 + a + 6 

52. 3(a - x)« - X + a 55. x* - 32 a^x 

53. x" +1 56. 21 X* - 40 x«y - 21 xY 

54. 27000-1 57. a^ - 2 a^b - i ab^ -\- S b* 



30 DURELL'S ALGEBRA: BOOK TWO 

10. Highest Common Factor and Lowest Common Multiple. 

Ex. 1. FindtheH.C.F.andL.C.M.of 12a2-12aJ + 662, 
9a2-962, andl2(a- J)8. 

Qa^ ^ 12ah + Qb^ = e(Q - by 

12(a - by = 12(a - by 
Hence, H. C. F. = 3(a - b), Ans. 

L. C. M. = 36(a + b)(a -^ by. Ans, 

Ex. 2. Find the H. C. F. and L. C. M. of a?(3 a^^y- Sxf)\ 
and 6 x(j/ — a?)^. 

x\^ z^y - 3 zy^y = a:«[3 xy(a: - y)]^ = 9 x*y%x - y)« 
6x(y - xy = 6a:[-(x - y)]8 = - 6a:(ar— y)« 
Hence, H. C. F. = 3 x(x -yy. A ns. 

L. C. M. = 18 xY(x - yy. A ns. 

SZSRCISS 18 

Find the H. C. F. and L. C. M. of 

1. 3a?y, 9xi/\ 12a?y2 4. ^^ - 3 a\ a*- 9 

2. 24 a%^ 56 a%^ 12 a%^c 5. 6(258 - yS), 9(a? - y^) 

3. 5a6(?, 10a2J2c2, ISaSJc^ 6. 18(a: - 1)2, 24(a^ - 1) 

7. 6(a2-l), a2-a, 4a2-8a + 4 

8. 9aj8-9a:, 6aj*-12a;3+Ca;2 

9. a^fi-a;, a^ + a^-2ic2, a?* + 3a^+ 2a:2 

10. 1 - ic2, (1 + xy, 1 + a;3 

11. 16 0x2(2? - y)2, 3 Ja;y(a? - f^, 9 a;(a; + y)* 

12. (x^y-xy^y.oflfix-'yy 

13. (3 a?^ - 3 a:y)8, 36 a?(a;2 _ ^2)2 

14. (a - h)(x + y), (h -aXx- y) 

15. (a - 6)2(a; + y), (6 - a)2(a: - y) 

16. a?(5a;-5a)2, 5x(a2-a:2) 



FACTORING 31 

17. Define common factor. Also highest common factor. 
Give an example of a common factor which is not a 

highest common factor. 

18. Define common multiple. Also lowest common multiple. 
Give an example of a common multiple which is not a 

lowest common multiple. 

19. Write two expressions whose H. C. F. is 3(a — 6). 

20. Write three expressions whose L. C. M. is 24a?y* 
(a - 6)2(a + 6)8. 

21. Wlio first used the sign + to denote addition and 
when? (See p. 267.) 

22. Give some other symbols used to represent addition 
before the sign + was invented. Discuss, as far as you 
can, the relative advantages of these signs. 

23. Answer the questions in Ex. 21 for the subtraction 
sign. 

24. Answer the questions in Ex. 21 for the sign x . 

25. For the sign -s-. 

26. For the sign = . 



CHAPTER III 

REVIEW EXERCISES IN FRACTIONS; FRACTIONAL 
EQUATIONS 

11. Important Properties of Fractions. 

(1) If the numerator and denominator of a fraction are 
loth multiplied or divided by the same quantity^ the value of 
the fraction is not changed. 

(2) The signs of any even number of factors of the nu- 
merator and denominator of a fraction may be changed 
without changing the sign of the fraction ; 

But if the sig'ns of an odd number of factors are changed^ 
the sign of the fraction must be changed. 

Ex. 1. Reduce — -^ to its lowest terms. 

(2 — X) 

(2^xf (2-x)(2-x) (a;-2)(a:-2) x-2' 

Ex. 2. Reduce — ^ — J-? to a mixed number. 



a;2 + 2 

Z^- X^-^ X 

x» +2x 



x2 + 2 



-x^-x 
-x^ -2 
~x + 2 

Hence, __±_ = x- 1 - -j-^. A»s. 



FRACTIONS 33 

KZBKCISB 14 
Reduce to its simplest form : 

54 64aV»y 8g»y 

81 72a*«6 ■ &x^-93»y 

27a8aV 9ay-12y' 18(a-6)' 

■ 86a*c*y* * 12a?-16a!y ' 46(a«-^ 

a:>-4 j3 6^a«-7aA + 26a 



aa_4a; + 4 6a»-aJ-26» 

8 ag-27 ,- 16 -a? 

8. -r—z — : —T 14. 



6aa-24a+18 a!»-7x + 12 

g 8a^+4a!-4 ^ p»-27 

i» + 8a! + 2 ■ 9-6/> + ^ 

10. t^^^ 06. 4-(x + y)« 

0^-16 (a,_2)«-y» 

8:r(a;-y)« (a:- l)(2-a:X3 -a:) 

' (8a?-8a:y)a ' (1 - x)(a! - 2)(a; - 4) 

^ a' + y-c« + 2a5 a;»-(p + 9)a; + Pg 

19. State which of the following can be simplified by 
striking out cfi : 

ch; a^ + x ahe x + a' c?(x + y) 

a^ cfi + y a'+y 6 + 3a' 6 a* 

Reduce to a mixed quantity : 

6 X 2a+l 

23. ^-^ 24. ^^~^ + ^ + ^ 



34 DURELL'S ALGEBRA: BOOK TWO 

Reduce to an improper fraction : 

25. 13* 28. a^-x + 2~ ^<^f"^> 

31. Define /racf/(?w. Improper fraction. Mixed number. 

32. Define integral ea^resaion. Fraction in its lowest 
terms. 

33. State some of the uses of fractions. 

34. How many figures are used in writing fjf ? How 
many in writing § ? Compare the labor in writing these 
fractions. 

What other advantages are sometimes connected with 
using f instead of §^ J ? Why is it sometimes an advan- 
tage to use f If instead of | ? 

35. The labor of writing SJ^n~-,.» , is about how 

12 a^— 16 aft 

q 1 

many times that of writing — ? What principle makes 
this economy possible ? 

12. Fimdamental Operations with Fractions. 

Ex.l. Simplify ^--4-+ * 



a^—b^ a + b b ^a 

a^^b^a + ba-b a^-b^ 

__ a^ + b^-a^-\-ab-ab-b^ 
a^-b^ 




a^-b^ 



•=0. Ans. 



FRACTIONS 35 

The given expression 

a^ + b^ ^ a ^ (fl-6)(qa + aft + &a) 
a(a^ + aft 4 6^) a-b a« - aft + 6« 

= a + ft. Ana. 

SXERCISS 16 
Simplify : 

6 10 ■*■ 16 

4a:-7 7a;~3 3a;-2 
' 7a? 9a; 3a? 



6. 



11. 



12. 



a + 22-a a2_4 



* + ' 



a+3 a+2 a+1 
y «2 a'y' 



3a + t Za-h 14 a6 
' a-26 a + 26 4J2_a2 

2 r-3 »f_ 

• ;. + 4 r2-4r+16 r« + 64 

o P + 3 p-2 p + 4 

• ^_2^-8 jpa-7/> + 12 j9«-j»-6 

,„ 28 or* ^ 9 ««* 21 a^x 



ISJ^c 8 A; 10 6c8 
h J . 1 



4+462 2 + 2 J* 8-86 8 + 8i 
^ bxy )\^ + y) llO^y^ 2xy\ 



36 DURELL'S ALGEBRA: BOOK TWO 

a^-y^ y-x z + ff 

a + 6-tf a«-(6 + 0^ (a + J)'-c» 
3 , fi 7 



18. 



19. 



8-8a; 4a;+4 8x»-8 

/ 'a+2ft a-26\ /a + 2b a-2b\ 
\a-2b'*' a + 2bj\a-2b a + 2bj 

1_ f aJ»-6a;-3 T 1 x 1] 

2 1 2a;»-2 U-1 ar» + a; + ljj 



ao. ^ + ^ + ? 

(a;-y)(a;-«) (y_a)(y-a:) (a_x)(«-y) 

21 l+« I- 1 + ^ I ! + <' 



22. 
23. 
24. 
25. 



(a-6)(a-c) (J>-cXf>-a) (c-a)(e-i^ 

1 + 1 1 

(6-a)(a-3) lO-Ta + a* (o-2)(3-a) 
/^ 1_L 5 Va' . g . l^. / 2a'^ + 8g ^ 

l"-^+^u-'4+i2r(T^T3-j 

p + 1 2 p - 1 ^ 



6j?-6 12j» + 12 Z-Zp^ 12p 

3 ri 3a^-4a;-l / 4 o\"| 
■ x-2 \_x a?-3x + 2 \x-l Jj . 

28. How many examples in Exercise 12 (p. 28) can you 
work at sight ? 



FRACTIONS 37 

13. Bednotion of Complex Fraotions. 

17 1 3x 3ar 3x-2 9z^ 8 . 

a: — • 



Ex.2. 



9a; 9x 

1 



2 ~ 2 



a-3 

1 /. _ a 



a-2 a--2 

1 1 a-3 



Simplify : 




A- 


2. 


Z 








a \ — a 


3. 


1 + a a 




a 1 — a 




1 + a o 




o»_6a_ca 


4. 


2 6c 




aa + fta_^ 




2a6 


5 


1+1 1-1 

X y X y 




2f+£ ^_£ 



2g-4 g« - a - 4 a*- a- 4 
■*" 0-3 0-3 

EXERCISE 16 

x-\-y x-y 

x-y x+y 

7. V g '•^^ 



8. 



o + 



1+^ 



9. 2x-l =— :^ 

2 £— 

a: 
a; — 

y 1+a; 



38 DURELL'S ALGEBRA: BOOK TWO 

1 1 1 

10. -l±^+t±l+hil 



1- 



l + a 1 — a 1 + a 



11. 



12. 



x-2- 



x-2 



x-2 



x-5 
l-27a« 



X — 



X-4: 



X — 



a;-4 



1 + 



8a 



1- 



3a 



1- 



l + 3a 



1 + 



2a 



l-2a 



13. ..„, -»- 



11 

3 X 



3^ + 27 • ^2+9 



3 



14. 



ri+-i-+ ^ Ti '^—1 



IS. 



~ a; ~ 


-J- 


x-l 


x+1 


4 



a;+l 



-1 



x-l 



+ 1 



x-1 



+ 1 



x+1 



-1 



16. Define complex fraction. Also continued fraction. 
Give an example of each of these. 

17. State some of the ways in which factoring is an 
aid in simplifying fractions. Illustrate. 



FRACTIONAL EQUATIONS 39 

Fractional Equations 

14. An Equation is a statemeut of the equality of two 
algebraic expressions. 

Define members of an equation and illustrate by an ex- 
ample. 

A root of an equation is a number which, when sub- 
stituted for an unknown quantity in that equation^ satis- 
fies the equation; that is, reduces the two members of the 
equation to the same number. 

Thus, 6 is the root of the equation dx — l = 22; + 5, since, when 
6 is substituted for ar, the given equation reduces to 17 = 17. 

Define degree of an equation (of one unknown), simple 
equation^ identity^ conditional equation^ fractional equation^ 
numerical equation^ literal equation, 

15. Aids in solving Equations. The following principles 
are frequently used as aids in solving equations: 

The roots of an equation are not changed if 

(1) The same quantity is added to both members of the 
equation. 

(2) The same quantity is subtracted from both members 
of the equation. 

(3) Both members are multiplied by the same quantity 
or by equal quantities (provided the multiplier is not zero, 
or an expression containing the unknown). 

(4) Both members are divided by the same quantity (pro- 
vided the divisor is not zero, or an expression containing 
the unknown). 

Other principles similar to these are used later as aids in solving 
equations. 

Explain the meaning of transposition^ and show for 
which of the above principles it is an abbreviation. 



40 DURELL'S ALGEBRA : BOOK TWO 

Ex.1. Solve --i 3^ = §^. 

Rewriting the second fraction (why?), 

4 3 ^ 8a: + 3 

3 + a:'^3-2: 9 - x* 

Multiplying each member by 9 - x^, the L.C.D. of the fractions 
involyed (see Art 15, 3), 

4(3-a:)4-3(3 4-a?)=8a?+3 
12 - 4a: + 9 + 32? = 82:+ 3 
-4a: + 32:-8a:=-12 -9 + 3 
-9x = -18 
x = 2. Root 

Check, -i ?_- = _i L. = ^+3=^ 

3 + x 2:-3 3 + 2 2-3 5 5 

8x + 3 ^ 16 + 3 ^19 
9_aJ» 9_4 5 

Removing the parentheses, =^ - =^ = -^ + -t" 

3 3 5 5 

Hence, 10x-10a« = 9x+9a« 

X = 19 a«. Root. 
Let the pupil check the work. 

EXERCISE 17 

Solve for x and check : 

,0 Q 24-2a; n 
!• 2 a; — 8 = 

2. |(l + a;)-|(2a;+.l) = 

3. 5[3a: + (a;-4)]-4(a;-2)-36 = 

4. a? + 2 = a;-.8-2}8-3(5-a;)-a;{ 

5. §£^-^-l(4a; + l) = |(a:-l)-3 



FRACTIONAL EQUATIONS 



41 



6. 
7. 

8. 

9. 
10. 
U. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 



20. -7 



.3a?-.14ar=.012a; + .692 
3.2a;~3.4^ .6a; + 4 



4.5 



2.5 



2J 



a? — 6 x — a 



^0 



x^ a_ 2 _ a? — <? 



5a;-[8a;-3}16-6a;-.(4-5ar)}]=6 



a;+l I g^-3 ^Q 



2 + a; 2-a? 4-a;2 



8a:+3 4^3 



2 



+ 



8+x a!-3 
1 1 



= 



x + 2 
3 



= 



a; — 1 x + l 

l_3a; 1-2! 



a!-2 ar» + 2a;+4 8-a? 
a;-3 a!»+6 a; + 3 



»0 



2a: + 6 6a?-54 3a:-9 " 
a; + 2a a; - 2a 4a6 


2i-x ' 26 + a: 462-a:a 

3-a; 6-a; ^ xa-2 
l-a; 1-x 7-8a; + a!» 


X 

1 

4 


— a a? — J 


= 1 



3ar 



42 DURELL'S ALGEBRA: BOOK TWO 

x2-4a;H-5 



21. 



x^-^Qx-^lO 






p{q — x) g^r — x) pir — x) 



24. 






25. £±*-M=l + 



a; — 6 x-{-b P-- x^ 

Find the value of the letter in each of the following : 

p—i p—o 
27 4a-7 ^ 7a + 2 11 
■ 6a + 18 lOa + 30 45 

28. -^ I 5« + l ^0 

30. If r= ?t^;A, r= 480, h = 2.5, «; = 36, find I 

Do you know the meaning of this process in connection with the 
study of volumes? 

31. If i ==prt,i = $lS2,p = $ 720, t = 3f , find r. 

What is the meaning of this process in connection with the subject 
of interest in arithmetic ? 

32. In the formula (7= | (jP-32), find F when 
= 200. 

What is the meaning of this process in connection with the subject 
of temperatures ? 

33. If r=7riJ(iJ + i), 7= 2288, 7r = ^\R=z 14, find L. 



FRACTIONAL EQUATIONS 43 

34. Make up and solve an equation containing fractions 
with the denominators 6, 9, 12. Can you form this equa- 
tion so that the root shall be 1 ? 8 ? 5 ? 

35. Make up and solve an equation in which the de- 
nominators shall be 3, 6, a; — 2. 

36. How many examples in Exercise 13 (p. 30) can you 
work at sight ? 

16. Special Methods of solving Fractional Equations. 
Ex.1. Solve _5__2ia:-8^^jJJL_ll:.-f 5, 



7-x i 8 16 

, the L. C. D. of the monomial 
-9a:+12 = 2a: + 22 -liar- 



Multiplying by 16, the L. C. D. of the monomial denominators, 
80 



7-x 

Hence, -?5_ = 5. x = - 9. Root. 

7 — X 

Let the papil check the work. 
Ex. 2. Solve 



x—l a: — 3 X — 6 x—7 



x—2 a;— 4 ar — 6 a: — 8 

In this equation, it is best to combine the fractions in the left- 
baud member into a single fraction, and those in the right-hand 
member also into a single fraction, before clearing of fractions. We 
obtain 

-2 ^ -2 

x^-Qx + S ,x^-Ux + 4:S 

1 1 



Diyidingby-2, ,. _ g^ ^ « " x« - 14x + 48 
Hence, x = 5. Root. 

Let the pupil check the work. 

17. Two Equivalent Equations are equations which have 
identical roots ; that is, each equation has all the roots of 
the other equation and no other roots. 

Thus, x^-x^-6x = and x (x + 2)(x - 3)= are equivalent, 
since each is satisfied by the values x = 0, 3, — 2, and by no other 
values of x. 



44 DURELL'S ALGEBRA: BOOK TWO 

If we multiply the two members of an equation by the 
same expression, the resulting members are equal, but the 
resulting equation may not be equivalent to the original 
equation. 

Thus, if we take the equation :r = 5 and multiply each member by 
X — 3, we obtain x(x — 3) = 5(a: — 3) or 

(x-3)(x-5)=0, 
which is not equivalent to the original equation, since it has the root 
X = 3, which the original equation does not have. 

In general, if the two members of an integral equation are 
multiplied by x^a^ the root a i% introduced and the result- 
ing equation is not equivalent to the original equation. 

18. An Eztraneons Root is a root introduced into an 
equation (usually unintentionally) in the process of 
solving the equation. 

The simplest way in which an extraneous root may be 
introduced is by multiplying both members of an integral 
equation by an expression containing the unknown num- 
ber. See the example in Art. 17. 

A more common way in which extraneous roots are in- 
troduced during a solution — and one more difficult to 
detect — is by a failure to reduce to its lowest terms a 
fraction contained in the original equation. 

Thus, in solving ^^^ = 1, the first step should be to 

^ (ar + l)(x-3) ^ 

reduce the fraction to its lowest terms. K this is 

2 
done, we obtain the equation = 1, whence x = 1. 

X + 1 

If, however, we should fail to reduce the fraction to its lowest 
terms and should multiply both members by (x + l)(a? — 3), we 
obtain 2x— 6 = x^ — 2x— 3, whence x" — 4 x -f 3 = 0, or 
(x - l)(x - 3) = 0, and x = 1, 3. 

On testing both of these results, we find that the extraneous root 
3 has been introduced. 



FRACTIONAL EQUATIONS 45 

Often the fraction which can be reduced to simpler 
terms occurs in a disguised and scattered form. In this 
case it is best to solve the equation without attempting to 
collect the parts of the fraction. An extraneous root 
may then be detected by checking the results obtained. 

Thus, the fraction in the above equation might be 
changed in the following way so as to make it difficult to 
detect its presence in the equation : 

We have — 2x-6 — ^ ^ 

(2r+l)(j;-3) 

Whence, 2x±2 8 ^ ^ 

(a:+l)(x-3) (2r+l)(x-3) 

Whence, ^ - -^^ -^ = 1 

There is nothing in the appearance of this last equation to indi- 
cate that it contains a fraction which should be simplified before pro- 
ceeding with the solution proper. 

Hence, it is important constantly to remember that a root of an 
equation is its root because it satufiea the original equatioriy not because 
it is the result of a series of operations, as clearing an equation of 
fractions, transposition, etc. 

19. Losing Roots in the Process of solving an Equation. 

If both members of the equation (a; + 3) (a: — 2) = are divided 
by a: — 2, we obtain x + 3 = 0. 

The resulting equation is not equivalent to the original equation 
dnce it does not contain the root x = 2, which the original equation 
contains. 

Hence, in general, 

If both members of an equation are divided by an expression 
containing the unknoivn quantity, write the divisor expression 
equal to zero, and obtain the roots of the equation thus formed 
as part of the answer for the original equation. 



46 DURELL'S ALGEBRA: BOOK TWO 





EXERCISE 18 


Solve for x and check : 


1. 


3-2a; a; 2-3a: l-6a: 
4 ^6 9 Ib^lx 


2. 


X 2a:-3 7a:-15 3a;-l 4a;-7 
4 12a;-ll 60 30 ' 15 


3. 


1111 
a;-4 a:-5 a?-2 ir-3 


4. 


2x+^ 13a;-2 a; 7a: a;+16 
9 17a:-32'3 12 36 


5. 


a:-l a;-3 a:-2 a:-4 
a:-2 a;-4 a?-3 x-b 


6. 


2a:-lJ 2a:+3J a:-i 7 3a:-2i 
3 6 |a;+ll 12 9 


7. 


x-\-b a:+6__a:H-2 a: + 3 




a: + 4 a: + 5 a; -f 1 a?+2 


8. 


ex — dx cx-\- dx C'\-d x 
cd cd <^— cd c 


9. 


x—5 a: — 7_a:— 4 a; — 8 
a:— 6 x—S x — 5 x—9 


10. 


P ? P-^^ , f-f 




X p X p^+pq 


n. 


2 3 3 2 



2a: + 3 3a: + 2 3a;-2 2a:-3 

12. Multiply each member of the equation a: + 2 = 1 by 
a: — 3. Is the resulting equation equivalent to the original 
equation ? Why ? 

13. Make up and work an example similar to Ex. 12. 



FRACTIONAL EQUATIONS 47 

14. Multiply each member of the equation a; == 3 by a; — 2. 
Is the resulting equation equivalent to the original equa- 
tion ? Why ? 

15. Divide each member of the equation a:^ — 16 = a; — 4 
by a; — 4. Is the resulting equation equivalent to the orig- 
inal equation ? Why ? 

16. Make up and work an example similar to Ex. 15. 

^ 2 

17. Solve the equation— = 1 after first reducing the 

ar — 4 

fraction to its lowest terms. Now solve the equation with- 
out reducing the fraction to its lowest terms. Do the two 
methods of solution give the same result ? Which result 
is correct ? Why ? 

18. Make up and work an example similar to Ex. 17. 

Solve each of the following, check each result, and point 
out each extraneous root, giving the probable reason for 
the occurrence of such a root : 



19. 



X 



aj4-2 ar-f-2 2 



^. , 15 3 



(a;+3)(a;-2) a;+3 

21. 2 2__j 

(a;-l)(a;-2) x-2 

22. ^3:4-3 = ^ 

23. -^ ^ = 3- 2^ 



x-2 a! + 2 ifi-^ 

24. Form an equation in which 4 is the extraneous root. 



48 DURELL'S ALGEBRA : BOOK TWO 

EXERCISE 19 

Oral 

Solve orally for x : 

1. 12 = -4x 3 ^^a 17 _a^^ 

X bx 

10.2 = 1 18.4 = ^ 






6z = l 



a 



X 3 



bx 



^' = 1 11.1 = 2 19.:iJ = 9 

a: 5 ^ 

*• 4^ = 1 12-3-8 20. ax- ab=ac 

5. b = ax ' ^^ 21— = — 

13. ax + bx = c '4 16 

6. 1 = 4 l*.px = 5-qx 22. a+b='-±^ 

7. ax-e = d IS. e = a + dx ^^ p-q^g + b 

8. «=1 16. A = 4 ■ 4 

X 2x 24. ix = t 



CHAPTER IV 

SIMULTANEOUS EQUATIONS 

20. Methods of Elimination. 

Ex. 1. By the method of addition and subtraction^ solve 

9a:-8y = 6 (1) 

15a:+12y = 2 (2) 

TheL.C.M. of 8andl2i8 24. 

Multiply in eq. (1) by 24-5-8, or 3; and in eq. (2) by 24-S-12, or 2. 

Hence, 27 a: - 24 y = 15 

30a?4-24y= 4 

By adding 57 a; = 19, or a; = -. Root. 

o 

Substituting J for a: in eq. (1), 3 - By = 5, or y = — -. Root. 

4 
Let the pupil check the work. 

Ex. 2. By the method of substitution^ solve 

9a;-8y = 5 (1) 

16a:+12y = 2 (2) 

From eq. (1), 9a: = 5 + By, or x = ^ \^y - 

Substituting for x in eq. (2), 

15(5 + 8y) ^ i2y = 2, whence y = - 7. Root. 
9 4 

Substituting for y in eq. (1), we find a: = -. Root. 

o 

Ex. 3. By the method of comparison^ solve 

9a;-8y = 6 (1) 

15a:H-12y = 2 (2) 

49 



50 DURELL'S ALGEBRA: BOOK TWO 

From eq. (1), 9 a: = 5 + 8y, whence x = ^^ - 

From eq. (2), 15 x = 2 - 12 y, whence x = ^ "" ^^y . 

15 

Equating the two values of x, 

9 lo 4 

Substituting for y in eq. (1), we find x = r. iJoo^. 

o 

EXERCISE to 

1. Define 9imultanefm% equations. Also independent 
equations. Give an example of two simultaneous equa- 
tions which are independent. Also of two simultaneous 
equations which are not independent. 

In solving the following pairs of simultaneous equations, 
use that one of the three methods of elimination which 
is best adapted to each particular problem : 



2. 


3x+4y=l 


8. 


2x-3y + 17 = 




6a;-2y = -3 




y = 3 


3. 


x = 3 


9. 


y = 22 




8a; + 2y = 5 




y = K72 + l) 


4. 


p = '2q-Z 


10. 


a;-6y-2 




p^4q-7 




y=2a:-3 


5. 


4»»-3y = -l 


11. 


a; = 6y-3 




5m — 4y = — 1 




8-5y = a; 


6. 


3a: + 2y = 21 


12. 


y = K^-5) 




y = 2x 




y=|(2a;+l) 


7. 


4^ 3 


13. 


2 3 




5 a; 3 ,v _ 2 

8 "^ 2 




2-E = l 
4 9 



SIMULTANEOUS EQUATIONS 



51 



14. =-- 



22. 



+ 1 



= 1 



X 3^+1 ^. 



3 

17. 



15. \x^\y^\b 

16. \x^2 — \y 

3a; + 4y-25 = 



.5a;-.3y = 3.4 
.3a;-.5y= -.2 

K22:-y)-K3^-2y) = 5y + 2 
f(2:-4i^)=17-|(7:r-2y) 
2a; + y ZxA-\ a:+2y ^Q 

2 7 8 

J^ 3_^0 

aj+3 y+4 

<y-2)-y(a;-5) = -13 
.08 a; -.47 .02y + .17 ^Q 
.3 .9 

1.7a? + 3.2.y oi^ .08y + .35 .13a; + .29 
10 .15 .6 



18. 



19. 



20. 



21. 



(:i;-l)(y + 2)-(a:-5)(y + 3)=0 
a;y + 2 a: - x(jf + 10) = 72 y 

Solve for x and y : 

23. ax^-hy^d^^V^ 
bx-{-ay=^2ab 



24. 






27. 



1-3: 



2-3J, 

3 
2,v + 3 



25. a; 4- y = a + J 

a-{'X _ b 
b + y a 

26. (c — rf)a: -\- dy = c^ 
dx^ cy = cP 

-y = 13 



10a:4-l 



..3-1H-. 



62 DURBLL'S ALGEBRA: BOOK TWO 

28. aa;-Jy = J(a24-62) ^ __a ^ = 

(a — J)a; — (a + 6)y =sO ' J + y a— a? 

« * =0 



5 + 2! a — y 






^ • y ^ ' 



32. 



^_ + _J^^ 2_ 

6a!-5 _ a! + y + 6} ^ 3x-2 

10 6 a; + y 5 

2y-5 3 + 7x ^ 3.v-2 

8 10y-3a; 12 

— a 0—1 

'g-i I y+i^i 

1-a b b 

34. a(a;+a + l)-(a + 6)(y — 1) = 
i(y-6-l)-(a-JX^ + l)=0 

36. Make up and work an example similar to Ex. 12. 
To Ex. 23. 

21. Three or More Simaltaneous Equations. 

Ex. Solve 4a;-2y + 2 = -2 (1) 

3a;-y-22J = ll (2) 

6a:-y + 32 = -3 (3) 



SIMULTANEOUS EQUATIONS 53 

If we choose to eliminate z first, multiply (1) by 2, and use (2) 
unchanged. 

Then 8a:-4y + 2z = -4 (4) 

3a:-y-2z=ll (5) 

Add the corresponding members of (4) and (5). 

llx-5y = 7 (6) 

Also multiply (1) by 3, and set down (3) unchanged. 

12a:-6y + 32=-6 (7) 

Subtracting, 6a?- .v + 3z = -3 (8) 

6x-5y =-3 (9) 

From (6) and (9), 5 a: = 10, hence a: = 2. Root. 

By substitutions in (9) and (1), find y = 3, and z = ~ 4. RaoU, 

Let the pupil check the work. 

22. Use of - and - as Unknown (Inantities. 

X y 

Ex. Solve A-.A = 41 (j^ 

3 5^ _ 23 ^2^ 

4a: 3y 2 ' ' ' ^ ^ 

Multiply in (1) by 8 (the L. C. M. of 8 and 2), and in (2) by 12. 
5 -12^32 (3) 

X y 

?_?2=138 (4) 

X y 

To eliminate y, multiply in (3) by 5, and in (4) by 3. 

23-«? = 410 . (5) 

X y 

2Z-«5 = 414 (6) 

X y 

Subtracting each member of (5) from the corresponding member 
in (6), 

- = 4, whence 2 = 4 a:, and a: = -. Root. 
X 2 



Substituting for x in (3), 
10 - 1? = 82, ^ 

y 

Let the pupil check the work. 



19 1 

10 - — = 82, whence y = - ^- Root. 



54 



DURELL'S 


ALGEBRA: BOOK TWO 






EXERCISE SI 






v^e and check : 










? + § = ! 




6. 


4a: + 52 = 8 




X y 






6a:-3y-h22=0 




§ + 4^2 






2a:+6y-2 = -l 




X y 










^tx-by-^r 32 = 


= 18 


7. 


4 y + 3 a: = a:y 




5a: — 3y — 42 = 


= 2 




2y-6a: = 3a:y 




3a;4-4y-52 = 

Zx by 
1.1 o 


= 32 










8. 


.8a: + .4 y — .52 = 


.5 






.5a:-.6 y + .22 = 
.9a:-|-.5y-.62 = 


.5 
.5 


•^ + :^— =8 










5a: 2y 




9. 


5a: + 3y = 21a:y 




3a: + 42 = -9 






5_3_j 




5a:-Jy = 4 






a: y 




y + 32 = -7 










2+1=1 
3:r^4y 




10. 


3a 26_^ 
a: y 




o'+/=2 






5a 46_^ 




2a: 8y 






a; y 





U, a: — 2y — 22 = — 2a 
a: + y — 3 2 = — 46 
3a:-3y-52 = -2(a + 6) 

^2. 1 + 1 = 1 "• 2a: + y + 2 = 2(a+6) 

' X y a 2y-f- a:4-2 = 2(a4-<?) 

1__1^^ 22 + a: + y = 2(6 + <?) 
a: y 

13. a: + y + 2 — v = 2 15. 3a:4-5y = — 9 

y + z + i;-a:=8 4y-22 = -22 

2 + v + a:— y = 6 32 + t; = 19 

t^4-a:4-y-« = 4 3a: — 2i;= — 2 



SIMULTANEOUS EQUATIONS 55 

16. ±:ZJ±JL^2 20. A+ 3 _ 4 ^jg 

%x 4y oz 

6a? 8y 4^ 

2x by^lOz 



17. 



18. 



19. 



^+3.y_2 


2y + « 


7« — a;_i 


y + 2 


6y + 2a_. 


3a:-9 


2 3, 


- + - = 7 


a; y 


3 4 n 


- + - = 9 


y « 


^ + 6^15 


X z 


111^ 


-+-+i=6 


a; y 2 


3 + 2^1 = 10 


X y z 


? + U? = 14 


X y z 


b , a a^b 


-+- = 


X y a 


a J_ b — a 




X y a 


a + 2 J . 2a-3J 


r 
X y 


3«-i=3a 6 



X y z 
a: z y 
y 2 a: 



a: y 



l + ^ = a + Ja 



23. 6y— 2a;=7a;y 
a?-22=0 

Sy + 5z==--yz 



24. ""^"^>""""" = 3a-6 



25. 4 y2 — 3 a;2 -f 5 a;y = — 16 a;y2 
4 a;y — 3 y2 + 2 a;2 = 3 a?y2 
4a»+7y2 — 3a;y = 4 a;y2 



26. How many examples in Exercise 5 (p. 15) can you 
work at sight ? 



56 DURELL'S ALGEBRA : BOOK TWO 

EXERCISE IS 
Oral 

1. Arthur has a marbles and his brother has 10 more than twice 
as many. How many has his brother ? How many do they have to- 
gether ? 

2. Of the n pupils in a certain school, x are boys. How many are 
girls? 

3. The difference between two numbers is 12 and the less num- 
ber is X. What is the greater ? 

4. The difference of two numbers is a and the greater is y. Find 
the less number. 

5. If a man is x years of age now, how old was he 10 years ago? 
How old will he be a years from now ? 

6. Name three consecutive numbers the smallest of which is x. 
Also three, the largest of which is x. 

7. Express 20 % of x as a fractional part of x. 

8. Express in algebraic language the number whose tens' digit is 
X and units' digit y. Also the number whose hundreds' digit is x, 
tens' digit is y, and units' digit is z, 

9. Express in algebraic language the number which exceeds x by 
30%. 

10. The dimensions of a given rectangle are x and y. State the 
dimensions of a rectangle which exceed those of the given rectangle 
by40o/o. ^ 

11. The dimensions of a given rectangle are x and 2x. State the 
dimensions of a rectangle which exceed those of the given rectangle 
by 20%. 

12. A man bought a house for x dollars and sold it so as to gain a 
dollars. For how many dollars did he sell it ? What per cent did 
he gain? 

13. How many hours will it take an automobile to go a miles at x 
miles an hour? 

14. At a cents a yard, how many yards of calico can be obtained 
in exchange for b dozen eggs worth c cents a dozen ? 

15. A boy has x half dollars and y quarters. How many cents 
has he? 



SIMULTANEOUS EQUATIONS 57 

16. State the interest on p dollars, at r per cent, for t years. 

17. How many square yards are there in the area of a rectangle a 
yards long and b feet wide ? 

18. Express algebraically the following statement : a divided by h 
gives c as a quotient and </ as a remainder. 

19. A man having 10 hours at his disposal rode x hours at the 
rate of 8 miles an hour and walked back during the remainder of his 
time at the rate of 3 miles an hour. State in terms of x the number 
of miles which he rode. Also the number which he walked. By the 
use of these expressions, what equation may be obtained for deter- 
mining X? 

23. Written Problems. 

Ex. The larger of two given numbers exceeds the 
smaller by 5, and the sum of the two numbers is 33. Find 
. the numbers. 

Solution by Use of One Unknotcn 

Let X = the smaller number 

Then a: + 5 = the larger number 

a: + z + 5=33 

^ Ans. 



a: = 14. 1 
; + 5 = 19. r 



Solution by Use of Two Unknowns 

Let X = the larger number 

y = the smaller number 
Then x + y = 33 .-. a: = 19. | ^^^^ 

x- y = 6 y = 14. 1 

2 a: = 38 
Let the pupil check each solution. 

EXERCISE 23 

In solving the following, use either one unknown or 
more than one, selecting the more convenient method for 
each problem : 

1. Find four consecutive numbers whose sum is 50. 



58 DURELL'S ALGEBRA: BOOK TWO 

2. A certain macadam road cost $36,000, of which 
the county paid tvC^ice as much as the state, and the town- 
ship three times as much as the state. How much did 
each pay? 

3. If the amount of potash in a given kind of glass is 
5 times as great as the amount of lime, and the amount of 
sand 3 times as great as the amount of potash, how many 
pounds of each will there be in 6300 lb. of glass? 

4. Four times the height of Mt. Washington is less 
than the height of Mt. Everest by 3842 ft. If the height 
of Mt. Everest is 29,002 ft., find the the height of Mt. 
Washington. 

5. Separate 4800 into three parts, such that the second 
part is three times the first, and one half of the third part 
exceeds the second by 400. 

6. For every dime of his savings that a boy spent for 
books his father gave him a quarter to spend for the same 
purpose. If the boy spent $78.75 in all, how much did 
his father give him ? 

7. New York and Philadelphia are 90 miles apart. 
Two bicyclists start from these places at the same time, 
travel toward each other, and meet in 7 J hours. If one of 
them travels 2 miles an hour faster than the other, what 
is the rate of each? 

8. Find a number, such that three fourths of it ex- 
ceeds two thirds of it by 12. 

9. The difference of two numbers is 12 and three 
times the smaller number exceeds twice the larger num- 
ber by 18. Find the numbers. 

10. A, B, and C together have $1285. A's share is 
$25 more than f of B's, and C's share is ^ of B's. Find 
the share of each. 



SIMULTANEOUS EQUATIONS 59 

11. If 5 bbl. of apples and 4 bbl. of flour together cost 
$40, while 4 bbl. of apples and 5 bbl. of flour cost $46.50, 
find the cost of a barrel of each. 

12. A given mass of metal, composed of lead and iron, 
contains 8 cu. ft. and weighs 4500 lb. If a cubic foot of 
lead weighs 700 lb. and a cubic foot of iron weighs 480 lb., 
how many pounds of each metal are there in the mass? 

13. A mass of metal composed of iron, lead, and alumi- 
num weighs 6280 lb. and contains 13 cu. ft. A cubic foot 
of each of these three metals weighs 480 lb., 700 lb., and 
166 lb., respectively, and the given mass contains 2 cu. ft. 
more of aluminum than of lead. How many cubic feet of 
each of the metals does the mass contain ? 

14. If the cost of a telegram of 13 words between two 
cities is 31 ^ and that of a telegram of 19 words is 43^, 
what is the charge for the first ten words in a message 
and for each word after that ? 

15. A pupil has worked 20 problems. If he should 
work 10 more and get 8 of them right, his average would 
be .80. How many problems has he worked correctly 
thus far? 

16. A ball nine has played 42 games and won 30. If 
after this it should win | of the games played, how many 
games must it play to bring its average up to .72? 

17. Find a fraction, such that if 1 is added to the nu- 
merator, the value of the fraction will become J ; but if 1 
is added to the denominator, the value of the fraction 
will be J. 

18. How much water must be added to 50 gallons of 
milk containing 6 % of butter fat to make a mixture con- 
taining 4 % of butter fat? 



60 



DURELL'S ALGEBRA : BOOK TWO 



19. A mass of silver and copper alloy weighs 128 lb. 
and contains 12 lb. of silver. How many pounds of silver 
must be added to the mass in order that 8 lb. of the re- 
sulting alloy shall contain 1 lb. of silver? 

20. In an athletic meet, the three winning teams made 
scores as follows : 



Tkam 


l8T PlACS 


2d Placs 


8d Plaos 


Total Sooas 


A 


6 


3 


2 


41 


B 


4 


4 


1 


33 


C 


2 


2 


5 


21 



What did each first, second, and third place in an event 
count in this meet? 

21. A farmer one year made a profit of $840 on 15 acres 
planted with wheat and 12 acres planted with potatoes. 
The next year, with equally good crops, he made a profit 
of i996 on 12 acres planted with wheat and 15 acres 
planted with potatoes. How much per acre, on the 
average, did he make on each crop? 

22. The difference of two numbers is 6. One half of 
the smaller number equals one third of the larger. Find 
the numbers. 

23. The difference of two numbers is 28. If the larger 
number is divided by the smaller, the quotient is 3 and 
the remainder 4. Find the numbers. 

24. A boy's average in four of his studies, is .885. 
What grade must he get in a fifth study to bring his 
average up to .90? 

25. If the wheat crop of the United States during a 
period of 6 years averaged 680 millions of bushels, what 



SIMULTANEOUS EQUATIONS 61 

must it average during the next two years to bring the 
average up to 700 millions of bushels for the entire period 
of 7 years ? 

26. The freight charges on shipments between two 
places were as follows : 

600 lb. of 2d class + 300 lb. of 4th class + 400 lb. of 
5th class, $12.46. 

300 lb. of 2d class + 600 lb. of 4th class + 900 lb. of 
5th class, $14. 

500 lb. of 2d class 4- 800 lb. of 4th class 4- 200 lb. of 
5th class, $13.96. 

What was the rate per 100 lb. on each of these classes? 

27. A ton of fertilizer which contains 40 lb. of nitrogen, 
120 lb. of potash, and 80 lb. of phosphate is worth $15.60. 
A ton containing 60 lb. of nitrogen, 110 lb. of potash, and 
120 lb. of phosphate is worth $19.90, while a ton contain- 
ing 55, 105, 100 lb, of each, taken in order, is worth $18. 
Find the value of one pound of each constituent. 

28. If 100 lb. of sea water contains 2^ lb. of salt, how 
much water must be evaporated from 200 lb. of sea water 
in order that 12 lb. of the water remaining shall contain 
lib. of salt? 

29. In the United States, the gold dollar is nine tenths 
gold and one tenth copper. If a mass of metal weighing 
150 lb. contains 18 lb. of copper, how much gold must be 
added to it, to make it ready for coinage into gold dollars ? 

30. How much water must be added to 2 J gal. of alcohol 
which is 90 % pure to make a mixture 80 % pure ? 

31. If a bushel of oats is worth 50^, and a bushel of 
corn is worth 75 ^, how many bushels of each must be used 
to make a mixture of 80 bushels worth 60^ a bushel? 



62 DURELL'S ALGEBRA : BOOK TWO 

32. If two grades of coffee, worth 18^ and 28 ^ are to 
be mixed to make 100 lb. which can be sold for 30^ 
at a profit of 20%, how many pounds of each must be 
used? 

33. A farmer wishes to combine milk containing 4 % of 
butter fat with cream containing 60 % of butter fat in order 
to produce 30 gallons of cream which shall contain 30 % 
of butter fat. How many gallons of milk and how many 
gallons of cream must he use ? 

34. If a certain man can spade a garden in 2 days and 
a boy in 6 days, how long will it take the man and the boy 
together to spade the garden? 

35. If A, B, and C together can do in lOJ days a piece 
of work which B alone can do in 48 days, or C alone in 
32 days, how long will it take A alone to do the work? 

36. One pipe can fill a swimming pool in 24 min. and 
another pipe can fill the same tank in 36 min. How long 
will it take the pipes together to fill the tank ? 

37. Make up and work a similar example concerning 
two pipes which fill a given tank and another pipe which 
at the same time empties the tank. 

38. Of three pipes, the first and second running together 
can fill a swimming pool in 80 min. ; the first and third can 
fill it in 1 hr. 40 rain. ; and the second and third in 2 hr. 
How long will it take each pipe running alone to fill it? 

39. At what time are the hands of a clock together 
between 2 and 3 o'clock ? 

40. At what time between 2 and 3 o'clock are the hands 
of a clock pointing in opposite directions ? 

41. At what time between 6 and 7 o'clock are the hands 
of a clock at right angles to each other? 



SIMULTANEOUS EQUATIONS 63 

42. A man having 10 hr. at his disposal rides oat into 
the country at the rate of 10 mi. an hour and walks back 
at the rate of 3 mi. an hour. Find the distance he rides. 

43. The length of a given rectangle exceeds the width 
by 14 in. But if the length is diminished by 9 in. and 
the width increased by 4 in., the area remains unchanged. 
Find the dimensions of the given rectangle. 

44. A man has $40,000 to invest and from it wishes to 
obtain an annual income of $1800. If he invests part of 
his principal at 4% and the rest at 5%, how much must 
he invest at each of these rates ? 

45. A man invests f of his capital at 4 % and the rest 
at 3J%. The income from these investments is $304. 
Find his capital. 

46. A party of boys purchased a motor boat. They 
found that if there had b^en 2 more boys, they would have 
paid $10 apiece less; but if there had been 2 less, they 
would have paid $20 apiece more. How many boys were 
there and what did the boat cost? 

47. A 42-lb. mass of gold and silver alloy weighed only 
38 lb. when immersed in water. If the gold lost ^^ of its 
weight when weighed under water, and the silver -^ of 
its weight, how many pounds of each metal were there in 
the alloy? 

48. A mass of copper and tin weighing 300 lb. when 
immersed in water weighed 262.6 lb. If the specific 
gravity of copper is 8.8 and that of tin is 7.3, how much 
of each metal was there in the mass? 

49. The denominator of a certain fraction exceeds the 
numerator by 2. If a certain number is added to both 
numerator and denominator, the value of the fraction thus 



64 DURELL'S ALGEBRA: BOOK TWO 

formed is | ; while if this number is subtracted from both 
numerator and denominator, the value of the fraction is ^. 
Find the original fraction. 

50. In a given number of two digits, the units' digit is 
double the tens' digit. If the position of the digits is re- 
versed, the value of the fraction is increased by 36. Find 
the number. 

51. In a given number of three digits, the sum of the 
digits is 9. The units' digit exceeds the hundreds' digit 
by 1. If the units' and hundreds' digits exchange places, 
the value of the number is increased by 99. Find the 
number. 

52. A steamer can run 20 miles an hour in still water. 
If the steamer can go 72 mi. with a current in the same 
time that it can go 48 mi. against the current, what is 
the rate of the current? 

53. A clerk earned $504 in a certain number of months. 
His salary was increased 25% and he then earned $450 
in two months less time than that in which he previously 
earned 1504. What was his original salary per month? 

54. The sum of $7 is to be changed into dimes and 
quarters in such a way that the value of the dimes shall 
equal the value of the quarters. How many dimes and 
how many quarters are there? 

55. A and B together can do a certain piece of work in 
10 days ; but at the end of 7 days A stops working and B 
finishes it by working alone for 6 days. How long would 
it take each man, working alone, to do the entire work ? 

56. A farmer has enough feed for his oxen to last a 
certain number of days. If he sold 10 oxen, his feed 
would last 30 days longer. If, on the other hand, he were 



SIMULTANEOUS EQUATIONS 65 

to buy 10 oxen more, his feed would last 10 days less. 
Find how many oxen he has and for how many days he 
has feed. 

57. Given three metals of the following composition by 
weight: the first, 5 parts gold, 2 silver, 1 lead; the 
second, 2 parts gold, 5 silver, 1 lead ; the third, 8 parts 
gold, 1 silver, 4 lead. To obtain 9 ounces of a metal con- 
taining equal quantities by weight of gold, silver, and lead, 
how many ounces of the first, second, and third must be 
melted together? 

58. The sum of two numbers is a, and five times the 
smaller number exceeds four times the larger by 5. Find 
the numbers. 

59. If A can do in p days a piece of work which B can 
do in q days, how long will it take them together to do the 
work? 

60. Find three consecutive numbers whose sum is a. 

61. Generalize Ex. 11, p. 59, by using a letter for each 
number in the example. 

62. Generalize Ex. 81, p. 61, by using a letter for each 
number in the example. 

63. Who first used the letters a, 6, <?, to represent known 
numbers? (See p. 257.) Tell all you can about this man. 

64. Before the use of a, 6, (?, what algebraic symbols 
were used to represent known numbers? Discuss the rela- 
tive advantages in these different sets of symbols. 

EXERCISE Si 

1. Given A = Iw. find I in terms of A and w. Also 
find w in termis of A and I. 

2. Given a =ip-hprt^ fin(r^jp"in terms of the other 
letters. Also solve for t For J*. 



G6 DURELL'S ALGEBRA: BOOK TWO 

In the following formulas used in geometry, find each 
letter in terms of the others. Give the meaning of each 
formula obtained. 

3. -K'^JJA 5. S^ttRL 

4. ir=i/<6-|-6') 6. C^2irR 

7. Given C^irD and iJ=---, eliminate D (that is, 

obtain a formula which gives (7 in terms of ir and R). 

8. In T= irRQIt + i), find L in terms of the other 
letters. 

Find each letter in terms of the others, in the following 
formulas used in mechanics and physics. If possible, give 
the meaning of the formulas obtained : 

9. S=vt 11. (7=1(^-32) 13. iJ=-21_ 

ff-i-s 

10. LW^lw 12. (7=^ 14. ^=:^ + l. 

R f p p 

15. In the electric furnace, a temperature as high as 
4000° C. has been obtained. What is the equivalent 
temperature on the Fahrenheit scale ? 

16. What temperature is numerically the same on both 
the Centigrade and Fahrenheit scales ? 

SuG. Eliminate C between the equations C = ^(F — 32) and 
C = F. 

17. Find that temperature on the Fahrenheit scale 
which is numerically double the equivalent temperatare 
on the Centigrade scale. 

18. Also find that temperature on the Fahrenheit scale 
which is numerically one half the equivalent temperature 
on the Centigrade scale. 

19. How many examples in Exercise 6 (p. 18) can you 
now work at sight ? 



CHAPTER V 
GRAPHS 

24. A Variable is a quantity which has an indefinite 
number of different values. 

A function is a variable which depends on another 
variable for its value. 

Thus, the area of a circle is a f UDction of the radius of the circle ; 
the wages which a laborer receives is a function of the time that the 
man works. 

A graph is a diagram representing the relation between 
a function and the variable on which the function depends 
for its value. 

A function may depend for its value on more than one variable. 
Thus, the area of a rectangle depends on two quantities — the length 
of the rectangle and the breadth. The present treatment of graphs, 
however, is limited to functions which depend on a single variable. 

In algebra, we study only those functions which have a definite 
value for each definite value of the variable. 

25. Uses of Graphs. A graph is useful in showing at a 
glance the place where the function represented has the 
greatest or least value and where it is changing its value 
most rapidly, and in making evident similar properties of 
the function. 

Graphs of algebraic equations are useful in making evi- 
dent certain properties of equations which are otherwise 
difficult to understand. A graph also often furnishes a 
rapid method of determining the root (or roots) of an 
equation. 

67 



68 



DURELL'S ALGEBRA: BOOK TWO 



26. Framework of Reference. Axes are two straight 
lines perpendicular to each other which are used as an 
auxiliary framework in constructing graphs; as X.X! 
and YT'. 

The X-axis, or axis of absoissas, is the horizontal axis ; as 
XXK The j^axifl, or axis of ordinates, is the vertical axis ; 

as YT. 
,p The origin is the point 

at which the axes inter- 
sect ; as the point 0. 

The ordinate of a point 
is the line drawn from 
the given point parallel 
to the y-axis and termi- 
nated by the a?-axi8. 
The abscissa of a point 
is the part of the 2;-axis intercepted between the origin 
and the ordinate. 

Thus, the ordinate of the point F is AF^ and the 
abscissa is OA. 



The ordinate is 
sometimes termed 
the "y" of a 
point, and the ab- 
scissa, the "a; " of 
a point. 



Q 

I 
( 

z'l — I — I — I — !— h 



(-2,-2)"- 



Ordinates 
above the 2;-axis 
are taken ^s 
plus; those be- 
low, as minus. 
Abscissas to the 
right of the origin are plus ; those to the left are minus. 



P 



I 



H — I IJC 



\S 
—"(1-4) 



GRAPHS 69 

The coordinates of a point are the abscissa and the ordi- 
nate taken together. They are usually written together 
in parenthesis with the abscissa first and a comma between. 

Thus, the point (2, 4) is the point whose abscissa is, 2 and ordi- 
nate 4, or the point P of the figure. Similarly, the point ( — 3, 2) is 
Q; (- 2, - 2) is /? ; and (1, - 4) is 5. 

The quadrants are the four parts into which the axes di- 
vide a plane. Thus, the points P, Q, i2, and S lie in the 
first, second^ thirds 3,nd fourth quadrants, respectively. 

EXERCISE as 

Draw axes and locate each of the following points : 

1. (2, 3), (-3, 2), (4, -2), (-4, -3), (-5, 3), 
(2,-6). 

2. (5, 0), (3J, 0), (0, 4), (0,_- 3), (0, - i). 

3. (3, - V2), (2V3, 0), (V2, -3V3), (-3, - JV2). 

4. Construct the triangle whose vertices are (3, 4), 
(-2,3), (-1,-4). 

5. Construct the quadrilateral whose vertices are 
(-3,4), (-2, -5), (4, -3), (1,4). 

6. All the points on the aj-axis have what ordinate ? 

7. All the points on the y-axis have what abscissa ? 

8. Plot the following pairs of points and find the 
distance between each pair : 

(1) (2, 2), (5, 6) (3) (- 3, 5), (2, - 7) 

(2) (- 3, - 2), (5, 4) (4) (- 4, - 6), (- 1, 3) 

9. Construct the rectangle whose vertices are (—4, 4), 
(3, 4), (-4, -2), (3, -2), and find the number of 
square spaces in its area. 

10. Construct the triangle whose vertices are (3, — 3), 
(—4, — 3), (—1, 5) and find the number of square spaces 
in its area. 



70 



DURELL'S ALGEBRA: BOOK TWO 



11. Construct the triangle whose vertices are (—1, 4), 
(5, — 4), (1, — 1), and find the length of its sides- 

12. Make up and work an example similar to Ex. 9. 

13. In which quadrant are the abscissa and ordinate 
both positive ? Both negative ? 

In which quadrant is the abscissa negative and the 
ordinate positive ? In which is the abscissa positive and 
the ordinate negative ? 

Graphs of Equations op the First Degree 

27. To construct the Graph of an Equation of the First 
Degree containing Two Unknown Quantities, as x and y. 




Let X have a series of convenient values, as 0, 1, 2, 8, etc., 
- 1, -2, - 3, etc.; 

Find the corresponding values of y ; 

Locate the points thus determined, and draw a line throvgh 
these points. 



GRAPHS 



71 



X 


y 





1 


1 


-1 


2 


-3 


3 


-5 


etc. 


etc. 


-1 


3 


-2 


5 


etc. 


etc. 



Ex. Construct the graph of the equation y = 1 — 2 a;. 

Construct the points (0, 1), (1, -1), (2, -3), 
(3, — 5), ( — 1, 3), ( — 2, 5), etc., and draw a line through 
them. The straight line AB \s> thus found to be 
the graph of y = 1 — 2 x. 

28. Linear Equations. It will always be 
found that the graph of an equation of the 
first degree which contains not more than two 
unknown quantities is a straight line. Hence, 

A linear equation is an equation of the first degree. 

29. Abbreviated Method of constructing the Graph of a 
Linear Equation. Since a straight line is determined by 
two points, in order to construct the graph of an equation 
of the first degree, it is sufficient to construct any two points 
of the graph and draw a straight line through them. 

Ex. 1. Graph 3a:-2y = 6. 

Whena:=0,y=-3; 
when y = 0, X = 2. 
Hence, the graph 
passes through the 
points (0, — 3) and 
(2, 0), or CD is the 
required graph. 

The greater the dis- 
tance between the 
points chosen, the 
more accurate the con- 
struction will be. It 
is usually advisable to 
test the result obtained by locating a third point and observing 
whether it falls upon the graph as constructed. 

If the given line does not pass through the origin, or near the 
origin on both axes, it is usually best to construct the line by 
determining the points where the line crosses the axes as above. 




72 DURELL'S ALGEBRA: BOOK TWO 

Ex. 2. Graph 5y - 7a; = 1. 

When X = 0, y = t; when y = 0, x = — |. Hence, the graph passes 
close to the origin on hoth axes. Hence, find two points on the 
required graph at some distance from each other, as by letting x = 0, 
7, and finding y = i, 10. Let the pnpil construct the figure. 



EXSSCISE S6 


Graph the following: 




1. y=x + l 

2. y = a;-3 


^^- 2 =2 


3. 3a;4-4y=12 


11. a;=3 


4. 3a;-4y4-12 = 


12. a;= -2 


5. 3a; + 2y=9 

6. y = 3a; 

7. 2y = a; 


13. y=4 

14. y= -4 


8. 3a; + 5y = l 


15. a; = 


9. a;=2(y-l) 


16. y = 



17. Construct the triangle whose sides are the graphs 
of the equations 4a; + 3y+2=0, 3a;— y + 5=0, 
2a; - 5y 4- 14 = 0. 

18. Graph 2a; — 3y = a when a = 1. On the same 
diagram, graph the equation obtained by letting a = 2. 
Also a =3. 4. -2. 

19. On one diagram graph y = b when 6 = 1; 6 = 2; 
6 = 3, - 2, - 3. 

20. An equation of the form y = 6 represents a line in 
what position with reference to the axis of a; ? To the 
axis of y ? 

21. An equation of the form x = a represents a line in 
what position with reference to each of the two axes ? 



GRAPHS 



73 



22. Make up and work an example similar to Ex. 6. 
To Ex. 17. 

23. How many examples in Exercise 7 (p. 20) can you 
work orally as sight examples ? 

30. Oraphio Solution of SimnltaneooB Linear Equations. If 

we construct the graph of the equation 2y + 3a; = — 5 
(the line -4.J8) and the graph of iy -{-x= 5 (the line 
(72>), and measure the coordinates of their point of 
intersection, we find this point to be (— 3, 2). 



x^ 



Y 




A 


\ 


c-^^ 


(-M>y-"^^_ 


>. ^-^^ 


\ "^"^ 


V 




^ 


s^ 


B 


Y' 



If we solve the pair of simultaneous equations 

^ "I by the ordinary algebraic method, we 

find that a; = — 3 and y = — 2. 

In general, the roots of two simultaneous linear equations 
correspond to the coordinates of the point of intersection of 



74 DURELL'S ALGEBRA: BOOK TWO 

their graphs; for these coordinates are the only ones 
which satisfy both graphs, and their values are also the 
only values of x and y which satisfy both equations. 

Hence, to obtain the graphic solution of two simulta- 
neous equations, 

Draw the graphs of the given equations^ and measure the 
coordinates of the point (or points) of intersection. 

Graphing two simultaneous equations is a conven- 
ient method of testing or checking their algebraic solu- 
tion. 

31. BimnltaiieoaB Linear Equations whose Graphs are 
Parallel Lines. Construct the graph of 3a;— 2y = 6 and 
also of 3 a; — 2 y = 2. 

You will find that the graphs obtained are parallel straight lines. 
Now try to solve the same equations algebraically. You will find 
that when either x or ^ is eliminated, the other unknown quantity 
is eliminated also, and that it is therefore impossible to obtain a 
solution. The reason why an algebraic solution is impossible is 
made clear by the fact that the graphs, being parallel lines, 
cannot intersect ; that is to say, there are no values of x and y 
which will satisfy both of these lines, or both equations, at the same 
time. 

32. Graphic Bolntion of an Equation of the First Degree of 
One Unknown Quantity. By substituting for y in the first 



equation of the pair 



the two equations reduce 
[2^ = ^ 



to 2 a; — 5 = 0. Accordingly, the graphic solution of an 
equation like 2 a; — 5 = can be obtained by combining the 
graphs of y =2 a;— 5 and y =0. In other words, the root 
of 2 a; — 5 = is represented graphically by the abscissa 
of the point where the graph of y = 2 a; — 5 crosses the 
a;-axis. 



GRAPHS 75 

EZ£RCISE 27 

Solve both algebraically and graphically : 

1. 2a;-7y = 9 5. 8a;-f2ya:21 
5a? + 3y = 2 y = 2a? 

2. 8a;-2y = l «• a^~^^=5 

3. a; + 3 + oy = ^ 
7a; + 8y = 6 7. 6a:— 5y=3 

5a;-6y=8 

4. ii;=3 a. i/ = Sx + 9 

2y + 3a;=5 2a; + 7y + 6 = 

9. Construct the graphs of j "" ^ "■ 
^ ^ |9a;-6y = 4 

Can you solve this pair of equations algebraically? 
Give reasons for your answer. 

10. Construct the triangle whose sides are the graphs 
of the equations 3y + 8a; = 26, 4y — 9a; = 15, 7y — a;4-18 
= 0, and find the coordinates of the vertices of the 
triangle. 

U. Construct the quadrilateral whose sides are the 
graphs of the equations y4-2a;=10, 5y + 2a; = 26, 
2^ + lla; + 16== 0, Qy-5z+25 = 0, and find the co- 
ordinates of the vertices of the quadrilateral. 

12. Make up and work an example similar to Ex. 1. 
To Ex. 9. 

13. Also an example similar to Ex. 10. 

14. How many examples in Exercise 9 (p. 24) can you 
now work at sight ? 



76 



DURELL'S ALGEBRA: BOOK TWO 



33. Oraphio Solution of Written Problems. 

I. Bailway Problems. 

Ex. Two places, A and B, are 120 miles apart. At a 
given time a train leaves each of the two places and travels 
toward the other, the train from A at 40 miles an hour 
and the train from B at 20 miles. In how many hours 
will they meet and how many miles from A ? 

The train dispatcher represents the distance between the stations 
by the line A By each space denoting 10 miles. Each space on ^ / 

represents 1 hour. He 
locates E four units to 
the right of A and one 
unit above ABf and F 
two units to the left of B 
and one unit above AB. 
He produces AE and BF 
to meet at C, and draws 
CD perpendicular toAB. 
He obtains the distance 
from A at which the trains meet, by measuring AD to scale (and 
hence determines the siding at which one train must wait for the 
other). Here AD =: SO miles. He obtains the time that elapses 
before the trains meet, by measuring CD to scale. Here CD = 2 
hours. 

The advantage of the graphical method is that in this solution it 
is easy to make allowance for any waits which trains may make at 
stations. Hence, railroad time-tables are often constructed entirely 
by graphical methods. 

II. Problems in the Mixture of Materials. 

Ex. In order to obtain a mixture containing 25 % of 
butter fat, in what proportion should cream containing 
30 % of fat be mixed with milk containing 5%? 

Graphical Solution 

We construct a rectangle, and write in two adjacent comers 
(here the left-hand corners) the per cents of fat (30 and 5) in the two 




30 


25 


20 


5 




5 



GRAPHS 77 

given fluids; and in the middle of the rec- 
tangle we write the per cent (25) desired in 
the mixture. The differences between the 
number in the middle and the numbers in 
the corners (20 and 5) are then found and 
placed as in the diagram. The differences thus found show the rela- 
tive amounts of the given fluids to be used, viz. : 5 parts of milk, and 
20 of cream. 

Now solve this problem algebraically by the method used in solving 
£x. 33, p. 62. 

By an examination of this algebraic solution, discover for yourself 
the reason for the above graphical solution. 

EXESCISE S8 

1. The distance between New York and Philadelphia 
is 90 miles. If a train leaves New York at noon and 
goes 30 miles an hour, and another train leaves Phila- 
delphia at the same time and goes 20 miles an hour, at 
what time will they meet ? How far from New York will 
they meet ? 

2. The distance from New York to Boston by a certain 
route is 240 miles. If a train leaves Boston at 2 p.m. and 
goes at the rate of 45 miles an hour, and another train leaves 
New York at the same time and goes 35 miles an hour, 
at what time and how far from New York will they meet ? 

3. The distance from A to B is 36 miles. At 
9 A.M., a boy starts from A and walks toward B at the 
uniform rate of 4 miles an hour. At the same time, an- 
other boy starts from B on a bicycle and rides toward A 
at the rate of 12 miles an hour, but at the end of each 
hour of riding he rests J hour. By means of a graph, 
determine where and when the two boys will meet. 

4. In order to obtain a mixture containing 20% of 
butter fat, in what proportion must cream containing 
28 % of fat be mixed with milk containing 4 % ? 



78 



DURELL'S ALGEBRA: BOOK TWO 



5. In order to obtain a mixture containing 24 9& of 
butter fat, in what proportion must cream containing 
80 % of fat be mixed with cream containing 20 % ? 

6. In what proportion must coffee worth 24 ^ a pound 
be mixed with coffee worth 16 ^ a pound to make a mix- 
ture worth 18 ^ a pound ? 

7. Make up and work an example similar to Ex. 4. 

8. Also an example similar to Ex. 1. 

9. Also an example similar to Ex. 3. 

EXBRCISS S9 
Review 

1. State the degree of each term of 

3 x«/ - 5 a:» - 3 zy + 7 - 8 a:y 2 + 1 1 y - 7 a: 

2. Remove the parentheses and collect terms in 

3 x2 - 6(z - 1)(2 X + 3) -{- 3 a: + [4 a: - 3x-l - 6(x - 2)]} 

3. Divide 8 a' + 6' + c^ — 6 abc by 2 a + 6 + c and verify your 
result (1) by substitution of numerical values for a, b, and c; (2) by 
multiplication of the quotient by the divisor. 



4. Factor : (1) a* + 4 a^x^ + 18 x* 



(2) 1-1-fi 
^ ^ x^ xy^y^ 



Simplify : 



(3) a2 + 2 a6 + 63 - 3 a - 3 6 

(4) a* - 62 - c2 - 2 6c + 2 ax + x« 



6. 



(ja — 52 a + 6 b — a 

\x-y x-k-yl \x-y x-\-yl 



■y 

6 



(a-6)(a-c) 

l + 8x» 



(c-6)(6-a) 



1 -- 



2x 



1 + 



2x 
l-2x 



9. 



(c-a)(c-6) 



i+i+T 



-^(x-2) 



GRAPHS 79 



Solve: 

10. t(3-2x)-Ka:-3)-l = i(a?+0)-J 



U. 
12. 
13. 



-i|f-(^'---^)l=K*-i)-i*l' 



X — 2 X — 4 X — 3 X — 5 

6- Ix 23 _ 2r+5 5x + 18 

6 12 







3 4(x- 


.9) 


14. 


1^ 
x 

1 


= 7+? 

y 




15. 


10 
2. 

X 


4^4y 


ay 



17. ax = c — by 
qyzz^r-px 

18. (/x4-(a + 6)y = l 
(a + 6)x + rfy = 1 

19. (a + 6)x = 3aft + (a-% 
(a-6)x = ai+(a + 6)y 

16. l-2x_+j^ 5x + ^^Q 

11 13 20. 3x + y + 32 = l 

3y-x-?I^L±il=2 3y-5z = l 

^ 3 9x + 102=l-8y 

21. State and illastrate some of the advantages connected with 
the use of algebraic symbols. (See DurelVs Algebra^ Book One, pp. 
248-250.) 

22. State and illustrate some of the advantages connected with 
the use of elementary algebraic processes. 





EZ£SCISE 80 


Give the value of 


Ghal Review 


1. i— J- 

8z 9z 


3 x_±j( x-j( 
2 2 


2. ^ + ^ 


*• ^'s 


Expand: 




^ (!-!)■ 


- (5-^)" 



80 DURELL'S ALGEBRA: BOOK TWO 

Factor: 



7. :r«-2£ + i, 

y y' 






x^ ory ya 


8. i-± + i 






10. a.-i 


11. Divide each of tbe foUowing by 2: |, |, |, |, |, ^, ^ 


X 3 a a + x 3a 
2y' x' 6* 2 • 26' 


26 
3^' 


56« 
a 


662 
56' 



12. Divide^ by 3 a. By 4a. 36. 6c. 66. 

5 

13. Divide 1 by each of the foUowing fractions: % ?, ^, -, — , 

2 3 6 y 2 6 

ox a -h 

14. Give the reciprocal of each of the fractions named in Ex. 13. 

15. Give the value of - whenx=2. When x = i; |; 4; -; |- 

16. Simplify those of the following fractions which can be reduced 
to lower terms : 

3x 3xy Sx Sx 4 + ar 

3x + y' 3(x + y)' 3y' 3 + x' 4 + y' 

Solve each of the following for x : 

17. i = | 20. ? = 6 
X 5 a 

18. 5 = 4 21. ^ = c 

X X 

19. J- = 4 22. U-? 

2x X 4 

When a = 2, 6 = l,x = 0, give the value of 

26. 3a6x 30. (3a + 6)x 

27. 3a + 6x 31. ax« + 6x 

28. 3 063 

32. (3a + x)« 

^* Jb 33. (3a-x)« 



ax + 6 


a(x + y) 


23. 


-3 = ? 

X 


24. 


1^4 
2 X 


25. 


c_ a 
rf~x 


34. 


(a + 6)x 

aa 


35. 


5a6 

a H-x 


36. 


3a + f. 




CHAPTER VI 
INVOLUTION AND EVOLUTION 

Involution 

34. Involution is the operation of raising an expression 
to any required power. 

Since a power is the product of equal factors, involution is a 
species of multiplication. In this multiplication, the fact that the 
quantities multiplied are equal leads to important abbreviations of 
the work. 

Powers op Monomials 

35. Law of Exponents or Index Law. 

Since a^=^a y.ay.a^ 

(cfiy =:Qa X a X a) (a X a X a) C^ X a X a)(ja X a X a) 

In general, in raising a** to the w'* power, we have the 
factor a taken m x n times, or 

(«»)"• = flr» I. 

Also, (aJ)* = oft X aJ X ai . . . to n factors 

= (a X a X a ••• to w factors) (b xb xb ••• to n factors) 

.-. (a6)'» = a'»6» 11. 

This law enables us to reduce the process of finding the 
power of a product to the simpler process of finding the 
power of each factor of the given product. 

36. Law of Signs. It is evident from the law of signs 
in multiplication that 

(1) An even power of a (juanUty (whether plus or minm) 
is always positive. 

Thus, (-5a6)2 = 25a^2 

81 



82 DURELL'S ALGEBRA: BOOK TWO 

(2) An odd power of a quantity has the same sign as the 
original quantity. 

Thius, (-2ay = -128aT 

37. Involntion of Honomials in General. Hence, to raise 
a monomial to a required power, 

Raise the coefficient to the required power ; 
Multiply the exponent of each literal factor by the index of 
the required power ; 

Prefix the proper sign to the result. 

Ex. ( - 5 ah^y = 625 ofix^. Ans. 

38. Powers of Fractions. By a method similar to that 
used in Art. 35, it can be shown that 

Hence, to raise a fraction to a required power. 
Raise both numerator and denominator to the required 
power ^ and prefix the proper sign to the resulting fraction. 



Ex. f-i^Y^-^I^. Ans. 



Powers of Binomials 

39. General Method. In obtaining a required power of 
a binomial, it is possible to abbreviate the work even more 
than in the involution of a monomial. 

It is sufficient, in taking up the subject here for the first 
time, to obtain several powers of a binomial by actual mul- 
tiplication, and by comparing them, to obtain a general 
method for writing out the power of any binomial. A 
formal proof of the method is given later. (See p. 221.) 
(a + 6)2 = a2 + 2a6+62 



INVOLUTION 83 

If b is negative, the terms containing odd powers of b 
will be negative ; that is, the second, fourth, sixth, and all 
even terms, will be negative. 

Comparing the results obtained, it is perceived that 

I. The nnmber of terms equals the exponent of the power 
of the binomial, plus one. 

II. Exponents. The exponent of a in the first term equals 
the index of the required power, and diminishes by 1 in 
each succeeding term. The exponent of b in the second 
term is 1, and increases by 1 in each succeeding term. 

III. Coefficients. The coefficient of the first term is 1; 
the coefficient of the second term is the index of the re- 
quired power. 

From any given term, to find the coefficient of the next 
term, multiply the coefficient of the given term by the exponent 
of a in that term and divide by the number of the given term, 

IV. Signs of Terms. If the binomial is a difiference, the 
signs of the even terms are minus ; otherwise the signs of 
all the terms are plus. 

Ex. (a + 6y = a7 + 7a«6 + 21a«6a+35a*J«+35a»5* 

The coefficient of the third term = ^-^ = 21. 

2 

The other coefficients are determined similarly. 

Observe that the coefficients of the latter half of the expansion are the 

same as those of the first half in reverse order. 

40. Binomials with Complex Terms. If the terms of the 
given binomial have coefficients or exponents other than 
unity, it is usually best to separate into two steps the 
process of writing out the required power. 



84 DURELL'S ALGEBRA : BOOK TWO 

Ex. (4a«-i6«)6=(4a«/-5(4a«)«(J6«) + 10(4aa)8(JJ8)» 

- 10(4 o2)a(i i«)»+ 5(4 a«)(J6»)* - (J 68)6 

= 1024 a«> - 640 cfiV^ + l^cfilfi - 20 0*6* 

Check the work by letting a = 2, 6 = 2. 

41. Application to Polynomials. 

Ex. (a?-3y+2«)8=[(a?-3y) + 2a]8 

= (a?-3 y)8+ 3(a^-3y)a(2 2) 

+3(a;2_3y)(2g)2+(22)8 
=a^-9a!*y+27a!2y»-27y«+6 a^ 
-36a:V«+54y%+12a^-36y22+8g8, ^1„,, 

Let the pupil check the work. 

xxssasE SI 

Write the square of 
1. -\a% 2. -^ 3. %\ai* 



4. 3a;"y»7i 5. -^a^y»-" 

Write the value of 
6. (3x^2)8 12. (.005)8 



17. 



V 6 6W 



7. (-4aJ)* ^ /2 a86a \» 

8. (-fo'J2)2 ■ I 3 j 18. id)* 

9. (.05)2 14. (-ia^ja)* 19. (3J)2 



10. 



(.005)» "• (-2«y 20. (2i)8 

11. (.05)8 "• (-J^)' 

21. Give the value of 3 x 2". Also of (3 x 2)8. 

22. On squared paper show the meaning of (.4)^, or 
.16. Also of (1.3)8. Of (.3)8. Of (2.3)8. 

23. To 5 c8d8 add the square of one half of -|— 



INVOLUTION 85 

24. Find the value of ar^"^ when a =s |, r = 3, and 
n=5. 

25. Show that 2^ X 6^ = lO'^. Is there any advantage 
in knowing this relation ? 

Expand and check : 

26. (b + xy 31. (Ja + 2J)fi 36. (3aa-2J)* 

27. (b-xy 32. (a-iy 37. (Ja« + 3J)« 

28. (a + a:)8 33. (|?2-2y)« 38. (2a2-jy 

29. (b-^xy 34. a + 3cP/ 3^^ /g^^Y 

30. (ib + 2xy 35. (3r^-l)* ' \ V 
40. (a^2-ir+l)8 41. (l + 3a;-a?)8 

42. (a:2+a.y+y2)4 43. (3-.a;+2aj2)8 44. (a?-a;-l)* 

45. How many terms are there in the expansion of 
da + xy? OiCa + xy? Of(a + a:)"? What is the num- 
ber of the middle term in the expansion of (a + a;)^? 

46. Write out the last three terms in the expansion of 
(a + a;)i8. Also of (a - a:)i8. 

Give orally the value of the various powers of 

47. 2 up to 2'^ 49. 4 up to 4^ 

48. 3 up to 3* 50. 6 up to 6* 

51. 6 up to 6* 

52. The square of each number from 1* up to 29^. 

53. Give the value of 2«, 3«, 2®, 5\ 1«, 2^ 6*. 

54. Give at sight the value of ofi^x when a; = 2. 
Also when x = l. — 1. 0. J. 

55. Make up and work an example similar to Ex. 12. 
To Ex. 37. 

56. Also an example similar to Ex. 40. To Ex. 46. 



86 DURELL'S ALGEBRA: BOOK TWO 

Evolution 

42. A Boot of a given quantity is a quantity which, 
taken as a factor a certain number of times, will produce 
the given quantity. 

43. Evolntion is the process of finding a required root 
of a quantity. 

What is the radical or rootsign ? What is the mean- 
ing of V9? Of<^? OfV7? 

44. ITumber of Boots. Taking a particular example, we 
find that ' V3 has two values, viz. : + 2 and — 2, for 
( + 2)2 = 4 and (-2)2 = 4. 

A number containing a square root of a negative quan- 
tity is termed an imaginary number. 

A real number is a number which does not contain an 
imaginary number. 

The nature of the square of an imaginary number, as of V— 4, 
is explained in Chapter IX (p. 119). 

If we include imaginary roots, it may be shown that when any 
root of a given number is extracted, the number of possible roots equals 
the index of the root to he extracted. 

Thus, in taking the cube root of 8, we find three possible roots, 
viz. : 2, - 1 + a/^, and - 1 - V^. 

45. The Principal Boot of a number is that real root of 
the number which has the same sign as the number itself. 

Thus, the principal root for Vi is 2; for \/27 is 3; for \^-27 
is -3. 

In this chapter, only the principal roots of the numbers are con- 
sidered. 

Evolution of Monomials 

46. Index Law. Since (a«)" = a«" (Art. 36, p. 81), it 
follows that 

"v^a""* = a"» I. 

where m and n are positive integers. 



EVOLUTION 87 

Hence, the process of finding the root of a quantity 
affected by an exponent becomes simply a division of 
exponents. 

Also, ^ah = -VaVh II. 

For, let Va = x, -y/b = y ; 

.-. a:» = a . . (1) y» = J . . (2) 
But a;»y» = {xyY (by Art. 35) 

Substitute a^ and y" from (1) and (2), 

a5 = (v^aV6)" (3) 

Extract the nth root of each member of (3), 

Vab = y/a^b 
This reduces the process of finding the nth root of a 
product to the simpler process of finding the nth root of 
each of the factors composing the product. 

47. Method. Hence, to extract a required root of any 
monomial, 

Eoctract the required root of the coefficient; 

Divide the exponent of each letter hy the index of the re- 
quired root; 

Prefix the proper sign to the result. 

How may the work be checked ? 

BXSRCISS 82 

Write the square root of 

1. 16 aV 3. ^aVy* g 25 ar^"^* 

2. 9afiy^ 4. ^^a^ ' 16 
Write the value of 



6. VSTM* 9- </l6a*b<> 12. il^^ 



4 



13 i\ 32aio 



16 

243 i6" 



88 DURELL'S ALGEBRA: BOOK TWO 

Find the value of x in each of the following equations : 

14. a?* = 81 16. a;7 = 128 

15. a;6 = -32 17. ir« = ^ 

18. Find the largest square factor in each of the foUow- 
^^g ' 64, 98, 320, 242, 460. 

19. Find the largest cube which is a factor of each of 
the following : 

40, 136, 88, 260, 432, 667, 686. 
Extract the square root of each of the following by tak- 
ing out pairs of like factors (or square factors) : 

20. 6184 22. 21x76x112 

21. 11664 23. 11x27x99x192 

24. Make up and work an example similar to Ex. 8. 
To Ex. 9. To Ey. 22. 

Square Root 

48. Square Boot of Polynomials. In order to determine 
a general method for finding the square root of any poly- 
nomial which is a square, we consider the relation between 
the terms of a binomial and the terms of its square; as 
between a -f- ft and its square, a^ + 2 crJ -f- ft^. This relation 
stated in inverse form gives us the required method. 
The essence of the method consists in writing a^ + 2 a6 + 6^ 
in the form a^ + J(2 a + 6). 

In squaring a trinomial, a -f- J + <?, we may regard a + 1 
as a single quantity, and denote it by a symbol, as p. We 
then obtain the square in the form p^ + 2pc + A 

Evidently we may reverse this process, and extract a 
square root to three terms by regarding two terms of the 
root, when found, as a single quantity. Similarly, a fourth 
term of a root, or any number of terms, may be obtained 
by regarding the root already found as a single quantity. 



EVOLUTION 



89 



Ex. Extract the square root of a^ + lOcfll + lda^^-- 
a* + 10 a86 + 19 a^b^ - 30 db* + 9 6< Igg + 5 aft - 3 &« . Root. 



2a2+ 6ab 



10a»6 + 19a2/i2 
10a»ft + 25a262 



2a2 + 10a6-362 



-6a26a-30a6»+96* 



The work may be checked by squaring the result obtained, or by 
numerical substitution. 

Let the pupil state the above process as a rule. 

49. Square Boot of Arithmetical ITnmbers. The same 
general method as that used in Art. 48 may be used to 
extract the square root of arithmetical numbers. 

The details of the method of extracting the square root of num- 
bers are explained in arithmetic (see DurelVs Advanced Arithmetic). 
As an illustration of the process, we give the following example : 

Ex. Extract the square root of 679.6449. 

6f9.64l9 |26.07 Root 

4 

46 



279 
276 



5207 



36449 
36449 



50. For extraction of Cube Root see Appendix, p. 267. 

BXERCISS 33 

Extract the square root and check : 

1. 29J2_266*4-1266-1468-106 + 96« + l 

2. 4|?«-126y + 20jt?8 + 96*-8062+25 

3. a2«2a6 + 62 + 25 + 10a-10J 

4. 24 a^b + 24 a66 + 9 ^6 ^ 9 j6 _ 8 a*62 _ 8 a%^ - 50 a^lfl 
5. ' 22 a:*- 20 a;8 + 4 - 4 iT + 17a:2^_ 9 2^ _ 24a:5 



90 DURELL'S ALGEBRA: BOOK TWO 

6. |a!*-3a:«y+16a:«^-10a;y«+25y 

8. ^+<l£+ll+6a^g 
a^ a X ar 

9. — H f--^— 2h ax 

x^ 4: a^ a 

10. ^ + -L + ^ + 2£ 2^_^^ 
9 ^yar2^;?2^ya; 8 ^ 3 j? 

Find to three terms the square root of 

21. cfi -\-6b 12. 1 — 4 a; 13. ir^ _ 3 ^y 4. ^a 

Find the square root and check wherever possible : 

14. 283024 16. 8042896 18. .64048009 

15. 9312.25 17. 4916.8144 19. 10.06475625 

20. 17.5 22. .4 24. ^ 26. 9.0042 

21. 7f 23. 9^ 25. .081 27. 176.5 
Compute to three places the value of 

28. V3TV2 29. V4Vr-V5 30. V3(V7+V^) 

31. Find the altitude of an equilateral triangle whose 
side is 27 in. 

32. Find the side of an equilateral triangle whose alti- 
tude is 27 in. 

33. Find the side of a square whose diagonal is 36 in. 

34. If a city park is 800 yd. long and 600 yd. wide, 
how much is saved by walking from a corner to the 
opposite corner along a diagonal instead of along the 
sides? How much time does a man save in a year by 
walking thus, if he walks at the rate of 4 mi. an^hour, 
and crosses the park 4 times a day on 300 days? 



EVOLUTION 91 



35. If Jr = V«(«-.a)(«-6)(«-c), a«126, 6 = 60, 
c = 148, s = J^(a + 6 + c), find K. What is the meaning 
of this process in geometry ? 

36. Find in feet the radius of a circle whose area is one 
square rod. 

37. Find in feet the length of the tether by which a cow 
must be tied in order that she may graze over two fifths 
of an acre. 

38. If 4 Trip = 200 sq. in., and tt = ^, find B. What 
is the meaning of this process as applied to the sphere ? 

39. The area of California is 168,300 sq. mi. Find the 
side of a square having an equivalent area. How may 
you then visualize the area of California ? 

40. By the use of the method of Ex. 39, visualize the 
area of the state in which you live. 

41. A bushel measure is to be a square 12 inches deep. 
Find in inches the inside edge of the square top. 

42. Make up and work an example similar to Ex. 20. 
To Ex. 34. 

43. Also an example similar to Ex. 11. To Ex. 37. 

44. How many examples in Exercise 14 (p. 33) can 
you now work at sight ? 



CHAPTER VII 
EXPONENTS 

51. PodtiYe Integral Exponents. Using o^ as a brief 
symbol for ax ax a^ and a"* as a brief symbol for a x a 

xaxa to m factors, we have already found the 

following laws to govern the use of positive integral ex- 
ponents: 

I. a« X a" = flT^^ III. (cry = a**" 

II. — =a«-", if m>n „ ;" 

a" V. (a6)'» = a'»6» 

52. Fractional and Negative Exponents. We have seen 
that by using fractions as well as integers, and negative as 
well as positive quantity, the field of quantity and opera- 
tion in algebra is greatly extended and some processes are 
made simpler, others more powerful. These same advan- 
tages are secured by the use of fractional and negative 
exponents. 

Let us suppose that the first and fundamental Index 
Law, a^ X a^'ss a**"^", holds for fractional and negative ex- 
ponents, and then inquire what meaning must be assigned 
to these exponents. 

We limit the fractional and negative exponents here treated to 
those whose terms are either positive or negative integers, and com- 
mensurable; that is, expressible in terms of the unit of quantity 
used in the given problem. 

Exponents like V2, as in a '^^, are not included in the discussion, 
though the student will find later that the same laws hold for these 
exponents. 



EXPONENTS 93 

53. L Meaning of a Fractional Exponent. 
Since by Index Law I, 

it follows that a* is one of the three equal factors which 

may be considered as composing a*; that is, a* is the cube 
root of aK 

So, in general, 

oi X a^ xc^xaq to q factors 

'4.?+€+- to q temu 
= »f t t 

Hence, in general, in a fractional exponent the numerator 
denotes the power of the base that is to be taken^ and the de- 
nominator denotes the root that is to be extracted. 

Ex.1. 27*=\/27* = 3* = 81. Ans. 

Ex. 2. aixaixai:=: a*^*^* = a«. Ans. 

Ex. 3. V2 af^^ . 2^3f'^ = 2^af^^ . 2*3^-^ = 2!h^ =4 a^. 

Ans. 

Note that in Ex. 1 it is best to extract the required rootfinU 
In the examples which involve letters, the work may often be 
checked by numerical substitutions. 

SXERCISS 34 

Express with radical signs : 

1. ic* 3. 3ji 5. 5ir*yi 7. 5a^ 

2. y* 4. 2a*6* 6. 3aM 8. ab\ 
Express with fractional exponents : 

9. W IL b</^ 13. -^Vy^ ^x-s/lj 

10. Vi 12. a^ 14. 3Va\^ * ab^Vc 



s 



94 DURELL'S ALGEBRA : BOOK TWO 

Find the value of 

16. 9* 19. </W 22. (-27)* 25. (^)* 

17. 16* 20. a/25« 23. (-32)* 26. (- ^V)* 

18. 81* 21. </W 24. (-125)* 27. (J4)* 
Simplify by performing the operations indicated: 

28. a* X a* 31. 3*a* x 8*a* 34. 2* x 2* 

29. 86* X 26* 32. a* x a* 35. ^^5 . VP 

30. a*y* X a*y* 33. a* + a* 36. 6*V^ • a:* V6 

37. 3^* . 3* . 3* «. 6»+»« . 6«-~ - 6«-» 

38. y«^*.y«-* 3 a* V? 4^Vi 

43. ^ — • :^^ — 

39. aJ»«-2& . aA.+26 6V6 3V6^ 

40. a:2a+5.^3«-5 V3\/5 3'\/6* 

41. a2+p . ^p-3 . ^8p-6 • -n/2-v^7 2*\/7^ 

45. Find the value of 5* to three decimal places. (See 
Art. 49.) Also of 5* or Vl25. Multiply the two results. 
Compare the amount of work in this combination of 
processes with that of finding the value of 6^. Which 
method gives the more accurate result ? 

46. What two parts are there to every power ? What 
is the difference between an exponent and a power ? 

47. Make up and work an example similar to Ex. 34. 
To Ex. 42. 

48. Also an example similar to Ex. 45. 

49. Work again Exercise 19 (p. 48). 



EXPONENTS 95 

54. n. Meaning of the Exponent Zero, or of (fi. 
By direct division, — = 1 

By subtraction of exponents, — = a® .-. a<^ = 1 

a"* 

Thus, a^ may be regarded as the result of dividing some power of 
a by itself. 

An expression like px^ + ^x + r is sometimes written jdx^ -\- qx-^- rsfif 
the advantage being that in the latter form every term contains an x, 

55. m. Meaning of a ITegatiye Exponent. 

a* 
By subtraction of exponents, = '»~*» 

By cancellation, 



Ex. 1, 
Ex. 2. 



Negative exponents are useful in enabling us to write 
certain decimal fractions in an abbreviated form. 

Ex. 3. Express .000000007 in a briefer form by the 
use of negative exponents. 

.000000007 = = -^ = 7 X 10-». Ans. 

1,000,000,000 10» 

56. Transference of Factors in Terms of a Fraction. It 

follows from the meaning of a negative exponent that any 
factor may he transferred from the numerator to the denom- 
inator of a fraction^ or vice versa^ provided the sign of the ex- 
ponent of the factor is changed. 



""■" a*+. - " 




a" 


a* 1 


a*+» a* 


X a* a* 


a" 




2-« = i = l. 

28 8 


Am. 


4f 8 


Ans. 



96 DURELL'S ALGEBRA: BOOK TWO 

Ex. 1. Transfer to the numerator the factors of the de- 
nominator of -• 

^^ =5a6-»gy<. Am, 

Ex. 2. Express with positive exponents ^^ 



3 6-ay-i 3 2^ 



BXERCISS 35 

Transfer to the numerator all the factors of the de- 
nominator in 



Express with positive exponents : 
5. aV>-^ 7. 5a-V* 9- 3(x~y)-2 

y 3a-86~* 1.11 



6 



Obtain the value of 



11. 



9-^ 16. —^ 21. (12J)-*-^C^)-i 

4"* 



-i 



13. -^ 18. 4-3+8-* 23. 40 + ("1)"" 

1 /4V* 2-8 . 3-2 . 4-* 

14. -J 19. - 24. 5 — 

15. P, 



(-f)- 



EXPONENTS 97 

25. Express .000000003 in a briefer form by the use of 
negative exponents. How many more figures and symbols 
are there in the first form than in the second ? 

26. Express .000000001 in a briefer form by the use of 
negative exponents. 

27. Show that 1 millimeter equals 10"^ meter. 

28. The micron is a small unit of measure equal to one 
millionth of a meter. Express it as a part of a meter by 
use of a negative exponent. 

29. The length of a wave of violet light is. 000016 in. 
Express this number as 16 in. multiplied by a power of 10. 

Give the value of 

30. 6o,500o,(lJ,(5)\(-7)0,(a + 6)o,|. 

31. 8 X 50, 5^)', 3 ao, (3 a)0, 9* -*- 80, 1. 

32. 1*, 1-4, 10, 1« X 1"*, 9* X 1*, 5 - 40 X 1"*. 

33. 8"*xl6*x20 34. 4-ax8* + 2-8 

35. Express 40 as some power of 4 divided by itself. . 

36. Express a^o as some power of x divided by itself. 

37. Express 4"^ as the quotient of two powers of 4. 
Express x"^ as the quotient of two powers of x. Ex- 
press a;""* in like manner. 

38. State the value of |40. Of $4-2. f 4"*. 

39. Which is the greater, (J)-2 or (J) "3? 

Simplify the following by performing the indicated 
operations, and reducing the results : 

40. 6a*x"*.aV 42. 8a-2-^2a-3 

41. 8ir"*y + 4xV *3. a^y/^^x'^ 



98 DURBLL'S ALGEBRA: BOOK TWO 



47. a»y"^-H3rV5^ 48. " J^^" 49. ^ ^^ 



50. Determine which of the following has the greatest 
value: (J)8, (|)-a (j)0, j. 

Give the value of 

51. 4"* + 8*-(J)"* 52. 40 + 2^-6x4-* 

53. Make up and work an example similar to Ex. 20. 
To Ex. 25. Ex. 38. Ex. 51. 

54. How many of the examples in this Exercise can 
you work at sight? 

55. How many examples in Exercise 12 (p. 28) can 
you now work at sight ? 

57. Meaning of (o^)" for Fractional and ITegatiYe Expo- 
nents. We now extend the law (a*")" = a*"** to fractional 
and negative exponents. 

.Ex. 1. Find the value of (9^)"*. 

(9*)-t = 9-s = l = 4. An. 

Ex.2, n^a-^^yf^n^^-^ 

81-^a:-* 

16*6 
- 8J • '*'"• 



EXPONENTS 99 

Hence, in general, to simplify a complex expression in 
exponents, 

Convert each radical tign into a fractional exponent; 
Convert each power of a power into a power with a tingle 
eaponent; 

Convert each negative exponent into a positive exponent ; 
Simplify hy cancellations and collections. 

EXERCISE S6 
Reduce to simplest form : 

1- (4')'* 6. (9 *-*)-» 11. (27*)"* 

a. («-«)* 7. 2"+»-2«-» la. (5a-*)» 

3. (6*)0 8. (2«+»)«-» 13. (-8a«H)-« 

4. (64*)* 9. (a;"*y*)-« 14. (ar»'+« • x*^)* 

5. (4 a*)* 10. (4a-«6-8a;)* 15. (49a!-*3^)-* 

17. 2x-^i</^H^-)~' 24. V-s/W^' 



18, ■-' •'-» 



{V(V^)-*}" (a"^')' 

"**• (a-)' 

vwxy/z 



f 8 g-y/g-g \ 



19. (a3'i-2')"i 
20. 

21. 
22. 



29. 8'* + 5a«-7(a: + 2y)<>-r* 



100 DURELL'S ALGEBRA: BOOK TWO 



30. 
31. 

32. 

34. 






35. By a numerical substitution, show that — does not 

2?** 

equal a?. Also that — does not equal ofl*. 

36. Write the square root of each of the following: 
9a;-*, 9ar-«,^a;"*, 9 a;-*, 16 a'*, 16 a"*. 

37. Solve :r"* = 27. 

SuG. Raise both sides to the power (— }). 

Then, {x"^)"^ = (27)"* 

Hence, x = 27"** = — = i .-. x = - . i4fw. 

27f 9 9 

Find the value of x in each of the following : 

38. a;* = 2 42. a;"* = - | 46. a:"* = — g"^ 

39. a;* = -27 43. a;"* = l 47. 3a;*=2 

40. a:-^- = 3 "• a;-- = 2 43. 4a;* = 9 

41. a;"* = 4 45. a;"" = — 3 49. 4a;"*=9 

50. Make up and work an example similar to Ex. 11. 
To Ex. 28. Ex. 41. 

51. Practice oral work as in Exercise 30 (p. 79). 



EXPONENTS 101 

58. Polynomialfl whose Terms contain Fractional or Negative 
Exponents. 

Ex. 1. Multiply 

x-^ — 2 x'^'ff^ + 4 y by x-^ + 2 x"^}/^ + 4 y. 

a;-2 _ 2 x~^yi + 4 z-^y 

+ 2 x-iyi - 4x-iy + 8 x~iyl 

+ 4 x-^y - 8 x-hji + 16 ?/« 

x-2 +4 z-i^ + 16 y\ Product. 

Ex. 2. Extract the square root of 

a-i + --2a"i+-^-8a-i + l. 
a Va 

Writing the expression by use of exponents only, 

a-i + 8 a-i - 2a-l + 16 a-i - 8 a't + l |a"i + 4a-i-l 



2a-l + 4a-* 



8a-i-2a~* + 16a-i 
8 rt-i + 16 a-i 



2a-J + 4a"i-l 



- 2 a"* - 8 a-i + 1 

- 2 a-J - 8 a-t + 1 



EXERCISE 37 

1. Arrange 5 x~^ + S + 4a^ — 3x-'X-^ in descending 

order of magnitude. 

_- - 

Also arrange a ^ — a"" + a" — 1 + a^ in ascending order. 

Multiply : 

2. 3 a:*- 2 a:* + 3 a:* + 4 by 2 a;* -3 

3. 3a-2--3a-^ + 2by 3a-2-2a-^ 

4. 3a*-2a*i*-46*by4a^-36* 



102 DURELL'S ALGEBRA: BOOK TWO 

5. 3a;*-5a:*+4by 3-4a;"* 

6. x"^ — x'^y + y^ by x''^ + x-^y + y^ 

7. a;* — ^y^ + 2 y* by y""* 4- 2;""*y"* + 2 a;"* 
Rearrange and then multiply : 

8. 3 a? - 4 a;-* + 3 by 2 a;-^ + 3 a; 

9. 3a:* + 2a:"*-a;*by a;"* + 2a;* 

,^ o 1 2Va , 6Va 3 , q 

10. 3 a —I- by h 9 a 

y Vy V^ y 
Divide : 

11. 6 a* - a + 1 by 2 a* + a* 

12. x~^ — y^ by a:"^ — y 

13. a-2 + a-iJ + i^ by a"! - ah^ + 5 

14. a:* 4- y~* by a;* 4- y~* 

15. 2a-i + 3 + a + 6aaby 2a"*-a* + 3a* 

16. a:^ + 2a;*-16a;"*-32a:-ibya;*-2a;* + 4a:"^-8a:"* 

17. a* + aft* 4- 5* - 2 a*62 _ ^ij i^y a* - 6* 4- a*i - aJ* 
Extract the square root of 

18. 4a?-12a:*4-7a:4-3a;*4-i 

19. a:* — 4 a;^y* 4- 4 a:y 20. Qxy-^ + Viy-^ + 4iX-^ 

21. 16a;*4-24A*-7a-12a*a:"*4-4a2a;"^ 

22. 13-126-*4-i*-66*4-4J-* 

23. ^cfi^lah^^\^aV>-^ah^ + ^dl^ 

2^ 7? a? X 

J 4 V6 



EXPONENTS 103 

EXERCISE 38 
Review 

1. Simplify (J)* + 4« - 8*. Also (t)-i . 4^ • 8*. 

2. Which is greater, (2»)* or 2' • 2*? How many times greater? 

3. Which is greater, (4~i) • or 4~a . 4* ? How many times greater ? 

4. Does (x^y equal x''^ ? Illustrate numerically. 

5. Divide 9 a* - 21 a« vi - s^ax-h 12 a-^xi by J a» - 4 ari 
Simplify : 

g x^(x^-iy 9. (a;«)«+6 . (a*)— » -i- (!:•+»)• 

^"■^^ ■ ^'^ 10. Solve 3 xi = 32 - 5 art 

7. (a:«-*)«+* -5- a;-** 

12. Arrange and extract the square root of 

x-^ + yi + 2 ar~iyi - 2 a: " iy - ar-iyJ 

13. Find the value of (x * - y * ) "^ (^ - y') 

,., a*fa-^ Factor: 

14. Simplify ^VVr i 

15. Simplify "" " ^«i^F^ 18. x* - 5a:J + 6 

19. a;5 — w« 

16. Simplify [(a») »]»-i 20. Divide a^ - 6» by Va-Vi 

21. Find the difference between the value of (})"^ and that of 
(- 2) -2. Also between (- 2)-2 and - 2-2. 

22. Does 2a""« equal — -? Why? Give a numerical illustration. 

23. Does ^^ t f '^ equal ^-±-^ ? Why? Illustrate by giving a, 

5 a'6 

6, x, and y convenient numerical values. 

24. Does a~i + b-^ equal -t ? Explain as in Ex. 23. 

25. Make up an example similar to each of the three preceding 
examples. 



104 DURELL'S ALGEBRA: BOOK TWO 

27. Reduce (n - l)x«(a:" + n) " + (x» + n) * to -^^ \ 

28. Show that K^ + «r^-K^-«r^ = - (x+a)U(x-a)i 

(x + a)i - (x- fl)i 3(x2 - a^)* 

29. Express 8» as a power of 2. Also 4* • 8* . 8" J, 4» • 4»+2 . 8»-i 

30. Simplify ^"(-""')" 31. Expand (x* - 4x-i)» 

32. Expand {^/x-2 \/xy 

33. In the year 1910 the record time for 1 mile traveled on a 
bicycle was 1 min. 7 sec, which was 12J sec. more than twice the 
record for 1 mile traveled by an automobile. Find the latter record. 

34. A certain quantity of 18^ coffee was mixed with 30^ coffee to 
make a mixture of 100 lb. worth 22^ a pound. How many pounds of 
each grade of coffee were used ? 

35. Who first suggested the use of a fractional exponent and 
when ? Who first showed that such exponents could be used accord- 
ing to mathematical laws ? Who first used zero as an exponent and 
when? 

EXERCISE 39 

Oral 
Give the value of each of the following : 

1. 4*, 4-J, 4-2, 40 (4-2)-i, (4-2)i» _!. 

2. «A(jn«)"^4i+(t)* 

3. a:«+* • x^~^ - (j:«+*)«-* 

n4_l n-1 

5. X 2 X 2 

6. 2»+i ^"-^ 

7. 2^ • 2*, 3^ . 3"i 

8. 4^.42, 4^.40, 7a:o + 5a0 

9. (a^ + b^)(J-h^) 



10. 


(a* + 6*) (a* -6*) 


11. 


(J _ 5*) ^ (a* - fti) 


12. 


(x-n + yXx-n - y) 


13. 


(ai + 6*)2 


14. 


(a~^ + 6^)2 


15. 


a:-8 - r* 



EXPONENTS 105 

Factor: 

16. a*-d* 18. a-«-6-« 

17. a* -6* 19. a;-*-9 

QO Q QO 

20. Give the value of each of the following : — , — , — , 3® x 5, 
3 X 5^ 30 X 50, 30 + 50, 30 - 5. 

21. Give the value of 16* • 16^. Of (16*)*, 16"^ • 16"*, (16~*)"*. 

22. Give the value of ^ . Of :^ . Of «-+» • x^-^ • x^-: 

X** 2n 

Give the value of x when 

23. x* = 3 25. a;~* = 2 27. x"^ = 3 

24. a:* =4 26. x"* = 4 28. x"^ = 8 

29. Give the square root of each of the following: 4x"*, 4x », 
4x-», a-%\ ax*^""*, Ja'ft*. 

30. Give the value of each of the following : (.3)-2, I-2 x 1^ x 1"^, 

(.01)*, 1"* ^ 40. 

31. Give the reciprocal of 2. Of f, - }, 4-2, 8"*, 1"*. 



CHAPTER VIII 
RADICALS 

59. Indicated Soots. The root of a quantity may be 
indicated in two different ways : 

(1) By the use of a fractional exponent ; as a*. 

(2) By the use of a radical sign ; as -Va, 

For some purposes, one of these methods is better ; for 
some, the other method. 

Thus, when we have bi x bi -»- 6~i, where the quantities are alike 
except in tlieir exponents, it is usually better to use fractional ex- 
ponents to indicate roots. But if we have 7Vl2 — 6V3 + 2V48, 
where exponents are alike, but coefficients and bases unlike, it is 
usually better to use the radical sign to indicate roots. 

In the preceding chapter we considered exponents ; we 
have now to investigate the properties of radicals. 

60. A Radical is a root of a quantity indicated by the 
use of the radical sign ; as Va, "v^H. 

The radicand is the quantity under the radical sign. 
In treating radicals, we deal only with principal roots 
(see Art. 45, p. 86), unless the contrary is stated. 

61. Surds. An indicated root which may be exactly ex- 
tracted is said to be rational; as V27, since the cube root 
of 27 is 3. 

A surd is an indicated root which cannot be exactly ex- 
tracted; as V5, V4. 

62. The Coefficient of a radical is the number prefixed to 
the radical proper, to show how many times the radical is 
taken. 

Thus, the coeflicient of 3V7 is 3 ; of 5(a + b)y/x is 5(a + 6). 

106 



TRANSFORMATIONS OF RADICALS 107 

63. Entire Surds. If a surd has unity for its coefficient, 
it is said to be entire. 

64. The Degree of a radical is the number of the indicated 
root. 

Thus, V5 is a radical of the third degree. 

65. Similar Kadioals are those which have the same 
quantity under the radical sign and the same index. 
(The coefficients and signs of the radicals may be unlike. 
Hence, similar radicals must be alike in two respects, and 
may be unlike in two other respects.) 

Thus, 3V7 and — 5"\/7 are similar radicals. 

66. Fundamental Principle. Since a radical and a 
quantity affected by a fractional exponent differ only in 
form, in investigating the properties of radicals we may 
use the properties obtained for fractional exponents. 

1 11 
Thus, since (aft)" = a«^ 

... ^y^=^.^ 

Transformations of Radicals 

67. Simplification of a Quantity under the Eadical Sig^ 
Ex. 1. Simplify Jg\/250aV. 

A v^250 a^3fi = iV v^l25 ar* x 2 a^ = J x^y/2d^. Am. 



Ex.2. Simplify ^^JM 



27 a 



2 



£^^2a /8a:g^8a^2a^^I^ 
U a:^27a 3a. x^Sla' 



x6a=^V(3a. Ans. 



108 DURELL'S ALGEBRA: BOOK TWO 

EXERCISE 40 

Express in the simplest form : 
1. </16a*a;/ 5. fVSl^^p^ 9. </8{hfiy^ 



2. V16 a*x^2 6. |V54a%V lO. axVcfl^ 

3. ^lt)a*xya 7. jV25a«P' 11. </M^i^ 



4. V54a362^ 8. -jV25a86 12. V729 6V 



13. -Vrd'Ia^i^ ^ J12aV 



14. V43^a=*z*'» 



26 6* 



15. -v/a»+V» 19. V72 X 40 



16. VC4a^ + 4.+ l)y P£2^3J2 

17. V(a + 6)(a2-6a) ' '>' 49(a;-y)2 
Simplify : 

21. lOVf 26. ■v'l *'|a^ 



„ 3^/20a2 ,, «/9a2 

24. VSy 29. •V'l 

'8 a; ^4 a* '(a + 6)* 



•1^+1 

/gan-8 



''• Vf5 ^'- Mu±^» 






TRANSFORMATIONS OF RADICALS 109 

38. Given V8 = 1.73205+, find the value of Vl08 to 
four decimal places in the shortest way. Also that of 
Vly2. 

39. Using Vt) = 2.44948'*', make up and work an ex- 
ample similar to Ex. 38. 

40. Who first used the sign V ^o denote a root? 
Who first suggested the use of the vinculum instead of 
the parenthesis in this connection, as in using Va -h b in- 
stead of V(^ + ^) ' How were higher roots like ^/^ y/ 
first indicated after the invention of the radical sign ? 







EXERCISE 41 










Oral 




Simplify 


at sight 


;: 






1. Vl2 




5. 4\/i 


9. -I^ 


«./-f 


2. V? 




6. 10\/} 


10. ^ 


"■ yi 


3. 2Vi 

4. 6Vj 




7. x^V- 

8. Vj 


11. 5Vi 


16. 6>/ij 



68. Making Entire Surds. 

Ex. 2^6=\/8"x5 = ^I^. ^n«. 

69. Simplification of Indices. 

Ex. a/8 aWc^ = ^23a3j6^9 ^ ^/^^p?. ^n«. 

70. Reducing Radicals to the Same Index. 

Ex. Arrange V2, \/5, \/9 in descending order of 
magnitude. 

V2, ^, ^ = 2i, 5*, 9* 

= 2^ 5", dA 

= \^, v'125, \/8T. Hence, v^, v^9, V2. ilrw. 



110 DURELL'S ALGEBRA: BOOK TWO 

EXERCISE 41 

Express as entire surds : 

1. 5V3 A^/M^ 11 fl^+ar fT" 

2. 5^ • 5a^ 2 "2 ^a-hx 

3. 5^/3 12. I^Sf 

7. fVJy^ ^ a* \ xV 

' ar ' a— 

Simplify the indices of the following : 



17. V^V 20. V646V2 23. V16a*»y8» 

18. -v^S a«6V 21. ^729 a«J9 24. Va;»t/2"z8« 

19. ^^sp? 22. ^/STPP 25. ^'32(a-J)6 

26. \/1024ai<>(a:-y)6 27. yf^^ 

Reduce to equivalent radicals of the same (lowest) 
degree : 

28. ^2, -s/b 30. </^, ^ 32. ^/2, ^^6, <^ 

29. V5, </2 31. V2, </3, -H 33. </^, ^^, V5 
Show which is greater : 

34. V5, Vl 36. 2V2, -^Vt 38. </%, 2-v/J 

35. V^, ^4 37. i\^, |V5 39. J^, fVf 

Arrange in the ascending order of magnitude : 

40. V3,^^^_ 42. Vl,</1</^ 

41. VU, V2, a/80 43. 3, 2V3, ^V6 

44. Make up and work an example similar to Ex. 14. 



OPERATIONS WITH RADICALS 111 

45. Also an example similar to Ex. 23. To Ex. 86. 

46. How many examples in Exercise 31 (p. 84) can 
you now work at sight ? 

Operations with Radicals 

71. Addition of EadicalB. 

Ex. 6Vii-10V3J-5V| + 7V24 

= 6V^ - lOy^ - 5V| + 14Vtf 
= 5>/6 - 8\/5 - 2>/5 + 14>/6 
= 19>/6- lOVo. Ans. 

72. Multiplication and Involution of Eadicals. 
Ex. 1. 3^5 X 4 V2 = 3^/25 X 4<^8 

= 12v^00. Ans. 

Ex.2. (V3)8 = (3*)8 = 3* = 81. Atw. 

73. Division of Eadicals. 

Ex. 8V2-!-2V5 = 4V| = |VlO. Ans. 

EXERCISE 43 

Simplify and collect : 
1. 2V27-3Vi8 4- VaoO- V162 
2- f^|-i^^-^-^48 

3. a</2^ + a/128 cflx-- v^250 a^x 

d 

4. fVT62^+20a;V4j-^V2^-a?^ 



'6 ■ 6 ^6 ^""3 ■ 2a 



6. 5V(a - 6)2a;- 3V(2 a - 3 6)2a; + Va^a: + 2 a6x+6% 
8. 4Vj-5V6-3V|+2\/36 



112 DURELL'S ALGEBRA: BOOK TWO 

9. Compute to three decimal places the numerical value 
of VSU + V98 •" V72 without first simplifying and collect- 
ing the radicals. Then simplify first and compute. Com- 
pare the amount of work in the two processes. 

10. State some of the advantages in being able to sim- 
plify radicals. 

Multiply : 

U- M by J V| 14. V3, </2, and </} 

12. V5 by ^ 15. aVa — bVb by SVab 

13. 2^/3byV}, 16. VS-VTby VS + VT 

17. fV8-h2V32-jV48by 2V8-jV32-hVl2 

18. 10Vi-4VJ-hiV500by V|-h8V2--jV5 

19. Va: + 2-Va;-2by Va;+2 + Va:-2 

20. jV?^=T2 + iV^M^byiV^23p«^VSr4^ 

21. In the shortest way find the value of 

(3 V2 - V5)(4V3 + V7)(3 V2 + V5)(4V3 - V7) 
Divide : 

22. f Vis by f V72 24. Vol by VFc 

23. jVSbyiVlO 25. \f by\f 

26. 15Vi2-12V40-4Vl05by 3V15 

27. Compute in the shortest way the value of (V2)®. 
Also of (V2)io, (V3)i2, (- Vsy, (JV2)«, (V6)«. 

Expand : 

28. (2V3-V2)2 31. (V2«\/3)6 

29. (V3-2)8 32. (ViTy-Vi^)2 

30. (JV3-2V2)* 33. (V3-V2 + l)2 



OPERATIONS WITH RADICALS 113 

34. State the rule for multiplying one radical by 
another. 

35. Make up and work an example similar to Ex. 9. 
To Ex. 13. 

36. Also one similar to Ex. 17. To Ex. 23. To 
Ex. 27. 

37. How many examples in Exercise 32 (p. 87) can 
you now work at sight? 

74. Nationalizing a Denominator. 

7 7 -J/2 7 - 

E^ 2 —5 5 ^3V ^+2Vg 

3V^-2Vb 3Va-2V6 3Va+2VJ 

^15Va+10V5 ^^ 
9a-46 
EXERCISE 44 

Reduce to an equivalent fraction with rational denomi- 
nator : 

4 > _8_ J Va- b 

■s/2 ' Va-b 

*• ' v^T" 



3V7 
5 
3V2a: 
4V5-2V2 


3V6+ 4 V2 
3_Va; + 2 


3 + Va;-|-2 
V8-Va;-»-2 



Va-J 4V3 

10. 



Va2 + 62 + Va2-J2 



V2-V6-2 

V6-V2-2 

4V5 

V3 + ViT2 ~" 3V5-2V3 + 2 



8. ' 11. 



12. 



114 DURELL'S ALGEBRA: BOOK TWO 

^^ Va4-6-Va-V6 ^ 3+V& 8-V5 

Va + 6-Va + V6 * 3-V6 3 + V6 

14. ^^ --^^^^ ,e. 1 1 



Va-Va2-6a ' V7+V3 V7-V3 

17. State the rule for rationalizing a monomial denom- 
inator. 

18. State the rule for rationalizing a binomial denom- 
inator. 

Use the process of rationalizing the denominator as an 
aid in finding to four decimal places the numerical value 
of 

.o 2 ^, 8 ^, 3-V5 

19. — -: 21. , 23. — 

V6 V200 2-hV5 

^5 ,, 2V7 ^^ 3V6-2V6 

20. 22. 24. 

2V7 3V5 2V6-hV6 

25. Make up and work an example similar to Ex. 3. 
To Ex. 10. 

26. Also one similar to Ex. 22. To. Ex. 23. 

27. How many examples in Exercise 13 (p. 30) can 
you now work at sight ? 

Square Root op a Binomial Surd 

75. A Quadratic Surd is a surd of the second degree ; as 
V3 and Vab. 

A binomial surd is a binomial expression, at least one 
term of which contains a surd ; as V2 + 5 VS, or a -h V6. 

76. A. 77ie product of two dissimilar quadratic surds is 
a quadratic surd. 

Thus, V2 X V6 = Vi2 = 2V3 

Or Va6 X VaJc = aJ Vc 



SQUARE ROOT OF A BINOMIAL SURD 115 

Pboof. If the surds are dissimilar, one of them must 
have under the radical sign a factor which the other does 
not contain. This factor must remain under the radical 
sign in the product. 

77. B. The sum or the difference of two dissimilar quad- 
ratic surds cannot equal a rational quantity. 

We use X db y as a short way of writing x + y and x — y. 

Proof. If Va±VJ can equal a rational quantity, c, 
squaring, a ± ^^ah -f J = c^ 

± 2 VaJ = c2 - a - J 

But VoJ is a surd by Art. 76 ; hence, we have a surd 
equal to a rational quantity, which is impossible. 

78. C. Jfa + V* = x + Vy, <A«na = x, * = y. 
Proof. If a + VJ = a: + Vy 

transposing, Vft — Vy = a: — a 

If h does not equal y, we have the difference of two 
surds equal to a rational quantity, which is impossible; 
hence. 

In like manner, show that if 

a—yfh^x — yfy^ then a = a?, J = y. 



79. D. 7f Vjc + Vy = Vfl-fV*, eAmx + y=a 

Squaring the given equals, x + y 4- 2>/xy = a + V6 

Hence, x-\- y -^a (Art. 78) 

In like manner, if Vx — Vy — V a — yJh 

it may be shown that x ■\-y — a 

Also, since 2Vxy = yfh (Art. 78) 

X + y — 2"\/xy = a —y/h 
and Vx — Vy = v a — VS 



116 DURELL'S ALGEBRA: BOOK TWO 

80. Extraction of the Square Boot of a Binomial Sard. 

Ex. Extract the square root of 6 + 2 VsT 



Let \/5 + \/^ = V5 4-2\/6 .•.x + y = 6 (Art.79) 

Then, V5 - Vy = Vs -2V6 (Art. 79) 

Multiplying, a? — y = V25 - 24 .•. a? — y = 1 

.-. a? = 3 
y = 2 
.-. Vx + \/y = V3 + \^ 



.-. V5 + 2V6=V3+\/2. Ans. 

81. Finding the Square Boot of a Binomial Surd by In- 
spection. 

By actual multiplication we may find 

(V2 + V6)a= 2 + 2VT0 + 6= 7 +2V10 

In the square, 7 + 2 VIO, 7 is the sum of 2 and 5, 10 is 
the product of 2 and 5. Hence, in extracting the square 
root of a binomial surd, 

Transform the surd term so that its coefficient shall be 2 ; 

Find two numbers such that their sum shall equal the 
rational term^ and their product equal the quantity under the 
radical; 

Extract the square root of each of these numbers^ and con- 
nect the restUts by the proper sign. 

Ex. Find the square root of 18 + 8V5. 

18 + 8V5 = 18+2V80 
The two numbers whose sum is 18 and product is 80 are 8 and 10. 



.-. Vl8 + 8V5=V8+VlO 

= 2v^+Vl0. Root. 



SQUARE ROOT OF A BINOMIAL SURD 117 

EXERCISE 49 
Find the square root of 

1. 17-12V2 8. 77-24V10 

2. 23 + 4VT5 9. 87-36V5 

3. 35-12V6 10. 14 + 3V3 

4. 9-6V2 11. 8-J5<tV2 

5. 42 4-28V2 12. 5J + 3V3 

6. 73-12V85 13. 4J-|V3 

7. 26 4-4V30 14. 2?n + 2V7na-7i2 



15. 10a24-9 + 6aVa2+l 

Find the fourth root of 

16. 28-16V3 18. 193- 132 V2 

17. 97-56V3 19. ^ + ^^V6 

Find by inspection the square root of 

20. 3 + 2V2 23. 23-6VIO 

21. 9-2V14 24. I8-I2V2 

22. 2I + I2V3 25. 7 + 4V3 

26. Prove that Va ± Vb cannot equal Vc. 

27. Prove that Va cannot equal b + Vc. 

2a Make up and work an example similar to Ex. 2. 



CHAPTER IX 



IMAGINARY QUANTITIES 

82. An Imagmary Qnantity is an indicated even root of 
a negative quantity; as V— 4, \^— 3, and V— a. 

The term "imaginary" is used because, so long as we 
confine ourselves to plus quantity and to its direct 
opposite, minus quantity, there is no number which mul- 
tiplied by itself will give a negative number, as — 4, for 
instance. All the quantity considered hitherto, that is, 
all positive or negative quantity, whether it is rational or 
irrational, is called real quantity. 

If we extend the realm of quantity outside of positive 
and negative quantity, imaginary numbers are as real as 
any others, as will be shown in the next article. 

A complex number is a number part real and part imagi- 
nary ; as 3 + 2V— 1 and 
a + J V^^. 

83. Heaning of V^^. 

If O^ = + 1, and OA^ is of 
the same length, but lying in 
the opposite direction from O, 
0.4' = - 1. 

Hence, we regard the opera- 
tion of converting a plus quan- 
tity into negative quantity as 
equivalent to a rotation through 
an angle of 180^ If we divide 
this rotation into two equal 
rotations, each of these will be a rotation through 90°. 

Hence, V— 1 must be equivalent (geometrically) to the result of 
rotating the plus unit of quantity through 90°. Hence, V— 1 on our 
figure will be represented by OB, 

118 



J 
o 




*. J 




-1 I 


r 
+ 1 



B' 



IMAGINARY QUANTITIES 119 

Hence, it is easy to see, also, that V— 1 x V— 1 = — 1. 

We thus perceive that the introduction of imaginary quantity 
enlarges the field of quantity considered in algebra from mere quan- 
tity in a line to quantity in a plane. This gives a vast extension to 
the power of algebraic processes and introduces many economies in 
them, as will be found by the student who pursues the study of 
mathematics extensively. 

In taking up the subject for the first time, we consider only a few 
of the first properties of iraaginaries, so called. 

84. The Fundamental Principle in treating imaginaries is 
that V^^ X V^n^ = - 1. 

Using < as a symbol for V — 1, this principle is f x t= — 1, 
or t2= — 1. 

Considering this matter algebraically, if we use the law 
of signs in the most general form, 

( V^a = V^ X V^^ = VT = ± 1 

Now, if we extract the square root of -f- 1, we shall not 
have V— 1. But if we extract the square root of — 1, 
we shall have V— 1. 

Hence, we must limit the product V— 1 x V— 1 to — 1. 

Likewise, V— a x V— 6=VaV— 1 x VftV— 1 

= VaV6( V^nr)a= - VaJ 

Or, using the symbol f, ai xbi = — ab. 

85. Powers of V— 1. 

(V3T)2=_i .•.|2 = -i 

(>/:ri)* = ( v'3i)8vzi = + 1 ... ,-4 = 1 

(See OA' of figure in Art. 83.) 

Thus, the first four powers of V— 1 are V— 1, —1, — V— 1, 
+ 1 ; and for the higher powers, as the fifth, sixth, etc., these four 
results recur regularly. The same fact is clear from the figure in 
Art. 83. 



120 DURELL'S ALGEBRA : BOOK TWO 

86. Operations with Imaginaries. It follows from Art. 
84 that, in performing operations with imaginaries, we tise 
all the ordinary laws of algebra^ with the exception of a 
limitation in the use of signs, which may be mechanically 
stated as follows : 

The product of two minus signs under the radical sign of 
the second degree gives a minus sign outside the radical sign. 
But in dividing first indicate the division and afterwards 
rationalize the denominator. 

Ex. 1. Add V^r9, -. 3 + 2 V^^, 7 - 2V^=T6. 

- 3 4- 2\/^^ = - 3 + 2 V'^l 

4 -. 3 V^3. Sum. 

Ex. 2. Multiply 2 V^=^ + 3 V^^ by 3 V^^ - bV^^. 
2\/3^+3V3b 

6(_3)-9>/T8 

+ 10V6 + 15\/l2 



- 18 - 27 \^ + 10V6 + 30V3. Product. 
Ex. 3. Divide - 2 V6 by V^2. 



- 2\/6 - 2V6 ^ 



y/Zr^ yT^ y/^^ 



Ex. 4. Extract the square root of 1 + 4V— 3. 

l4.4V33 = l + 2v'^=n^ 
The two numbers which multiplied together give — 12, and added 
together give 1, are 4 and — 3. 

.-. Vl 4- 4V^=r y/i + yf^ = 2 + V^^. Root. 
Let the pupil work the above examples using t instead of — 1. 



Collect : 



IMAGINARY QUANTITIES 121 

SZESCISE 46 



1. 7V:^ + 3V^^49-10V-9 

2. -v^irji-v:^! + 2^/^^+^/^^^5 



3. 6V:rj_3V-| + 4V-50-V^r200 



2 



4. 2V^=^-3aV-4 + -V-16a*-JV-36aa 



a 



a + 5 Vrr _ 6 _ a V3i _ V^rp + 2 V^T^ - a 



6. (a-2i)V^nr-(2o + J)V-l 

7. Express in terms of t the results obtained in Exs. 1-6. 
Multiply : 

8. V^n by V^^ 11. - V^:^0^ by - V^^Hl" 

9. V^ by - 2V^ 12. 2 V^^^Ol by - 2^^^^ 
10. -V^^by-VT 13. -5-/^by-2V^ 
14. — Va; — y by Vy — x 



15. — a VI — a by — V(a — 1)' 

16. V^^ + V^^ by V^Ti _ 2 V^^ 

17. 3V^^-2V^^by 2V^ + 3V'^^ 

18. 2V2-2V^r2by 3V2 + 3V^^ 

19. 3V^=^-V^^by 2V3-V^^ 

20. V^-V^^ + V^^by V^ + V^^ + V^ 

21. V^a + V^by VB + V^^ 

22. a;-2 + V^^bya;-2-V^ 

23. a-^—a + b-y/— h by aV— a — 6V— 6 

24. x — 1 — V^ by a; — 1 + V— 1 

25. Multiply g- ^ ~ "]f~^ by g- -^ "^ "^f"^ 

26. In the shortest way find the value of 

(VITs + V^^)( V3 + V2)( V^ - V^::2)(V3 - V2) 



122 DURELL'S ALGEBRA: BOOK TWO 

Divide : 

27. - Vl8 by V^^ 29. - 6 V^^^ by 2 V^ 

28. -V^Il2by-V^ 30. 8V^^r^by-2Va 
31. 2V^T8~4V^=n^ + 10V30by -2V^=r3 



32. ay/ --a — 2 aV— 6a— a*V3 a^ by — aV—a 
Express with rational denominators : 

33. 1 37 3V^-V32 

2V2+V:r2 2-3V^l 



g + iV^T 8V2+2V^-V-10 

a-bV-1 8V2-2V-7+V^^10 

Find the square root of 

41. 3-6V^^ 44. 12V10-38 



42. 1-2V^^ 45. -29-24V^ 

43. 12V^^-6 46. 7 + 40V^^2 

47. By use of t (or V^^), factor a* + 6". Also a* + b*. 
aJ«+2. aV+J2. aJ« + l. a^ + 1. 



48. For what values of x is V2 — x imaginary ? V2 — a;^? 

49. Find the value of (a/^^)6. Of (-V^)*,(V^)«, 



IMAGINARY QUANTITIES 123 

Ex. SimpUfy 3( V^^ + 2)a - (2 a/^^ - 1)^. 

Substitute t forV— 1. 

3(» + 2)2 - (2t - ly = 3i« + 12t + 12 - 4t8 + 4t - 1 
= -t8 + 16t + ll 

= -(-l)+16V'^ + ll 
= 12 + 16\^in[. Ans. 
Simplify : 

50. i2 + 3£*-2t8 52. i^xi^xSf 

51. i8-5f + 4i* 53. (f-l)8_(f-l)24.3(i_l) 

54. (V^T-l)8-(V^n:-l)a+2(V^n-l) 

55. (V^T-2)(3V^n + l)-(V^ri-3y-(V^)a 

56. (V^ri_l)4 4.3(V31_l)8^4(V:n-l)2 

57. If 2? = -!— ^^~^,findthevalueof3a?-62J+7. Of 

a:8-5a?4-2a:-l. 

58. Simplify i^+\ Also i^+2, t^+», f*«+* 

59. Who first discussed imaginary quantities and 
when? Who first put the use of these quantities on a 
scientific basis ? Who invented the symbol % for V— 1 ? 

60. Make up and work an example similar to Ex. 2. 
To Ex. 17. To Ex. 61. 

EXERCISE 47 

Review 
Simplify and collect : 

1. 3V5Ta-10aA/i-5Jip? + oVi5^» 

^o a ^ 5 

2. 2V^=r50+5>/^-6V}-2\^ 

3. 3\/J-2v^-5^/^ + vI2 



124 DURELL'S ALGEBRA: BOOK TWO 

Multiply : 



5. 2V3by4^ 7. V4-V5by V4+V5 

6. 3V^by2V^ a V3-2>A=^by V3 + 2Vi:5 

Rationalize the denominator of 

ft ^ 

9. -4- 12. 



8v^ ' Va+ y/b 

10. _?_ 13. 2^±J!i±i 

11. y/'=^ + V^ 14, V^^ 



15. 



6 . 5 



h+y/i^-sc^ h^Vb'i^x^ 



16. ^^ + y + ^^-y ^ Vg-f y 



— Vx— 1 



y/x-\-y-y/x-y Va? + y + Va: - y 
Determine which is greater : 

17. 4V3or5V2 18. jv^oriv^ 

Find the square root of 
19. -.7-2\/Id 20. 33+20V2 

21. Simplify (SV^^T + 2)» - 4(3 - V^T) + 5. 

2 

22. Solve Vx - Vx- 8 = 



Vx-8 

23. Solve |^^4^H1 = 4. 

24. When a = 2, 5 = 8, c = 4, find to two decimal places the value 



25. Divide 1 — Vx — yVx + xy by yVx — y. 



26. Find the value of x« - 2 x + 1 when x = "" ^ ^ ■ i 

27. Also when x = ^ — ^^L — Z — 

28. Collect 3(a + 5)vf±| + 2(a- 6) ViH|_2^±|:ivp3F. 



IMAGINARY QUANTITIES 125 

29. Simplify (1) (v^i, (2) \/V^, (3) \^V^^. 

30. Find the value of 5 - 8 1» + 2 1« -f 7 1 - 4. 
Rationalize the denominator of 

31. 2 32. ^+^_ 

VV7 + 3- VV7-.3 x + y+Vory 

33. Solveforxandy: f BVx + l + 2v7=3 = 13, 
\4>/a? + 4-3Vy-l=:6. 

£X£RCIS£ 48 
Oral Review 

1. State which of the following are imaginary: V— 3, — VjJ 

- Vi, ^^, ^J^^, ^J^^^is, ^^. 

Reduce to simplest form : 

2. 2Vi, ^, V3T8, V|, S^ITJ, ^, J^, 5Vj 

3. V^:^ X V^ \/^=^ X >/3, - V^=T X V3, Vi X V5, 
>/^ X V— a, — Via X V— a 

4. -VB X V^^, t« X tS - t8 X »«, 3i X 2i, - 5t X 3t 

5. 1-V^, l-4-iV2, J + i\^, 2+V^ 2+V3j 

6. (VITl + Vir2)( V33 _ V3^), (V5 + V2)(\/6 - v^), 

(Vr^ 4. vr^)(Vir^^ yTTv), (5 + Vir2) (5 - V32), 

a+.iV-2)(i-iV-2) 

7. 1 1 1 1 iV^=^ 
V3-V2' VS-Vft' jVd' V3-Vir2' i>/ir2 

a v^-VIS; 2^1 + ^8, v^+\^, (v^)8, (V:r3)6 
9. Does (a + «)" equal a* + a:*? 

10. Does y/a + x equal v^ + Vx? 

11. Is (ox)* equal to a*a:*? 

12. Is Vax equal to Vay/xl 

13. What advantage is it to know the principle contained in Ex. 11 ? 

14. That contained in Ex. 12 ? 

15. Which is greater, (3*)**^ or 3* • 3^*^ ? How many times greater ? 



1 






U-a? 




eorJc 


. . ' 1^ 







CHAPTER X 

QUADRATIC EQUATIONS OF ONE UNKNOWN 
QUANTITY 

87. Need and Utility of Equations of the Second Degree. 

Ex. A certain field of grain 

is 60 rd. long and 40 rd. wide. 
^j^J I How wide a strip must be cut 

off along two adjacent sides in 
order that 5 acres may be left 
^^ uncut? 

Let X = width of strip in rods 

Then (60 - «)(40 - a:) = 5 x 160 

Hence, a:* - 100 x + 1600 = 

Hence, in order to find the value of x, it will be necessary to solve 
an equation of the second degree. 

88. A Quadratic Equation of one unknown quantity is an 
equation containing the second power of the unknown 
quantity, but no higher power. 

A pure quadratic equation is one in which the second 
power of the unknown quantity occurs, but not the first 
power. 

Ex. 9a:2__25 = 0. 

A pure quadratic equation is sometimes termed an incomplete 
quadratic equation. 

An affected (or complete) quadratic equation is one in 
which both the first and second powers of the unknown 
quantity occur. 

Ex. 9a;a_5^_25 = 0. 

126 



PURE QUADRATIC EQUATIONS 



127 



Pure Quadratic Equations 

89. Solation of Pare Qnadratici. Since only the second 
power, a?, of the unknown quantity occurs in a pure 
quadratic equation, in solving such an equation, we 

Reduce the given equation to the form ofl = c; 

Extract the square root of loth members. 

3-2:2 ^ 0^2 + 5 



Ex. Solve 



3. 



11 6 

Clearing of fractions, 18 - 6 a^» = 198 - 11 a?^ - 55 

Hence, a:^ = 25 

Extracting the square root of each member, 
a: = +5, or x = — 6 

That is, since the square of + 5 is 25, and also the square of — 5 is 
25, X has two values, either of which satisfies the original equation. 
These two values of x are best written together. 

Thus a? = db 5. Roots. 



Check. For a: = 5. 
3 _> a;2 3-25 



11 



11 



= -2 



3-^J:^ = 3-?5Jl5 = -2 
6 6 



Check. 
S-x^ _ 
11 



For a: =—6. 
3-25 



3 _ ^_±_5 = 3 _ ?2jhi = _ 2 
6 6 



Solve and check ; 
1. 6a?-24 = a? + 40 



EXBRCISS 49 



a;2 



x + l 10 x-1 



x + S 
4 



. 3a: 



x-S 
9x-7 



Sx + 1 1-Sx 9a?-l 



= 



128 DURBLL'S ALGEBRA: BOOK TWO 

6. _J_ + ^_+^_ + -J_ = 

a: — 8 x—Q x+6 x + 8 

7 aa?-ft ^ 10 y ax + b 
cuc + b ah? — 6* b — ax 

8. 3(2; + 3)2 = 270-3(2: -8)2 

9. 2(^px-qy-(^px-qy + (^p + qxy=^0 

15 5 

10. If 2? = T, find the value of x when v = --• 

2t; — 1;2 4 

9 8 

11. If 2)2 — ^ — find aj when v = — - . 

3 + 2v-iv^ 5 

12. State Ex. 1 as a problem concerning an unknown 
number. 

13. State Ex. 4 in like manner. 

14. A certain fioor is to be four times as long as it is 
wide and is to contain 10,000 sq. ft. Find its dimensions. 

15. The width of a certain field is one fourth of the 
length. If each side of the field is increased by one fourth 
of itself, the area of the field is increased by 22,500 
sq. yd. Find the dimensions of the field. 

16. Who first formed the idea of absolute or indepen- 
dent negative numbers? (See p. 262.) How was nega- 
tive number used before this ? How did the Arabs treat it? 

17. Make up and work an example similar to Ex. 6. 
To Ex. 14. 

Affected Quadratic Equations 

90. Completing the Square. An affected quadratic equa- 
tion may in every instance be reduced to the form 
a? +px = q. 

An equation in this form may then be solved by a 
process called completing the square. This process con- 



AFFECTED QUADRATIC EQUATIONS 129 

sists in adding to both members of the equation such a 
number as will make the left-hand member a perfect 
square. The use of familiar elementary processes then 
gives the value of z. 

Thus, to solve a:» + 1 a: = 24 

take half the coefficient of x (that is, 5), square it, and add the result 
(that is, 25) to both members of the original equation. 

We obtain 

xa+10« + 25 = 49 

Or (x + 5)2 = 7« 

Extract the square root of both members, 

x+6=±7 
Hence, x = — 5 ± 7 

Thatis, x = -5 + 7=2. '] ^ 

Also, x = -5-7 = -12.| ^'^' 

Hence we have the general rule : 

By clearing the given equation of fractions and parentheies, 
transposing terms^ and dividing ly the coefficient of x^^ reduce 
the given equation to the form x^ + px^q; 

Add the square of half ^^^ coefficient of xto each member 
of the equation ; 

Extract the square root of each member ; 

Solve the resulting simple equations. 

Before clearing an equation of fractions, it is important 
to reduce each fraction to its simplest form. 

Ex. 1. Solve Qqi? + 1x^Z. 

Dividing by 6, x^ + } x = J 

Completing the square, x« + } x + (A)^ = i + ^ = Hi 
Extracting the square root, a: + i^ = ± H 

X = J, — f . Roots, 



Check. For x = J. 

6x« + 7x = 6xJ + 7xJ 



Check. For x = — j. 
6x2 + 7x = 6x|-7xt 



130 DURELL'S ALGEBRA : BOOK TWO 

Ex. 2. Solve ^^Ili = 5. 

If we fail to reduce the fraction ^ ~y to its simplest form before 

clearing the equation of fractions, we obtain 

2z-4 = 6x2-20 
2^ -2x-lQ = 
Whence, a: = 2, - 1 



Check. For x = 2. 
2x-i 4-4 



Check. For x = - f. 
23:-4 __ -Jif-4 _ -y _g 
x'i-4: Ji-4 -It 



5. 


r"- 


.5x 


+ .06 = 


= 


6. 


a^ + .45i 


r + .02 = 


= 


7. 


5x 


= 3?- 


2 





xa_4 4-4 
Hence, a: = 2 is not a root. 
The root x = 2 was introduced (see Art. 18) by a failure to reduce 
the given fraction to its simplest form. 

Let the pupil give the correct solution of Ex. 3. 

EXERCISE 60 

Solve and check : 

2. 9a:2_i8a; + 8 = 

3. 6a?-7a:-5 = 

4. a:a+8ia:+10f = " 6 

9. (a:-l)2(2:-|-3)-a:(a:-|-5)(a:-2)=0 
2a:-i-3 1-x ^ 7~3a; 

■ 2(2a:-l) 2(a:-hl) 4-3x 

11. 2(2a:-l)(2a:-3)-2(a:-|-l)(a:-h3) 

+ (a:+3)(2a:-S)=0 

12. 3(a:-2)-32:(a:-h2)-4(a^J-4) = 

13. ^::ii = ^ ?_ _^ 15.5-^=0 

'2 x-l b-x ^ X 



15. 



16. 



2 


■ 


3x- 


-4 


a; — 


1 


2a: 





4x-U ^ 2a!-l 
l-ar^ a;+l 

„ ._£±2 
a:+2 " 2a; 



17. 



AFFECTED QUADRATIC EQUATIONS 131 

Z-x x-8 bx-1 



2-|-3a: 3a:-2 4-9a^» 

18. J(13-5a:)-Ka^»-a:)=0 

19. rr2+5a:+2 = 20. 3a?-2a: + 7 = 

21. ^±I + 2^±l4.i^±3 = 
a:-2 x^-1 l-x 

22. 6(a: + l)(a:-3)-2(a;-l)(2a;-6)=0 

23. =: 6 

X-1 2?--l 

24. In the equation a:^ + 5ir— 7 = 0, find the values of 
X to three decimal places. 

25. Find the values of x to three decimal places in 

26. The square of a certain number diminished by 5 
times the number equals 6. Find the number. 

27. Find three consecutive numbers the sum of whose 
squares is 245. 

28. State Ex. 3 as a problem concerning an unknown 
number. State Ex. 6 in like manner. 

91. Literal Qnadratio Equations are solved by the methods 
employed in solving quadratic equations with numerical 
coefficients. 

Ex.1. Solve oa^ + l^ ^"^"^ 

ax 

Clearing of fractions, a^x^ + aa; = 3 — a^a^ 
Hence, 2 aH'^ + aa; = 3 

2, JC ^ 3 

2a 2a2 



^2a^\ial 



Let the pupil check the work. 



16 a^ 

1 

a 2a' 



1 3 
a: = -, . Roots. 



132 DURELL'S ALGEBRA: BOOK TWO 

Ex. 2. Solve (^p + y)a? - (2 p + y)a: +|? = 0. 

;> + ? p-^q 






p-^q 



:,, 2/+y =^ 



2(;> + ^) 2(1, + ^) 



x = 1, — ^ . 22ooto. 
p + q 

Let the pupil check the work. 



EXERCISE 61 

Solve and check : 

2. a^-'8cx=10(^ 



15 rrfi 



mx 



8a2 4a 
7. a? — (^a— h)x = aJ 

3. a?=6a262-.aJa; s. x^- ax-ff-hx^ ah 

4. 2r^-h;?2;=10;?2 9 a?-f-3a2;4-26a:=-6 

5. 3c^ = 4(?c?2; + 4cP lO. jt?j' = a:2_jj^^^. g,^; 



11. 


26V-h2aa = 5aJa; 


12. 


3Z2ar2+3Zpa;-5Za;=5jt? 


13. 


aJ(?a:2 _ (^2j2 ^. ^)^ + aJ(? = 


14. 


2a + a; a-2x_% 
^a-x a-\-2x 3 


TS 


a-2J 1 1 




8a:2_262 22: + 6 2a 


16. 


aJJ- 6 a(?rc + a2(9 c2- 4 J2)= Q 


17. 


, + -6a= , 



a — b b—a 

18. aa?-f-Jt?a: + y=0 19. a^ + bx+c = 

20. x^-*^) + *(^-^^ + 0+^*^ = o 



AFFECTED QUADRATIC EQUATIONS 133 

21. Make up and work an example similar to Ex. 2. 
To Ex. 8. 

92. The Factorial Method of solving equations consists 
in transposing all terms to the left-hand side, factoring the 
resulting expression, and letting each factor = 0« 

Ex. 1. Solve 2:8 + 8 = 0. 

Factoring, (x + 2) (x^ - 2 a: + 4) = 

X + 2 = 0, gives X = - 2. Root. 

Also, x«-2x + 4 = 

Whence, x^ — 2 a: = — 4 

x = l± V^Ts. Roots. 

Let the pupil check the work. 

The factorial method of solution is especially helpful in 
solving certain literal quadratic equations. 

Ex. 2. Solve (p + q)a? -(2p + q)x -h p =^ by the fac- 
torial method. 

We obtain [(;? + q)x - p] (x - 1) = 

Hence, x = 1, — ^ — . Roots. 

Let the pupil check the work. 

This example is the same as Ex. 2 solved on p. 132. On comparing 
the two solutions, we observe that at least three fourths of the labor 
of solution is saved by use of the factorial method. 

EXERCISE sa 
Solve and check: 

1. a^-Qx + S^O 3. 2ar2 + 5a:-12 = 

2. a^^6a^+8x=i0 4. Sa^+S = 10x 

5. <2:-l)(a: + 2)(2:-3)=0 

6. 0=32:(a?-4) 9. a:^ = 16a: 

7. 2:8-27=: 10. a?*-5aV + 4a* = 

8. 2:8 = 27 11. a;S + a?*-a;-l=:0 

12. 3(a?-4)=5(a:-2) 



134 DURELL'S ALGEBRA : BOOK TWO 

13. a:(2a:-l)(3a?-4a;-4) = 

14. 5a:(rc2-4)-h2(a^^-2) = 

15. ^-Ix -6 = 

16. aJ»-3a?-10a:+24 = 

17. 6a:«-23a:24.i6a;_3 = o 

18. a:8-8a?» + 17a:-10 = 

19. Find the six roots of ic* — 64 = 0. 

20. Find all the values of ^\. Also of -v^. 

21. Obtain a complete solution of the equation a;i =27. 

22. Solve a? — (2a-|-56)a:-f 10ai = by the method 
of Art. 90. Also solve the same equation by the factorial 
method. Compare the amount of work in the two proc- 
esses. Why do we not solve all quadratic equations by 
the factorial method? 

Solve by the factorial method : 

23. a? + 3aa;-10a2=0 24. x^^. 2a;- 3 Ja:= 6ft 

25. cd:i^^iNi-\-d?x-\-ed^^ 

26. ^=^ 27. -^ + ^±^= * 

0^ jcr h—X X — X 

28. Make up and work an example similar to Ex. 2. 
To Ex. 5. 

29. An example similar to Ex. 8. To Ex. 16. Ex. 22. 

30. How many examples in this Exercise can you solve 
at sight ? 

Equations in the Quadratic Form 

93. Simple Unknown Qnantity. An equation containing 
only two powers of the unknown quantity, the index of 
one power being twice the index of the other power, is an 
equation of the quadratic form. It may be solved by the 
methods already given for affected quadratic equations. 



EQUATIONS IN THE QUADRATIC FORM 136 

Ex. 1. Solve a?* - 10 r^ = - 9. 

Adding 5^ to both members will make the left-hand member a 
perfect square. Thus, 

a:4_ 10^:3^.25 = 16 
Hence, x' — 6 = ±4 

a:^ = 9, or 1 
a? = ± 3, ± 1. Roots. 
Let the -pupil check the work. 

This equation might also have been solved by the fac- 
torial method. 

Ex. 2. Solve 2</^^ - SV^^ = 2. 

Using fractional exponents, 

2a:"* -3a:"* = 2 

Whence, x~» — j x' = 1 

^"* - ( ) + A = « 

a:-*-}=±t 

^"* = 2, -i 

Whence, a:i = J, -2 

x = J, -8. Roots. 
Let the pupil check the work. 

94. Compound Unkiiowii Qnantity. A polynomial may be 
used in place of a single quantity as an unknown quantity. 

Ex. 1. Solve 3V^Ty - 2\^xT9 = 8. 

This equation may be written 3(a: + 9)*- 2(a: + 9)i= 8. 

Let (a: + 9)i= y ; then (x + 9)*= y^ 

Hence, substituting, 3 y* — 2 y = 8 
Whence, y = 2, - | 



v'a:+9 = 2 


^x + 9 = -t 


a: + 9 = 16 


a: + 9 = ^y¥ 


x = 7. Root. 


a: = - 4^. Root. 



Let the pupil check the work. 



136 DURELL'S ALGEBRA: BOOK TWO 



Ex 2. Solve x^-\.5x+ SVa^-h 5a; + 2 = 26. 
Add 2 to both sides, 



a:2+5a: + 2 + 3Vxa + 5a: + 2 = 28 



Let V^T5xT2 = y; thenya+3y = 28 

Whence, y = 4, — 7 



Hence, Var^ + 5a; + 2 = 4 
a:2+5a:+2 = 16 

x = 2, -7. 

Let the pupil check the work. 



Vi2T57T2 = -7 
x2+5a: + 2 = 49 

z = -4±jV218. 



9. 


xi- 


3a!* + 2 = 


10. 


xi- 


3a;i + 2 = 


11. 


Sx-i 


+ 5a!-i = 2 


12. 


6 a;-' 


-1 = 12 

X* 



EXERCISE 83 

Solve and check: 

1. a?*-13a? + 36 = 

2. a?* + 2 = 3a:2 

3. 26a^J=a^-|-25 

13. 9a:^-f-4 = 13a;^ 

6. 2:8(2:3 -10) = -16 14. 7</F^-4^/^=3 

7. 2:* + 2:*-2 = 15. 2 x^ - 9-s/x -{- 4t == 

8. 3V5+4 = 8v^» 16. (a:+2)i-(2?4-2)* = 2 

17. (22:2-2:)a-4(22:2_^)_^3^0 

18. a^ + x- S^a^ + 2: - 2 = 

19. (3 2: -2)*- 4(3 a; -2)*+ 3 = 

20. ^32:-22:a-(3a;-22r2)*=2 

21. 2?-32:-h2 = 6V2;2-32;- 3 

22. (a; + 2)i-(2: + 2)* + 2 = 



RADICAL EQUATIONS 137 



23. 6(a? + a;)-7V3a:(2;+l)-2 = 8 

24. 3a:"* -7a;* = 4 25. 16 a;* - 22 = 3 a;"* 

26. 6(2a?-l)*-4 = \^(2a?»-l)-l 

6 CO ^ »« a:^,a;+16 

27. — — -~ = 5 — 2 a; — a:* 28. H ^ = - 

a^^-f-2a? a:+l t^ 2 

29. Make up and work an example similar to Ex. 1. 
To Ex. 9. 

30. Also an example similar to Ex. 16. 

Radical Equations 

95. Radical Equations Resulting in Affected Quadratic 
Equations. If an equation is cleared of radicals, the 
result is often a quadratic equation. 



Ex. Solve Va;+5-V2a;-7 = VS 



X. 



Transposing, Vx + 5 — Vx = V2a: - 7 

Squaring, x + 5 - 2 Vgg + 5x + x = 2 x - 7 

Hence, Vx*^ + 5 x = 6 

Whence, x* + 5 x = 36 

X = 4, - 9 

Substituting these values in the original equation, we find that the 
only value that verifies is x = 4, which is the root. The other value, 
X = -> 9, is not a root of the original equation, but is introduced by 
squaring in the process of clearing the equation of radical signs. It 
satisfies the equation, 

VxTl - V2X-7 = - Vx. 
(See the treatment of extraneous roots, pp. 43-47.) 

EXERCISE 64 

Solve and check : 

1. X + S-y/x +2-4 = 3. 2a;-5-3V^T2 = 

2. 16Vi-16a?-3 = 4. 4=V4a: + 17+V^Tl 

5. V2-|-a; + V2-a:=:V3 



138 DURELL'S ALGEBRA: BOOK TWO 



6. V2a:-|-5 + Vaa:-h4 = V5a: + y 



7. V2 2:+7 4-V;-i2;-18-V7 2:-f-l = 



8. Vl— a:-hVl — X — Va:= 1 



3 



10. V3a: + l~2V2a:+l + — = 

V2a;+1 



■ - Vrc - 2 

V2:-2 

12. Vi-V3+V3-V3¥+^=0 

1 V3 



13 



14. 



V3^r^ + V5 V34-a:-V3 ^ 
2 1 



V2;-3 -h V3a;-1 Vrr-l 

15. State the rule for solving radical equations. 

16. State Ex. 1 of this Exercise as a problem concern- 
ing an unknown number. 

17. Similarly, state Ex. 3. Ex. 6. 

18. Practice oral work with exponents as in Exercise 
39 (p. 104). 

96. Other Kethods of Solving Qnadratic Equations, besides 
those given in the preceding part of this chapter, may be 
used. One of these methods may occasionally be used to 
advantage for some special purpose. 

97. Hindoo Method to Avoid Fractions in Completing the 
Square. After simplifying the equation. 

Multiply through hy four times the coefficient ofafl; 
Add to hoth sides the square of the coefficient of x in the 
simplified equation. 



AFFECTED QUADRATIC EQUATIONS • 139 

The reason for this process is evident, since iia2^'\-bxssc 
is multiplied by 4 a, we obtain 

4 a^2? -\- 4 abx = 4 ac. 

The addition of J^ gives on the left-hand side 4 a^a:? + 4 abx 
+ 1^^ which is a perfect square. 

Ex. Solve 3a? — 4a; = 7 by the Hindu method. 

Multiply by 4 X 3 (or 12), 36 x^ _ 43 x = 84. 

To each member add the square of the coefficient of x in the orig- 
inal equation ; that is, add (-4)', or 16. 

36 x« - 48 a: + 16 = 100 
6a:-4 = ±10 
6a: =14, ~6 
a: = J, — 1. Roots. 

Let the pupil check the work. 

EXERCISE 66 

Solve and check : 

1. aj2_82:-h7 = 6. 6a;2 = i2J2_6^ 

2. 7 = 3a?-4a: 7. ir2+ (3 6 - 2)a:= 6 6 

3. Sa^-lx = 26 8. 4:a^+9 = Ux^ 

4. 3|a;2-22: = 4 9. 3 a:* -|- 4a:4 - 4 = 

5. a?-4ax = 12aa 10. (ia^--l}2^-\-(a^+l)x=^a 

98. Use of Formula. Any quadratic equation can be 

reduced to the form 

ax^-\-bx + c—0. 

Solving this equation by use of Art. 90, 



- J ± V62 - 4 a(? 



2a 

By substituting in this result, as a formula, the values of «, 
J, c in any given equation, the value of x may be obtained. 



140 DURELL'S ALGEBRA : BOOK TWO 

Ex. Solve 3 a:^ - 4 a: — 7 = by use of the formula. 
Here a = 3, J = — 4, c = — 7. 

Substituting for a, 6, c in the formula on page 139, 

^.i±vroiB.4±io_T _i, ^ 

Let the pupil check the work. 

BXERCISS 66 

Work the examples in Exercise 55 by use of the formula 



Solve and check: 

1. 12x« + a:=l 

2. 10aa = a:«-3aa: 

3. 2\/iT3-- Vx^^ = 4 



BXERCISB 67 
Review 



9. X-* - 3 z"* + 2 = 6 



10. 2Vx^^=5-- 



2 



4. a:*-8a: = 11. z' - a: = 



Vx-1 



6. a:«-9a;* + 8 = 



X 6 a 
13 —^ ^ _x— 6 



7. 3x2-x-3V3xa-x-3 = l 4 5 - a: a? 

a xa + 3fta:=6aft + 2aa: 14. x(a: - 2) (9 ar^ - 25) = 

15. V3 z + 10 = VIO a; + 16 + Vr+2 

16. -J_=l+1 + 1 
1+a+z a^z 

"• ('-a*-i('-i)- 

18. 2V67-v^2F31=:4£±l. 

V2z-1 

19. 3z2+ 15z-2Vz2 + 6z + l=2 

20. 3v^z+12+2Vz + 12 = 14 

21. V2 z - a* + Vz - Vz + 3 a^ = 



AFFECTED QUADRATIC EQUATIONS 141 

23 2a: -1 ^ ft^(x + 2) 
x-\-2 a2(2x-l) 

2a x-ft b\ x-h I 

25. Who, as far as we know, first solyed a quadratic equation, 
and at about what time ? (See p. 263.) 

26. How have the different cases in the solution of a quadratic 
equation been classified at different times ? 

27. Write (but do not solve) an equation of each of the principal 
kinds treated in this chapter. 

EXERCISE M 

1. One number exceeds another by 4, and the sum 
of the squares of the two numbers is 250. Find the 
number. 

2. The sum of the squares of three consecutive num- 
bers is 149. Find the numbers. 

3. Three times the square of a given number, dimin- 
ished by twice the product of the number and the next 
lower number, gives 143. Find the number. 

4. A certain number increased by four times its recip- 
rocal equals 8J. Find the number. 

5. The denominator of a given fraction exceeds the 
numerator by 3, and the sum of the fraction and its 
reciprocal is 2^^^. Find the fraction. 

6. The length of a given rectangle exceeds the width 
by 5 yd. and the area of the rectangle is 176 sq. yd. Find 
the dimensions of the rectangle. 

7. The base of a triangle is 4 ft. less than the altitude, 
and the area of the triangle is 48 sq. ft. Find the base of 
the triangle. 



142 DURELL'S ALGEBRA: BOOK TWO 

8. A rectangular lot is surrounded on all sides by a 
driveway 5 yd. wide. The lot is twice as long as it is 
wide. If the area of the lot and driveway together is 
6600 sq. yd., find the dimensions of the lot. 

9. A farmer has a field 60 rd. long and 40 rd. wide. 
How wide a strip must he cut around the field in order 
that 5 acres ma)^ be left uncut ? 

10. An open box is to be formed by cutting out equal 
squares from the corners of a square sheet of tin and fold- 
ing up the sides. The box is to be 10 in. deep and is to 
contain 2250 cu. in. Find the length of a side of the 
square sheet of tin. 

11. One baseball nine has won 6 games out of 15, and 
another has won 8 out of 13. How many straight games 
must the first nine win from the second, in order that the 
average of games won by the nines shall be the same ? 

12. A and B together can do a given piece of work in 
2 days. Working alone, A could do the work in 3 days 
less than B. How many days would it take each man 
working alone ? 

13. Two pipes together can fill a tank in 3^ hr. When 
running alone, one pipe takes 12 hr. longer than the other 
to fill the tank. How long does it take each pipe alone 
to fill it ? 

14. Two boats raced 24 miles. The first boat traveled 
8 miles an hour, and the second traveled 14 miles at a 
certain rate and then increased its speed 4 miles an hour* 
The second boat lost by 20 minutes. Find the rates at 
which the second boat traveled. 

15. A boatman can row 16 miles down a stream and 
back in 6 hours. If the rate of the stream is 2 miles an 
hour, find the rate of the boatman in still water. 



AFFECTED QUADRATIC EQUATIONS 143 

16. In a number containing two digits, the left-hand 
digit is four times as large as the right-hand digit. The 
product of that number and the number obtained by 
inverting the digits is 2296. Find the original number. 

17. In a given number, the tens' digit is one third of 
the units' digit, and the product of the number obtained 
by inverting the digits is 403. Find the number. 

18. In a given number, the units' digit exceeds the tens' 
digit by 3, and the product of that number by the number 
obtained by inverting the digits is 1300. Find the number. 

19. The side of a given square is 1 J ft. By how much 
must this side be increased, in order that the area of the 
square may be increased by 406 sq. in. ? 

20. Two men, A and B, can together do a piece of 
work in 12 days. B would need 10 days more than A 
to do the whole work. How many days would it take A 
alone to do the work ? 

21. A given field of grain contains 20 acres and is twice 
as long as it is wide. How wide a strip (in rods) must 
be cut around the field, in order that | of the grain shall 
be left uncut? 

22. A coal bin is to be 8 ft. deep and three times as 
long as it is wide, and is to contain 15 tons of coal. If 
40 cu. ft. is allowed for 1 ton, how long must the bin be ? 

23. One leg of a given triangle exceeds the other by 
2 ft. If the hypotenuse is 10 ft., find the legs of the 
triangle. 

24. Find the side of a square whose diagonal is a inches. 

25. The product of two consecutive numbers is b. 
Find the numbers. 



144 DURELL'S ALGEBRA : BOOK TWO 

26. The side of a given square is a feet. By how 
many feet must this side be increased, in order that the 
area of the square may be increased by h sq. ft. ? 

27. Make up and work an example similar to Ex. 6. 
To Ex. 12. 

28. To two other examples in this Exercise which you 
think are interesting or suggestive. 

EXERCISE 09 

1. Inir=7riP, findiJ. In jr= J7r2>a, find D. 

2. Find, in feet, the length of a tether by which a cow must 
be tied in order that she may graze over one half of an acre. 

3. In T^TrR(^R + L), if 7=440 sq. ft., 7r = ^4, 
i= 30 ft, find B. Also find a formula for R in terms of 
the other letters. 

4. In V^\irHiIP + r^+Rr), find r when r= 2662 
cu. in., jff= 21 in., and iJ = 10 in. 

What is the meaning of the above formula in Solid 
Geometry ? 

5. In 7= 27ri2(i2 -|- if), find R in terms of the other 
letters. 

What is the meaning of this problem in Solid Geometry ? 

6. In « = ^gt^, if ^r = 32 ft. and a? :a J mile, find t. 
What is the meaning of this problem when applied to a 

falling body ? 

7. In « = I ^r^ — vf, find t in terms of the other letters. 
As applied to projectiles, what is the meaning of the 

problem ? 

8. In h^a + vt-'\gt\ find t in terms of th^ other 
letters. 

Explain the meaning of the problem as ajiplied to 
projectiles. 



AFFECTED QUADRATIC EQUATIONS 145 

9. If an arrow shot over the top of a tower reaches 
the ground in 5} seconds, determine the height of the 
steeple (resistance of the air being neglected). Is the 
actual height of the steeple greater or less than the height 
as thus calculated ? (Use ^ = 82 ft.) 

10. Using the formulas 8=s^fffi and ^=s82.2 ft., find 
the distance a body will fall from the end of 2.4 sec. to 
the end of 4.2 sec. 

11. If ^ = 5.4 ft. on the surface of the moon and 32 ft. 
on the surface of the earth, how many seconds longer will 
it take a body to fall 1000 ft. on the moon than on the 
earth ? 



CHAPTER XI 
SIMULTANEOUS QUADRATIC EQUATIONS 

99. Need and Utility of Simultaneous Equations inyolving 
Qnadratie Equations. 

Ex. The area of a rectangular building lot is to be 
7500 sq. ft. and its perimeter is to be 400 ft. Find the 
dimensions of the lot. 



Let a: = no. ft. in length of lot 
y = no. ft. in width of lot 

Hence, 2x + 2y=400 .... (1) 
IB ■ xy = 7500. ... (2) 

By solving (1) and (2), x and y can be determined. (See Art. 106.) 

Try to solve the problem by use of only one unknown, 
as X. Even if you succeed in getting a solution, you will 
find the method awkward and inconvenient. 

100. Quadratic Equations containing Two Unknowns. The 
general quadratic equation containing two unknowns is 

ax^ -f hxy + cy^ -^-dx -\-ey +f = 0. 

By giving a, J, <?, etc., different numerical values, in- 
cluding zero, this general equation may be made to take 
many special forms. 

What values must we give a, 6, c ••• respectively, in order to 
obtain the equation 5x^-\- 3xy -{■ 2y^ = 6 from the general equation ? 

The absolute term of an equation is the term which does 
not contain an unknown factor, as / in the above general 
equation. 

146 



SIMULTANEOUS QUADRATIC EQUATIONS 147 

Simultaneous qnadratio equations is a brief term for 
simultaneous equations whose solution involves quadratic 
equations. 

Thus, equations (1) and (2) in Art. 99 are simultaneous quadratic 
equations. 

In general, the combination of two simultaneous quad- 
ratic equations by elimination gives an equation of the 
fourth degree in one unknown, which cannot be solved by 
the methods taught in this book. Two simultaneous 
quadratic equations can be solved by elementary methods 
only in certain special cases. 

101. A Homogeneous Equation is one in which all the terms 
containing an unknown quantity are of the same degree. 

Thus, 3x^ — 5xy* + y' = 18 is a homogeneous equation of the 
third degree. What is the degree of the equation xy = Q*i 

General Methods of Solution 
Case I 

102. When One Equation is of the First Degree and the 
Other of the Second, two simultaneous equations may always 
be solved by the method of substitution, 

2a;-.3y=3 (1) 

4a^^-7a:y = 15 (2) 

Eliminate y, since y occurs only once in equation (2). 
From(l), y = ^^ .... (3) 

Hence, from (2), 4 x^ - 7 x f^^""^ ] = 15 
Hence, 12 a:^ - 14 x^ + 21 a: = 45 

Substitute for x in (3), y = 1, 4. J 

Check. Check. 

For X = 3 and y = 1 For x = ^- and y = 4 

2x-3y = 6-3 = 3 2x-3y = 15-12 = 3 

4 x^ - 7 xy = 36 - 21 = 15 4 x^ - 7 xy = 225 - 210 = 15 



Ex. Solve 



Roots, 



148 DURELL'S ALGEBRA: BOOK TWO 





KXXRCISE 60 


Solve and check : 




1. 3a!-l = y 


2. a; + 2y = l 


6a?-y» = l 


a? + 4y3=6a! + ll 


3. Sx- 


-y = 3 


8a?- 


-3a:y+5a!-y=8 


4. 2a? + 3xy-4y» + 3a;-8y = 14 


7a:- 


-5y=14 


5. a!?-4y»-8a; = 


8. 1 + 1 = 2 


2 + 5i^ § = 


2 8 
2 3 « 


X X 


f + 2 = 2 


6. 1+1^ + 4 = 


a; y 


a? a: 


9- i-^=0 


a! + 2y + 4 = 


2 8 


7. 2a; + 8y = l 


x» + ya = 5(a; + y) + 2 


6 + 1 = 2 


10. y-4=Ky-^) 

a:y = 2y + x + 2 


11. 5a 


!-3v+l = 



2y2 + 3a:y-5a;a+7a;-6y = 4 

12. The sum of two numbers is 3, and twice the square 
of the second number diminished by five times the square 
of the first number gives 3. Find the two numbers. 

13. State Ex. 1 as a problem concerning two unknown 
numbers. 

14. How many examples in Exercise 42 (p. 110) can 
you now work at sight? 

Cask II 

103. When Both Equations are Homogeneoiu and of the 
Second Degree, two simultaneous quadratic equations may 
(dwayB he solved by the substitution y = vx. 



SIMULTANEOUS QUADRATIC EQUATIONS 149 

Ex. Solve 22^2- 3a;y + 4ya = 6. 
a;a+3ya=7. 

Substitute y = rx, 2 a:3 - 3 t'a:« + 4 i7«x2 = 6 (1) 

x»+3!;»xa = 7 (2) 

^~'"(^>' ^= 2-3^4.« (^> 

Fron.(2), *' = n^. ^*) 

Equate the values of o^ in (3) and (4), 
6 7 



2-3i; + 4»» l + 3t;a 

Hence, 6 -♦- 18i;« = 14 - 21 r + 28 1'« 

10t;3 - 21 1; -♦- 8 = .-. t? = J, J 



K t; = J, x* = 



1 + i 
. X = ± 2 
y=i;a;=J(±2) = ±l 



nr = t,xa=^ 



1 + W 
y = rx = f(± A>/3l) = ± /rVSi 



Hence, x = ±2, ±AV31.| jj^^^ 

y = ±l, ±AV31.J 

Let the pupil check the work. 

Two simultaneous equations of the kind treated in 
Case II may also be solved by eliminating the absolute 
term between them and factoring to find the value of one 
unknown in terms of the other, and then proceeding as in 
Case I. 

EZSRCISB 61 

Solve and check : 

1. y2 + 3a.y = 28 3. ya=5 + 2a;y 
4a^^ + a:y = 8 a;" + y2=:29 

2. 4a^^-3a'y+2 = 4. y2 + a:y = 21 
Sar^- 2^24-6 = 2a:y-ar^ = 8 



150 DURELL'S ALGEBRA : BOOK TWO 

5. a?-«y + 2^ = 21 7. a? + a?y + y2=,i 
a?-2a:y=:-15 22^ + Sxy + 4y^^S 

6. 2a?-4a?y+3y2 = 17 8. 4a? + 2a;y--ya = 41 
a?-y2^16 8a?-5iry + ya=58 

Solve and check the following miscellaneous examples : 
9. 2ir=:y-3 12. 46 + a;? = 2y2 

4a? + y2=,i7 a?=U-xf/ 

10. a? + 3y2=12 13. K^-l) = Jy 

6a:y-4y2 = ll y^- 7 = (2;- y)^ 



11. 



2y _ a;4-y 14. 4 y^ — a;y = a;^ — 16 

« — y — a = ofl X 



15. Point out the examples in Exercise 66 (p. 169) 
which come under Case I. Under Case II. 

16. Point out the homogeneous equations in Exs. 21-28 
of Exercise 62. 

17. Express Ex. 1 above as a problem concerning two 
numbers. 

18. Work the example of Art. 99 (p. 146) by the 
method of Art. 102. 

19. Find a number consisting of two digits such that if 
the number is multiplied by the left-hand digit, the result 
will be 105 ; but if the number is multiplied by the right- 
hand digit, the result will be 175. 

20. Make up (but do not solve) an example in each of 
the cases studied thus far in this chapter. 

21. How many examples in Exercise 44 (p. 113) can 
you now work at sight ? 



SIMULTANEOUS QUADRATIC EQUATIONS 151 

Special Methods of Solving Simultaneous 
Quadratics 

104. The methods of Cases I and II are the only general 
methods ^hich can be used in solving all simultaneous 
quadratic equations of a given class. Besides these, how- 
ever, there are certain special methods which enable us to 
solve important particular examples. 

Examples which come directly under Cases I and II 
are often solved more advantageously by one of these 
special methods. 

The special methods apply with particular advantage to 
symmetrical equations. 

105. A Symmetrical EqnatioiL is one in which, if y is sub- 
stituted for re, and x for y, the resulting equation is iden- 
tical with the original equation. 

Thus, each of the following is a symmetrical equation : 
x»4-3xV + y» = 18, x + y=12, ary = 6 

Case IH 

106. Addition and Subtraction Method (often in connection 
with multiplication and division). In this method the 
object is to find firBt, the values ofx+y and Jr — y, and then 
the valties of x and y themselves. 

x^y^l (1) 

^y = 10 (2) 

Here we have the value of x + y given, and the first object is to 
find the value of x — y. 

Square (1), x« + 2a:y + y' = 49 .... (3) 

Multiply (2) by 4, 4xy = 40 . . . . (4) 

Subtract (4) from (3), ar^ - 2 xy + yS = 9 . . . . (5) 
Extract square root of (5), x — y = ± 3 . . . . (6) 

Add (1) and (6), divide by 2, x = 5 or 2. 1 

Subtract (6) from (1), divide by 2, y = 2 or 5. j ^^^^• 
Let the pupil check the work. 



Ex. 1. Solve 



152 DURELL'S ALGEBRA : BOOK TWO 

Ex. 2. Solve 



Divide (1) by (2), 

Square (2), 

Subtract (3) from (4), 

Hence, 

Subtract (5) from (3), 

But 
Hence, 



Ex. 8. Solve 



Squaring (1), 



aJJ+y«=:28 . . 
a? + y=:4 . . 
x^ - ary + y« = 7 

3xy = 9 
xy = S 
x^^2xy + y^^^ 
.'.x-y=±2 
x + y = 4 
a: = 3, 1. 
y = 1, 3. 

i + V^s . • 

44-^ + 1 = 49 



Roots. 



xy y* 



Subtracting (2) from (3), 



xy 



= 24 



Subtracting (4) from (2), 1 - A + 1 = i 

x« xy y2 



(1) 

(2) 

(3) 
(4) 

(6) 



(1) 

(2) 

(3) 
(4) 
(5) 



Hence, 




X y 


But, from (1), 




i + i= 7 

X y 


Hence, adding, 




I =8.6 


Let the pupil check the work. 


.•.X=i, J.I jj^^ 

y = J. i- J 


Solve and check: 


EXERCISE 6S 


1. ir-fy=:2 




3. a? + y» = 34 


a:y = -15 




a;y = - 15 


2. a? + y = 5 




4. ar' + a;y + y« = 28 


x^ + y^=U 




a; + y-6 = 



SIMULTANEOUS QUADRATIC EQUATIONS 153 



5. 


9(n? + y») = 37 


13. 


1 + 1 = 9 




^y = -f 




aJ»^y8 


6. 


a^ + y8 = 28a« 




1 + 1 = 3 




a:4-y = 4a 




X y 


7. 


a^ + /=3J 


14. 


^ + f = -\xy 




a?+y = 2 




a; + y=J 


8. 


2:84.^=, 37 
^-a;y + y2=37 


15. 
16. 


a^ + a?y' + y* = 21 
a:a + a;y + y2=7 
a? + a;y + jr' = 133 


9. 


a? + 2^ = 2a2 4-262 


17. 


a:- Vxy+y^l 
a;y = 80 


10. 


^^2^ 




i_i=i 




y a; 




a: y 6 




a! + y = 7 


18. 


a? + y2 = 25 


11. 


a?+y^=56 




a;-y = 7 




j3y + a;y2=_i6 


19. 


aJ»-^ = 6a25+2 68 


12. 


Ul=7 




x-y=26 




a; y 


20. 


a'-y = |l>-2j 




-1 = 12 




.y = 2^^ 




a;y 




2 


Sol 


ve and check the following miscellaneous examples 


21. 


x = i/ 


25. 


3_l2 ft 
— 1 — —5 




3^ + 5xi/ = 5i 




a: y 

a; + y = 2 


22. 


a? + y2 = 13J8 


26. 


2ar« + ya-3y = 30 




a:y = 6 J2 




£ = 2 








23. 


2y2 = 15-xi/ 




y 




3:8-^2 + 8 = 


27. 


8a^ = 28-y2 


24. 


Ui=i 




l4-? + 2^ = l« 

y y' y' 




VA=25 


28. 


^ i^=2^ 




a:2^y2 




a;-y = 2 



154 DURELL'S ALGEBRA: BOOK TWO 

29. Find two numbers such that their sum is 12 and 
their product is 27. 

30. State Ex. 1 as a problem concerning two unknown 
numbers. In like manner, state Ex. 2. Ex. 8. Ex. 17. 

31. Make up and solve a problem concerning two un- 
known numbers such that the solution involves simultane- 
ous quadratic equations. 

32. Point out the symmetrical equations in the examples 
in this Exercise. 

33. Make up (but do not solve) an example in Case I ; 
one in Case II ; and five different examples in Case III. 

Case IV 
107. Solution by the Substitutions, jr = a + ^ and y=z a— b, 

Ex. Solve (:^-fy^ = 242. • • • (1) 

I a:-hy=2 . . . . (2) 
Substitute ar = a + byy = a-bm(l) and (2), 
Then, (« + 6)6 + (a -6)6 = 242 .... (3) 

a+b + a-b = 2 . . . . (4) 
From (3), 2 a^ + 20 a%^ + 10 a6* = 242 . . . . (5) 

From (4), 2a = 2 .... (6) 

Divide (5) and (6) by 2, and substitute 1 for a in (5), 

1 + 1062 + 56* = 121 
Hence, 6 =±2, ±y/-^ 

But a = 1 

Hence, a: = a + 6 = 3, - 1, 1 ± V- 6. 1 

y = a-b= -1,3,1T\/-6.J 
Let the pupil check the work. 

EXERCISE 63 
Solve : 

1. x5 + 2^=244 3. a:«-f/=211 
a: + y = 4 a;-fy = l 

2. 2?*+/ = 82 4. a:* + y* = 257 



SIMULTANEOUS QUADRATIC EQUATIONS 155 

5. a:«-y5 = 2 6. a^+16/ = 97 

x — y = 2 z + 2t/=: 5 

Work the following miscellaDeous problems : 

7. rr-h4y = 14 9. f + ^ = ll 

j,^4-4.= 2j,4.11 IJ^J^ 

5^2 + 3y2= 154.4 xy a^ + 2:2^2 ^ y4 ^ 9I 

11. Solve Ex. 1 by dividing the first equation by the 
second. 

12. Make up (but do not solve) an example illustrating 
each of the cases studied thus far in this chapter. 

Case V 

108. Use of Compound Unknown Quantities. It is often 
expedient to consider some expression (as the sum^ difference, 
or product of the unknown quantities) as a single unknown 
quantity^ and find its value^ and hence the value of the un- 
knotvn quantities themselves, 

Ex. Solve j.? + y« = 18-:r-y . . (1) 
1 xt/=6 (2) 

Multiply (2) by 2, 2^:^ = 12 (3) 

Add (3) and (1), x^ ■^2xy + y^ = S0 - x - y . . . (4) 

Let X -{■ y = V 

Then from (4), v^ = 30 - i? 

r = -6, 6 



Hence, x + y = — 6 

xy =:Q 

.•.a: = -3±V3. ] 

o AJ t Roots, 

y = - 3 =F v3- J 
Let the pupil check the work. 



also X + y = 5 
,xy = Q 



•.a: = 3, 2. 1 

y = 2,3.| 



Roots, 



166 DURELL'S ALGEBRA: BOOK TWO 

EZSRCISB 64 

Solve and check : 

1. ir + y + V^Ty = 6 6. a? + t/^ + x + i/=:24: 
icy = 3 a;y = - 12 

2. 2?y^ + xy=:6 7. 2^ + y^=:x-t/ + 50 
x + 2t/ = -5 a:y=24 

5. x-y + Vx-if = Q 10. a? + 3,2 + x + 5y = 6 
a'y = 5 a;y-2y = -2 

11. a;*yi(a;* + y*)=70 

a;V + ''^* + y* = l'*^ 

Solve the following miscellaneous examples : 

12. y-^ = i 16. a; + y + VF+y = 12 

a; ary = 20 

*y = ^ 17. a« + y2 = l25 

13. a;y + 3y2-20 = a:y = 22 
it?-3xy+8 = ^ £±2 = 3 

y — 1 

14. a^a + 3xy = 18 2a:a-xy=20 



a; + 2y = 5 



19. 



- + ? = ! 

a 6 



15. =1 -f ^ = — 1 

y a; 2 ^ + ^ = 4 

x+f/=:l X y 

20. Make up (but do not solve) an example illustrat- 
ing each of the cases studied thus far in the chapter. 



SIMULTANEOUS QUADRATIC EQUATIONS 157 

Case VI 

109. Factorial Solutions. The solution of a set of simul- 
taneous quadratic equations is often facilitated by the use 
of factoring. 

Thus, P'"'^^-^'/ = ^ 

may be solved by Case I, but the solution may be shortened by factor- 
ing in the first equation. The two following equations will then be 
obtained : 

\ 2ar + 3y = 7 

Since the first equation is satisfied either when ar — y = or 
X — 2 ^ = 0, we obtain the following two sets of equations as equiv- 
alent to the original single set or system : 



\2x + 3y = 7 |2x + 3y = 7 



Whence, a: = J. 1 . Whence, a; = 2. 1 . 

Let the pupil check the work. 

The solution of this example by the factorial method 
requires less than one fourth the labor involved in the 
solution by the method of Art. 102. 

In general, if -4, -B, (7, and D are algebraic expressions 
integral with reference to x and y, and if 

A'B' (7=0 

the given set of equations is equivalent to the following 
three sets : 



f^ = 
i> = 



5 = 
i>=0 



(7 = 
i)=0 



Hence, let the pupil state the sets of simple equations whose solu- 
tion IS equivalent to the solution of 

Ux-2y)(Sx + y)(x-Sy) = 
\ 4:x-y -6 = 



158 



DURELL'S ALGEBRA: BOOK TWO 



110. Factorial Method of Solution Aided by Bivision. 

If A, J8, (7, and D have the meaning given in Art. 109, 

A = B 

it is evident that this system of equations is satisfied 



either when 



or 



A = 
5 = 



Hence, the last two sets form a system equivalent to the 
original system. 

Note that the equation C = Z) is obtained by dividing the mem- 
bers of -4 • C = J5 • Z) by the corresponding members of -4 = B, 

Ex. Solve a:8^a: = 9y2 (1) 

a:2+l = 6y (2) 

Writing the equation (1) in the form x(a;*+ 1) = 6 y(f y), we obtain 
the two systems which follow : 



x2 + l=6y 
Whence, X =2 ±v/3. 1 ^^^ 
y = K2±V3>) 
Let the pupil check the work. 



a:* + 1 = 
6y = 

Whence, a: = ± V — 1. 

y = 0. 



Roots, 



EXERCISE 
Solve by the factorial method : 

1. a:--2y = 4 I 
7? —bxy + iy^ = 

2. rr— 3y=sl 
n? + 3a:y + 2y2=0 

3. x(x — y) = 
a^»-3a:y-4y2 = 

4. (2r-2)(y-2)=0 
3ar^- 4a;y + y2=:0 



(x-2Xy + 3)«0 
(a:-2y)(a:+2y) = 

6. 1 +a:8 = y« 

7. y(x+3)=9a;2-l 
y = 3a;-l 

8. y» + y = 92? 
y 2 + 1 = 6 a; 



SIMULTANEOUS QUADRATIC EQUATIONS 159 

9. Practice the oral solution of simple equations as in 
Exercise 19 (p. 48). 

10. Solve Ex. 1 of Exercise 61 (p. 149) by eliminating 
the absolute term in the two equations and applying the 
factorial method to the resulting equation. 

11. In the same way work Ex. 2 of Exercise 61 
(p. 149). Also Ex. 8. 

EXERCISE 66 

Review 

Id solving a system of simultaneous quadratic equations, the first 
thing to notice is the degree of the equation. Solve and check : 

1. x + 3y = 3 11. 2xa + 3x + y-2 = 
2y^ + xy = 2 Sx^ -j-2x -{■ iy + 7 = 

2. a:» + y»=117 12. xy+2y=o 
a? + y = 3 2xy-x = 3 

3. yi = Q-xy 23. x'^ + y-^ = 5 
6a:2=8-a:y ar-i + y-i = 3 

**.':!-n^^^'' 14. x+iHl = 2 

x-k-y = n x-\-2y 

5. x^ + y* = 17 x + 3y+l = 

^•^y = ^ 15. ar«y2-5xy + 6 = 

6. (x-l)(y-2)=0 6x + 3y=14 
(2.-6,)(3x-y + l)=0 le. ^^^^, 

7. l-i = l (a:~l)« + (y-2)« = 28 

^ y 

2xy+9 = 17. ? + -y=2A 



y X 

a:+y=7 

18. (x+y)2-(x + y)=20 
9. - + |=1 2a:2-3a:+4y=14 



8. J(^-y) = x-4 

a:y = 2x + y + 2 



a 6 

a 6^^ 19. x«-y« = -Jxy 

a: v x-y =1 



10. x+y + Vx + y=12 20. .y(y + 3a: - 7)= 

a:-y4-V'^^ = 2 (^ + x - 3)(y + 2x - 4)= 



160 DURELL'S ALGEBRA : DOOK TWO 



21. a: + y = 28 


31. 


2x-i + oy~i = 4 


^i + ^ = 4 




x-2--2x-iy"^ + y-2=l 


22. x'^ + y^ + x-^Sy^lB 


32. 


x» + a:=9y 


xy - y = 12 




xa+l = 6y 


23.i-? = 3 


33. 


x* = X + y 


X y 


34. 


y«=3y-x 
9x2+25y9=148 


24. a:« + xy + y« = 91 




5 = A 


X + Viy + y = 13 




xy 


25. x*-2yi = t 
x~8y = H 


35. 


x(x-y)=0 

x« + 2xy + y*-9 = 


26. 2x + y-^=l 

y 


36. 


^ + y = 3 
l-a:y 


3x + 4y~?^ = 2 




l+xy 3 


27. x« + 4y« + 80 = 15x + 30y 


37. 


x« + y» = 19 


xy = 6 




x + y + Va? + y = 2 


28. X* + y* = 17 


38. 


xy = 4 


a:y = 2 




X2 =9 


29. x« + y2 = ary + 7 




y2=16 


X ~ y = xy - 5 


39. 


xy + X2 = 5 


30. x-i+y-i = 5 




xy + y2 = 8 


(x+l)-i+(y + l)-i=H 




X2 + y2 = 9 



40. Make up (but do not solve) an example of each of the cases, 
and the principal sub-forms in each case, treated in this chapter. 

EXERCISE 67 

1. Find two numbers whose sum is 11 and the sum of 
whose cubes is 539. 

2. The difference of the cubes of two numbers is 279. 
Also the sum of the squares of the numbers increased by 
their product is 93. Find the numbers. 

3. Find two numbers whose sum is 16 and whose 
product is 55. 



SIMULTANEOUS QUADRATIC EQUATIONS 161 

4. Separate 12 into two parts, such that the sum of 
their squares exceeds their product by 63. 

5. Separate 10 into two parts, such that their product 
exceeds the square root of their product by 12. 

6. The sum of two numbers is 4 and the sum of their 
fifth powers is 244. Find the numbers, 

7. The hypotenuse of a right triangle is 20, and the 
sura of the other two sides is 28. Find the length of the 
sides. 

8. The area of a rectangular field is 750 sq. rd., and 
the perimeter of the field is 110 rd. Find the dimensions 
of the field. 

9. The area of a single tennis court is 234 sq. yd. 
If a margin of 15 ft. is added on all sides, the area is 
684 sq. yd. Find the dimensions of the court. 

10. A rectangular park is known to contain 6 acres. 
The path which leads diagonally across it is measured and 
found to be 52 rd. long. Find the length and width of 
the park. 

11. A rectangular piece of tin is made into an open box 
by cutting a 5-inch square from each corner and turning 
up the sides and ends. If a 3-inch square were cut from 
each corner, the box, made in the same way, would hold 
the same amount. The width of the tin is 4 inches less 
than its length. How much does the box hold ? 

12. There are two fractions : the numerator of the first 
is the square of the denominator of the second, and the 
numerator of the second is the square of the denominator 
of the first. The sum of the fractions is 6|, and the sum 
of the denominators is 5. Find the fractions. 



162 DURELL'S AI.GEBRA: BOOK TWO 

13. If a baseball nine should play tliree more games and 
win them all, its average of games won would be J. But 
if it should play 12 more games and win them all, its 
present average of games won would be increased by 
20 per cent of its previous average. Find the number of 
games it has already played and the number it has won. 

14. A laborer received $15 for a certain number of 
days' work. If he had received 25 cents less a day, it 
would have taken him two days longer to earn the same 
amount. How long did he work ? 

15. Two trains run 36 miles at uniform rates. One 
train travels 9 miles per hour faster than the other and 
makes the trip in 12 minutes less than the other. Find 
the rates of the trains. 

16. A rectangular court can be paved with 288 square 
tiles of a certain size. But if a side of the tile used is 
increased by 3 in., only 162 tiles are needed. Find a side 
of each kind of tile. 

17. A certain number contains two digits. If the num- 
ber is multiplied by the units' digit, the product is 216. 
But if the number is multiplied by the number obtained 
by inverting the order of the digits, the product is 2268. 
Find the number. 

18. A man can row 18 miles down a stream in 3 hours. 
But he can row 12 miles down and back in 5 hours. Find 
the man's rate and also the rate of the stream. 

19. A crew can row 16 miles down a stream and back 
in 6 hours. But if they row at half their usual rate, they 
can go only 5 miles down the stream and back in 6 hours. 
Find the rate of the crew and also that of the stream. 

20. Find two numbers, the sum of whose squares is 
225, and the difference of whose squares is 63. 



SIMULTANEOUS QUADRATIC EQUATIONS 163 

21. The area of a (double) tennis court is 312 sq. yd., 
and the length exceeds the width by 42 ft. Find the 
dimensions of the court. 

22. Telegraph poles are set at equal distances apart. 
In order to have two less to the mile, it will be necessary 
to set them 20 ft. farther apart. Find out how far apart 
they are now. 

23. A rectangular field of grain contains 30 acres. If 
a strip 10 rd. wide is cut around the field, 15 acres will be 
left uncut. Find the dimension of the field. 

24. The denominator of a certain fraction exceeds the 
numerator by 6. If the numerator of the fraction is 
increased by 3 and the denominator by 15, the value of 
the fraction becomes } of what it was originally. Find 
the fraction. 

25. The product of two numbers is 85. If the larger 
of the two numbers is divided by the smaller, the quotient 
is 3 and the remainder 2. Find the numbers. 

26. A company contracted to make 252 automobiles. 
Two factories, working together, can make this number in 
12 days. Working alone, one factory requires 7 daj's 
longer than the other to do this amount. Find the num- 
ber of days in which each factory alone can fulfill the 
contract. 

27. Find two numbers whose sum is b and the sum of 
whose cubes is a. 

28. The area of a right triangle is p and the hypotenuse 
is q. Find the other two sides. 

29. Make up and solve three examples similar to such 
problems in this Exercise as you think are most interesting 
or suggestive. 



CHAPTER XII 



GRAPHS OF QUADRATIC AND HIGHER EQUATIONS 
111. Graph of a Quadratic Equation of Two XTiikiLOwii 



X 


y 





6 


1 


2 


2 





3 





4 


2 


5 


6 


21 


-i 


etc. 




-1 


12 


etc. 





Quantities. 
Ex. 1. 
5 a; + 6. 



Construct the graph of y = a? — 



The graph obtained is the curve ABC, A curve of 
this kind is called a parabola. The path of a projectile, 
for instance, that of a baseball when thrown or batted 
(resistance of the air being neglected), is an arc of a 
parabola. 



X^ 



m 

m 



It will be noted that the above method of graphing is 
the same as that given in Art. 28 (p. 71), but that 

164 



GRAPHS OF QUADRATIC EQUATIONS 



165 



here it is sometimes advantageous to let x have frac- 
tional values, as J, \^ ^, |, etc. The observant pupil 
will also find methods of abbreviating the work in cer- 
tain cases. 

In general, it will be found that the graph of a quad- 
ratic equation of two unknown quantities is a curved line, 
and, in particular, either a circle, parabola, ellipse, or 
hyperbola. 

Ex. 2. Construct the graph of 4 a:^ - 9 y^ = 36. 



For negative values of x, the values of y are the same 
as for the corresponding positive values of x. Hence, the 
graph is a curve of two branches, ABC and A'B'C, of 
the species known as the hyperbola. 



X 


y 





imag. 


1 


imag. 


2 


imag. 


3 





4 


±1.7 


5 


±2.6 


6 


±3.4 


e 


tc. 



r 

_ _ 

-^ ^ 

-ig. aC 

£lZZZZZZZZZIIZZZZZZ£ 



166 DURELL'S ALGEBRA: BOOK TWO 

SXERCISS 68 

Graph the following : 

1. y = 2:2 - 2 14. y2 - a? = 4 

2. y = ar^ + 2a:-3 15. a^^-y2^4 

3. y = a;2- 2aJ-h 1 16. a:y = 3 

4. y = a? — 3 a: — 4 17. a;y = — 3 

5. y =^2^ IB. X -\-xy = 2 

6. y = a:2-h| 19. (a:-2)Hy2 = 9 

7. a:2^y^25 20. 16y2-a:2^_ig 

8. a^-hy2^| 21. y2 = 9^^i 

9. y2 ^ 9 a: - a;2 22. y^ = 9 

10. 16a:2^9y2^144 23. a^ + xy + f = 25 

11. ar2 + 9 y2 = 9 24. a;2 = 9 

12. 9a:2^y2^9 25. ar2-3a? + 2 = 

13. ya = 16a: 

Suo. Show that whatever the value of y, x always = 1 or 2 ; hence, 
the required graph is two straight lines parallel with the y-axis. 

26. On one diagram, graph the results obtained by 
letting a = 1, 2, 4, 9 in succession, in a^^ -h y^ = ^. 

27. Treat in like manner xy =s a. 

28. Make up and work an example similar to Ex. 1. 
To Ex. 2. To Ex. 9. 

29. Also an example similar to Ex. 14. To Ex. 15. 
To Ex. 25. To Ex. 26. 



Ex. Solve graphically 



112. Graphic Solution of Simultaneous Quadratic EquationB. 

a;2-f y2 = 25. 
y + 2a;-5 = 0. 

Constructing the graph of x^ + y^ = 25, we obtain the circle ABC 
(p. 167). Constructing the graph of y + 2ar — 5 = 0, we obtain the 
straight line FH. 



GRAPHS OF QUADRATIC EQUATIONS 



167 



Measunng the codrdinates of the points of intersection of the two 
graphs, we find the points to be (4, — 3) and (0, 6). 

These results may be verified by solving the two given simultaneous 
equations algebraically. 




113. Special Cases; Imaginary Boots. Construct the 
graphs of a:2 + y^ = 4 and x-{-i/=Sr You will find that 
these two graphs do not intersect. Then solve the given 
equations in the ordinary algebraic way. You will find 
that the roots are imaginary. If you treat the equations 
a^ -^ t/^ = l and 4:x^+9 i/^ = S6 in the same way, you will 
obtain a similar result. 

In general, imaginary roots of simultaneous equations cor- 
respond to points of non-intersection of the graphs of the given 
epilations. 

Remember that in solving a pair of simultaneous equations, the 
number of values of x (and also of y) is equal to the sum of the de- 
grees of the two equations. Hence, if two simultaneous equations 
are both of the second degree, their graphs should intersect in four 



168 DURELL'S ALGEBRA: BOOK TWO 

points ; and if their graphs are found to intersect in only two points, 
for instance, the other two points must correspond to imaginary 
roots. 

The pupil may illustrate this by graphing and also solving alge- 
braically y' = 4 X and x* + y^ = 25. 

SZSRCISS 69 

Solve both algebraically and graphically: 
1. y3=92; U. a:y = -l 

a. y^^Qx 12. «? + ya=:25 
y=s3x iry = — 12 

3. ya = 9a; 13. a? + y^=^25 
aJ4-y = a;-y = — 1 

4. y^ = 2x 14. «?-4y = 

5. a? + y3=:25 15. a? + y2-10y = 
3a: + y-5 = x=:2y 

6. a? + y^=25 16. a? + y* = 16 
y-2;=l a? + 9ya = 48 

7. a? + y2 = 25 ' 17. 4ya-3a? = 24 
4y-3a; = a? + ya = 6 

8. a;y = 4 18. 3a? + y3 = 3 
a; + y = 5 y=:a;-2 

9. ajy = — 4 19. a: — y — 6 = 
:{;-f-y=:3 y2--6a; — a:^ 

10. a;y = -3 20. a? + y2+ 32; + y = 18 

a: + y = — 2 a;y — a;=12 

21. Obtain both the algebraic and graphical solution of 
4 a? - 9ya = 36 and a^» 4- y^ = a^ when a = 1. Also when 
a = 3. When a = 4. Graph all results on a single dia- 
gram. 



GRAPHS OF QUADRATIC EQUATIONS 169 

22. Solve graphically rc2+y=s3 and a; + y^=s5. Also 
try to solve this pair of equations algebraically. 

23. Make up and work an example similar to Ex. 2. 
To Ex. 5. To Ex- 8. 

24. Also an example similar to Ex. 21. To Ex. 22. 

114. Graphic Solution of a Quadratic or Higher Equation of 
One Unknown Quantity. By substituting for y in the first 
equation, the pair of equations y^Qi?^bx + % and y = 
reduces to rc^— 5a: + 6=:0. Accordingly, the graphic 
solution of an equation like a;^ — 5a;+6=0 is obtained 
by solving graphically y = a?— 5a; + 6 and y = 0. 

In other words, the roots of a quadratic equation of one 
unknown quantity^ aa? + bx + c=0^ are represented graphi- 
cally by the abadnsas of the points where the graph of yss ax^ 
+ bx + c meets the z-axis. 

Ex. Solve graphically a? — 5a;+ 6=0. 

The graph of y = a;^ — 5 x + 6 is the curved line ABC of the figure 
in Art. Ill (p. 164). 

This curve crosses the x-axis at the points (2, 0) and (3, 0) 
.'. X = 2, 3. Roots. 

The same results are obtained by solving the equation x^ — 5 x + 6 
= algebraically. 

This method of solution applies also to a cubic equation 
or to an equation of one unknown quantity of any degree. 

Thus, to solve the equation x* — 3x^+5x— 2 = 0, graph the equa- 
tion y = x* — 3x*+ 5x — 2. The abscissas of the points where this 
graph crosses the x-axis have the same value as the roots of the given 
equation x* — 3x*H-5x — 2=0. 

SXERCISE 70 

Solve both graphically and algebraically : 

1. a?-4 = 3. 4a;2 + 8aj-6 = 

2. «2_3a._4:^0 4. a?-6a;+9=:0 



170 



DURELL'S ALGEDUA: COOK TWO 



5. a?-4a;-hl=:0 6. a^-2x=^0 

7. a:8-2a;-l = 
SuG. Solve algebraically by the factorial method. 

8. a:8-a?-6a; = 9. a:^- 2a:2- 2a;+ 4 = 

10. a^-5ic2-f4 = 

11. Make up and work an example similar to Ex. 1. 
To Ex. 2. Ex. 10. 

Some Applied Graphs 

115. Wider Application of Graphs. Besides their use in 
ordinary algebra, graphs may be used to represent the 
properties of a great variety of functions, in particular 
those occurring in the various departments of science and 
in business life. 

Sometimes it is found convenient to use a different scale 
in laying off magnitudes on one axis from that used on 
the other axis. 

EXERCISE 71 

1. Graph C = |(F — 32), making the scale on the 
C-axis one half as large as that on the F-axis. 

2. A thermometer reads as follows at different hours 
during the day : 



Hour . . . 


7 a.m. 


8 a.m. 


9 a.m. 


10 a.m. 


11 A.M. 


12 a.m. 


1p.m. 


2 p.m. 


Temperature 


50° 


51° 


54° 


69° 


65° 


71° 


75° 


78° 




Hour . . . 


3 p.m. 


4 P.M. 


5 p.m. 


6 p.m. 


7 p.m. 


8 p.m. 


9 p.m. 


10 p.m. 


Temperature 


78° 


77° 


71° 


65° 


60° 


57° 


65° 


51° 



Construct a graph showing the relation between the 
temperature above 60° (taken as plus) and that below 



GRAPHS OF QUADRATIC EQUATIONS 



171 



(taken as minus), and the hour of tlie day. Then point 
out some facts to be learned from this g^aph. 

3. Graph F = — , and on the graph obtained measure 

the value of F when P = 1.5. 

4. Construct the graph of « = 16.1 1^ making the «-8cale 
but one tenth as large as the ^-scale. 

5. The average temperature on the first day in each 
month for a period of thirty years in New York City was 
as follows. Graph these data. 

New York 



Date . . . 


Jan. 1 


Feb.l 


March 1 


April 1 


May 1 


June 1 


Temperature . 


31° 


31° 


36° 


42° 


54° 


64° 




Date . . . 


July 1 


Aug. 1 


Sept. 1 


Oct. 1 


Nov. 1 


Dec. 1 


Temperature . 


71° 


73° 


69° 


61° 


49° 


39° 



The corresponding temperatures in London were as 

follows : 

London 



Date . . . 


Jan. 1 


Feb.l 


March 1 


April 1 


May 1 


June 1 


Temperature . 


37° 


38° 


40° 


45° 


60° 


57° 




Date . . . 


Julyl 


Aug. 1 


Sept. 1 


Oct. 1 


Nov. 1 


Dec. 1 


Temperature . 


62° 


62° 


69° 


64° 


46° 


41° 



Graph these results on the same paper with the graph 
of the New York temperatures and then compare the two 
curves of annual temperature, and give three facts which 
may be inferred from these curves. 



172 



DURELL'S ALGEBRA : BOOK TWO 



6. The following table shows the number of years which 
a person having attained a certain age may expect to live. 
Construct a graph of life expectancy from the data. 



Age in Years .... 





2 


4 


6 


8 


10 


20 


so 


Life Expectancy in Years 


38.7 


47.6 


60.8 


61.2 


60.2 


48.8 


41.6 


34.3 




Age in Years 


40 


60 


60 


70 


80 


00 


100 




Life Expectancy in Years . 


27.6 


21.1 


14.3 


9.2 


6.2 


3.2 


2.3 



From this graph determine your life expectancy at the 
present time, and also that of several acquaintances of 
various ages. 

7. Graph the growth of the population of the United 
States, using the following table : 



Year 


1700 


1800 


1810 


1820 


1830 


1840 


1860 


Millions .... 


4 


6 


7 


10 


13 


17 


23 




Year 


1860 


1870 


1880 


1800 


1000 


1910 


Millions 


31 


30 


60 


63 


76 


02 



From your graph determine, as accurately as you can, 
the population in 1815. In 1835. In 1895. In 1905. 

From your graph determine, as nearly as you can, in 
what year the population was 35 millions. 70 millions. 
80 millions. 

8. The following table gives the amount of $1 at sim- 
ple interest, and also at compound interest at 4 % for 5, 
10, 15, 20, etc., years. On the same diagram draw (1) a 
graph of the amounts at simple interest, (2) a graph of 
the amounts at compound interest. 



GRAPHS OP QUADRATIC EQUATIONS 



173 



Years . . . 





5 


10 


16 


20 


26 


ao 


36 


Amoants at 
Simple Int. 


31 


J|^1.20 


$1.40 


$1.60 


$1.80 


$2.00 


$2.20 


$2.40 


Amounts at 
Com. Int. 


1 


1.22 


1.48 


1.80 


2.10 


2.67 


3.24 


8.96 



The amounts of f 1 at 5% for the same periods of time 
at compound interest are |1, $1.28, $1.63, $2.08, $2.65, 
•$3.39, $4.32, $5.52. On the same diagram make a graph 
of these amounts. 

9. The following table gives various wind pressures : 



Velocity of wind 
in mi. per hr. 


10 


20 


30 
4.6 


40 


60 


60 


70 


80 
32 


00 


100 


Pressure in lb. 

per sq. ft 


1.6 


2 


8 


12.0 


18 


2.46 


40.6 


60 



Graph the above table of facts. From this graph de- 
termine, as exactly as you can, the pressure when the 
velocity of the wind is 25 mi. per hour, 45 mi. 65 mi. 

10. From the table in Ex. 9, determine approximately 
the velocity of the wind when the wind pressure is 5 lb. 
per square foot. 10 lb. 30 lb. 

11. Graph y = a;*. 12. Graph y = ^. 

13. Construct the parallelogram whose sides are the 
graphs of the equations 3y— 4a:--13 = 0, 3y— 42;+19=0, 
y =s 3, y = — 1. Find the coordinates of the vertices of 
this parallelogram, and also its area. 

14. Graphs, or geometric pictures of numerical data, take 
many different forms beside the linear graphs treated in this 
book. For instance, the density with which national banks 
are distributed over the country may be indicated by dots 
on a map. Collect examples of different kinds of graphs. 



CHAPTER XIII 
GENERAL PROPERTIES OF QUADRATIC EQUATIONS 

116. Character of the Boots Inferred from the CoefSicientB. 

It is important to be able to infer at once from the nature 
of the coefficients of an equation whether the roots of tlie 
equation are equal or unequal, real or imaginary, positive 
or negative. 

Any quadratic equation may be reduced to the form 
03? + 6a; + <? = 0, in which a is positive. 

Solving 02? + 6a; + <? = 0, and denoting the roots by r^, 
rj (read, r sub-one, r sub-two), we obtain 

''^ Ta ' "» = Ta 

From these expressions we infer that 

1. If 6^ — 4 ac i% positive, the roots are real and unequal. 

2. If 6^ — 4 ac equals zero, the roots are real and equal. 

3. If 6^ — 4 ac is negative, the roots are imaginary. 

The roots are rational if 6^ — 4 ac is a perfect square or 
zero. 

Since the values of 6^ — 4 ac enable us to discriminate 
between different kinds of roots, this expression is termed 
the discriminant of aoi? + 6a; + c = 0. 

Ex. 1. Determine the character of the roots of the 
equation, a? — 2a; — 1 = 0. 

We have a = l, & = -2, c = -l 

69 _4ac = 4 + 4 = 8 
Hence, the roots are real and unequal. 

174 



PROPERTIES OF QUADRATIC EQUATIONS 175 



Ex. 2. Of a:2_o^^l^0 

Here a = l, b = - 2, c = 1 

Hence, the roots are real and equal. 

Ex. 3. Oi a?-2x + 2 = 
Here a=l, 6=-2, c = 2 

62-4a<; = 4-8=-4 
Hence, the roots are imaginary. 

The results obtained in Exs. 1, 2, and 3 may be con- 
vejniently illustrated by means of graphs. 

It is found that the graph of y = a:^ — 2 x — 1 is the curve (1) and 
crosses the or-axis at the two 
points A and B (correspond- 
ing to the two roots of 

The graph of y = ar^ — 2 a: 
+ 1 is the curve (2) and 
meets the a>axis at only 
one point (corresponding to 
the two equal roots of 
a:2-2a:+l = 0). 

The graph of y = x* — 2 a: 
+ 2 is the curve (3) which 
does not meet the x-axis at 
all (which illustrates the fact 
that the roots of the equa- 
tion x^ — 2x-|-2 = are im- 
aginary). 

117. Determining Coefficients so that the Roots shall satisfy 
a Given Condition. It is often possible so to determine the 
coefficients of an equation that the roots shall satisfy a 
given condition. 

Ex. Find the value of m which will give equal roots 
for the equation (m — V)Qfi + :nx + 2 m — 3 = 0. 











Y 






J 


































\ 










1 














i\\ 










// 














w 










/ 














\) 


\(3) 








'/ 














\ 










/ 














"1 


m. 








/ 


















s^. 


A 




f 


















^] 


[J 




















A 


( 


) 


IB 










X 










V. 


y 































































176 DURELL'S ALGEBRA : BOOK TWO 

By Art. 116, 2, in order that the roots may be equal 6^ — 4 ac = 0. 
In the given equation, a = ro — 1, 6 = m, c = 2m — 3 

.-. m3 - 4(m -l)(2m - 3)= 
m«~8m24-20w-12 = 

7 m2- 20m = -12 

m = 2, f . Ans. 

Chbck. Substituting these values for m in the original equation, 

x' + 2 X + 1 = 0, x^- 6 X + 9 = 0, 

in each of which equations the roots are equal. 

EXERCISE 7S 

Without solving, determine the character of the roots in 
1. x^-5x-^6=0 _ __ 9^:24.4 



2. 3a?-7a:-2 = 

3. 4a? = 4a:-l 



12 



4. 82? + 2a;+l = ^- ^ = K^' + 1) 

5. 22^-50^+3 = 1^- 35:r+18+122? = 

3 12. 7 2^+1 = 5a: 

13. Determine by inspection the nature of the roots in 

(1) a^-^x + 2=^0 

(2) a?-4a:-h4=:0 

(3) a;2-4a: + 6 = 
Verify your results by use of graphs. 

14. Make up and work an example similar to Ex. 13. 

15. In 3 a:^ — 2 a: + 1 = 0, determine the character of the 
roots by solving the equation. Now determine their 
character by the method of Art. 116. Compare the 
amount of work in the two processes. 



PROPERTIES OP QUADRATIC EQUATIONS 177 

Determine the value of m for which the roots of each 
equation will be equal : 

16. 2a^-2x + m=0 19. ^ofl + ^^mx 

17. ms? — 5x^^2 = 20. (m + l)afl'\-mx = l 

18. 23^-mx+12^ = 21. Qm+l)x^ + Sm^l2x 

22. (m + l^ofl 4- (wi - 1> + ?7i 4- 1 = 

24. What is meant by the root of an equation ? 

25. In Ex. 16, for what values of m are the roots 
imaginary? Real and unequal? 

26. Answer the same questions for Ex. 17. Ex. 20. 

27. Show that if one root of a quadratic equation is 
imaginary, the other root must be imaginary also. 

28. State and prove a similar fact concerning irrational 
roots. 

29. In a? — 6 a; + (? = 0, substitute (1) a value of c which 
will make the roots of the resulting equation equal ; (2) 
a value which will make the roots imaginary ; (3) a value 
which will make the roots real, unequal, and rational. 

30. How many of the above examples can you work at 
sight ? 

118. Belation between Boots and CoefBicients. In Art. 116 

a method was obtained of inferring, from the coeflftcients 
of a quadratic equation, the qualitative nature of the roots. 
A more exact, or quantitative^ relation between the roots and 
coefficients will now be obtained. This relation will enable 
us in any given equation to determine the sum or product 
(and often other functions) of the roots, without the 
labor of solving the equation. 



178 DURELL'S ALGEBRA : BOOK TWO 

Dividing both members of ca^ + bx -{-c^ by a, we 
obtain an equation in the form a? 4- joa: -h y = 0. 

SolviDg this equation and denoting its roots by a and p, 



«=-p+y^^,and)8=-/'-y-*? 

Adding the roots, a -\- p = " '^P z=-p 

Multiplying the roots, a/J = P^-^P^-^^) = g 

Hence, in general, 

(1) 2%« sum of the roots of jfl + px+ q = equals —p^ 
4>r the coefficient of x with the sign changed; 

(2) The product of the roots equals the known term q. 

Ex. 1. Without solving the equation, find the sum 
and also the product of the roots of 5(1 — 2 x) = 3 a:^. 

The given equation reduces toz^+J^a; — }=rO. 

Hence, sum of roots = — V, product of roots = — J. Ans, 



Ex. 2. Form the equation whose roots are - 



2 



The roots are - 1 -f vCTs - 1 - V-Ta 



Hence, sum of roots = - 1 + v^- 3 - 1 - y/^^ ^ ^ ^ _ ^ 

2 2 

Product of roots = ^""(~^) = M^ = 1 
4 4 

Hence, a:^ _|_ ^ + 1 = is the required equation. 
Checks for the above example may be obtained by solving the 
equations obtained. 

119. Factoring a Quadratic Expression. Any quadratic 
expression may be factored by letting the given expression 
equal zero, and using the property stated in Art. 118. 



PROPERTIES OF QUADRATIC EQUATIONS 179 
Ex. Factor 3 2:2 _ 4^ ^5 

Take 3(x2 -|a; + J) = 

Solve x^-ix-\-i = 

Whence, g= ^^^""^ l 

3 

Hence, the factors of 3 z^ — 4 a; + 5 are 



EXERCISE 73 

Find, by inspection, the sum and product of the roots 
in each of the following equations : 

1. T^+Sx + 5=^0 6. aV-aa; + 2=:0 

2. a?-a; + 7 = 7. 5a;-4a:a = l 

3. a:a-5x=10 8. 3-7a;=lla? 

4. 2a:2_6a;-3 = ^ £^a;-a 



.^ x-52> ^ « 4a:+l 

' "^"^ x^l 10. l-2(?a;-2aa:a = 3(? 

Form the equations whose roots are 

11. 2, 3 17. .08, -.2 2±V2 

22. ^;^ 

12. 3,-2 18. ab, - a 

13. — 1, - 

14. 5, .04 



1±V-T 



13. — 1, —5 ,^ a 6 23. 

19. T, -- z 

a 

-2±V-2 

15. -1,-1 20. 1 + V2,1-V2 24. ^ 



16. |, - t 21. - 3 ± VS" 25. \a± c V^ 6 

26. Form an equation whose roots are 2 ± i (see p. 119). 

27. Form an equation whose roots are 3, i, — i. 

28. If one root of the equation a;2_j^/|; — f = is 3, 
find the other root in two different ways. 



180 DURELL'S ALGEBRA: BOOK TWO 

29. If one root of the equation 8a? — 4a:+2=sO is 
"'' -, find the other root in three different ways. 



3 

30. Form the equation in which one root is — f and the 
product of the roots is — |. 

31. Make up and work an example similar to Ex. 28. 
Also to Ex. 30. 

Factor : 

32. 8a?-102;-8 35. a^ + U^6x 

33. a?4-2a;-l 36. 25a?+2-30a; 

34. x^^x-1 37. Sx-Sa^-1 

38. If r and 8 represent the roots of 3 a;^ — 8 a: + 5 = 0, 
find, without determining the actual roots, the values of 

r+s; rs; r^+8^; r-«; r^-^^; r^+ffl; - + i; i-i; 1-^. 

r s r s f^ t!^ 

39. Find the values of the same expressions for the 
equation 2a?-9a; + 7 = 0. Also for 6 a? - a;- 12 = 0. 

40. Find the values of the same expressions for the equa- 
tion aofl -h 5a; + <? = 0. Also for the equation a^+px +q=0. 

41. If m and n represent the roots of the equation 
10a;2 + 9a; — 7 = 0, form that equation whose roots shall 

be mn and m + n, m — n and —-}--. 

771 n 

42. In aa? + 5a; + <? = 0, if (? = a, show that one root is 
the reciprocal of the other. 

43. For what value of p does the equation a? +0p "* 8) 
a:— (5/) 4- 10)= have a zero root? Find the other root. 

SuG. K one root is zero, what does the product of the roots equal? 

44. How many of the examples in this Exercise can you 
work at sight ? 

45. Practice oral work with exponents as in Exercise 39 
(p. 104). 



CHAPTER XIV 

RATIO AND PROPORTION 
Ratio 

120. The Batio of two algebraic quantities is their exact 
relation of magnitude. It is the indicated quotient of the 
one quantity divided by the other, expressed either in 
the form of a fraction or by the symbol : placed between 
the two quantities. 

Thus, the ratio of a to 6 is expressed as ^, or as a : 6. 



121. TTtility of Batios. Ratios have the same uses as 
fractions, and also other uses obtained by selecting im- 
portant kinds of ratios, naming them (see Art. 123), and 
working out their properties once for all. Also proper- 
ties of equal ratios are worked out once for all and stated 
in such a form as facilitates their application to problems. 

122. The Terms of a Ratio are the two quantities com- 
pared. The antecedent is the first term. The conseqtient 
is the second term. 

The terms of a ratio must be expressed in terms of a common 
unit. Thus, to express the ratio of 3 qt. to 2 bu., either the quarts 
must be expressed as bushels or the bushels as quarts. 

If two quantities, as 5 in. and 2 bu., cannot be expressed in terms 
of the same unit, no ratio between them is possible. 

123. Xinds of Batio. An inverse ratio is a ratio obtained 
by interchanging antecedent and consequent. 

Thus, the direct ratio of a to 6 is a : 6 ; the inverse ratio of the 
same quantities is 6 : a. 

181 



182 DURELL'S ALGEBRA: BOOK TWO 

A oomponnd ratio is one formed by taking the product 
of the corresponding terms of two given ratios. 

Thus, acbdia the ratio compounded oi a:b and c : d, 

A duplicate ratio is formed by compounding a ratio with 
itself. 

Thus, the duplicate ratio of a:b is a^:b^. In like manner, the 
triplicate ratio of a : & is a' : &*. 

A oommeiUTirable ratio is a ratio that can be expressed in 
terms of two integers. 

oi 5 21 10 2it 

Thus, -2 equals -^--, or— -• Or-^isa commensurable ratio. 
5i ^ 2 4 21 5J 

An incommeiUTirable ratio is a ratio which cannot be 
expressed in terms of two integers. 

Thus, — - equals — =^. The fraction in the numerator can- 

5 5 

not be completed so that the numerator and denominator can be 

expressed as a ratio of integers in terms of the same unit; hence, 

V2 

— is an incommensurable ratio. 
5 

The properties of incommensurable ratios are obtained 

from those of commensurable ratios by an indirect method 

not discussed in this book. 

124. Fundamental Property of Ratios. If both antecedent 
and consequent of a ratio are multiplied or divided by the 
same quantity^ the value of the ratio is not changed. 

For, since ? = ^ 

b mb 

a : b has the same value as ma : mh. 

Proportion 

125. A Proportion is an expression of the equality of two 

or more equal ratios ; as ^ = ^, or a: 6 = c : d. 

b d 

The above proportion is read "a is to 6 as c is to d." 



RATIO AND PROPORTION 183 

126. Terms of a Proportion. The four quantities used 
in a proportion are called its terms or proportionals. 

The first and third terms are the antecedents. 

The second and fourth terms are the consequents. 

The first and last terms are the extremes. 

The second and third terms are the means. 

In a:b = c:d, d ia sl fourth proportional to a, J, and c. 

127. A Continued Proportion is one in which each conse- 
quent and the next antecedent are the same ; as 

a:b^b:c= €:d=d:e 

In the continued proportion a : 6 = 5 : (?, b is called a 
mean proportional between a and <? ; <? is called a third pro- 
portional to a and b. 

Two proportions of the form a: : y = a : J, and y:z = b:c 
may be combined in the form xiy :z= a:b:c. 

128. Eqnal Products made into a Proportion. If the 
product of two quantities is equal to the product of two other 
quantities^ either two may be made the means^ and the other 
two the extremes of a proportion. 

For, if ad = be 

Dividing by bd^ - = - 

a 

.«. a\b=ci d 

129. Fundamental Property of a Proportion. For algebraic 
purposes, the fundamental property of a proportion is that 

The product of the means is equal to the product of the 
extremes. 

For, if a:b=:c:d 

then T = J 

d 

Multiplying by bd^ ad = be 



184 DURELL'S ALGEBRA: BOOK TWO 

In like manner, if a : 6 = 5 : <? 

b'^= ac .•. 6 = -y/ac 

This property enables us to convert a proportion into an 
equation, and to solve a given proportion by solving the 
equation thus obtained. 

EXERCISE 74 

Simplify each of the following ratios: 

1. 5f:lf ^ a2^^y2 (2f/-^xy 

(x + 2i/y' 4y2_a? 

2. 2 ft. 4 in.: 8 in. ^ (.01:r)3;.2:r2 

3. 87J%:37^% 6. (2a8)*:(2a8X 

7. One rectangle is 5 yd. long and 7 J ft. wide ; an- 
other rectangle is 5 f t. x 2 ft. 6 in. Find the ratio of the 
areas of the two rectangles. 

8. Find the ratio of the volumes of two boxes whose 
dimensions are respectively 3' 6" x 2' x 1' 3" and 2' 9" x 
1' 8" X 1' 6". 

Find the ratio of 2; to y when 

9. 8a;— 8y = 2a;-|-5y 

10. bx—^y:2x-'by = ^ 

11. 6x2-hl5y2 = 19a:y 

12. 2^ + aby^ = (^a + b)xy 

13. Give the inverse of the ratio 7 : 3. Also otp : q. 

14. Determine which ratio is greater, 5 : 12 or 19 : 40. 

15. Show which of the following ratios are commensu- 
rable and which incommensurable : 

(1) 5J in. : 2f in. (2) Vl2 : V27 (3) VT2 : VI8 

16. If the death rate in New York City in the year 
1866 was 34 per thousand, and in 1912 was 14.1 per 



RATIO AND PROPORTION 185 

thousand, find how many lives in a population of 4,600,000 
were saved in 1912 by the reduced death rate. 

17. Explain the ratio called specific gravity. If 
1 cu. ft. of water weighs 62.5 lb., find the weight of a 
wagon load of dirt 6 ft. x 2 ft. x IJ ft. if the specific 
gravity of dirt is 2.1. 

18. Make up and solve an example similar to Ex. 17 
concerning a wagon load of brick. 

19. If a ratio is less than unity, does adding the same 
quantity to both terms of the ratio increase or diminish 

the value of the ratio ? By how much, if the ratio is j 

and is added to both terms ? 

20. Answer the same questions if the given ratio is 
greater than unity. 

21. If you do not already know it, ascertain the mean- 
ing of nutritive ratio. Mak6 up and work two examples 
concerning nutritive ratios. 

Find the mean proportional between 

22. Sa^ and ISa^y^ 23. .1 and .004 

24. (a2- 63)2 and (a + 6)* 

25. ^Zl6£+9 ^^^ ^H-a^-12x^ 

x+4: 4a:-12 

26. ^^-^^and 8V2 + 20 



2V6 + 5V3 3V6-4V3 

Find the fourth proportional to 

27. 3f , 3|, 5^ 28. (a + by, €? - J2, (a - 6)2 

Find the third proportional to 

29. a? -land (a: +1)2 31. 6-|andi-l 

o 

30. .3 and .09 32. a^ - 62 and (a - 6)2 



186 DURELL'S ALGEBRA: BOOK TWO 

33. If the first, second, and fourth terras of a propor- 
tion are 3, 12, and 8 respectively, find the third term. 

Solve for x and check : 

34. 2: 32;-l = 3a?:5-2aj 

35. 3a;-|-l : 5a;4-4=:a;-|-3 :4a;-|-6 

36. 3a; + 5 :a;-|-4 = l-a;: 2-a; 

37. a;-h5a:3a--a;=3a;4- 10 a:a;— 10 a 

38. What number must be added to each of the terms 
of 1^ to make the value of the fraction J ? 

39. A baseball nine has won 18 games out of 23. How- 
many straight games must it win, in order that its average 
of games won shall equal J ? 

40. Find the number which, when subtracted from each 
of the numbers 9, 18, 21, 48, will give results in proportion. 

41. Two numbers are to each other as 3 to 2. If 6 is 
added to the greater and subtracted from the less, the sum 
of the results thus obtained will be to the difference as 3 
to 1. Find the numbers. 

42. Two numbers are in the ratio c\ d. If a is added 
to the first and subtracted from the second, the results 
will be in the ratio 3 : 2. Find the numbers. 

43. Find two numbers whose difference is d and which 
are to each other as a : 6. 

44. A certain kind of brass is an alloy containing 2^ 
parts of copper and 1 part of zinc. How many ounces of 
copper are contained in 2 lb. 10 oz. of this brass ? 

45. The horse power generated by a stream falling over 
a dam is proportional to the height of the dam. If on a 
certain stream a dam 5 ft. high generates 200 H. P., how- 
much higher must the dam be made in order to get 500 
H.P.? 



RATIO AND PROPORTION 187 

46. That a door may look well, its height should be to 
its width approximately as 7 : 5. If a door is to be 7 ft. 
3 in. high, how wide should it be ? 

47. Convert each of the following into a proportion: 
(1) xy = ab; (2) (a: + y)2 = a^- ja. (3) 3a:y = 2a6. 

48. In a certain year the profits of a given business 
were f 15,600. Divide these profits into two parts which 
are as 8 to 5. Also into three parts which are as 6, 4, 3. 

49. Separate a : 6 : (? = 4 : 5 : 6 into two proportions. 

50. If air is regarded as a mixture of two gases, oxygen 
and nitrogen, whose volumes are as 21 : 79, find the num- 
ber of cubic feet of each of these gases in a room whose 
volume is 6000 cu. ft. 

51. If a given piece of land can be divided into 60 
building lots, each 30 ft. wide, how many lots 40 ft. wide 
would it make ? 

SuG. If X denotes the number of lots 40 ft. wide, 
30 X 60 = 40 X X 
or z : 60 = 30 : 40 

This problem can be solved either from the equation or 
from the proportion. 

A proportion of this kind is termed an inverse proportion. 

52. If 4800 shingles 4 in. wide are needed in building a 
house, how many 3-inch shingles would be needed ? 

53. If 12 yd. of cloth 36 in. wide are necessary for a 
dress, how many yards 27 in. wide would be needed ? 

54. If a trolley company reduces the hours of its em- 
ployees from 10 to 8 hours per day, by what per cent must 
it increase the number of its employees ? 

55. Make up and work an example similar to Ex. 30. 
To Ex. 39. 



188 DURELL'S ALGEBRA: BOOK TWO 

56. An example similar to Ex. 48. To. Ex. 58. 

57. Work again Exercise 48 (p. 125). 

130. Transformatioiis of a Proportion. Before converting 
a proportion into an equation, we may often simplify the 
proportion by the use of one or more of the following 
principles : 

If a : 6 = <? : d, then 

1. a: c=ssb:d (called alternation). 

2. b : a==d : c (inversion). 

8. a'\'b:b = C'\'d:d (addition). 

4. a — b:b = e'-d:d (subtraction). 

5. a-^bia — bssc + dic^d (addition and subtraction). 

For, from a : & = c : £?, we have ad = bc (Art 129); wheDce we ob- 
tain 1 and 2 (Art. 128). 

Also ? = ^, whence, ?+ 1 =^+ 1. 
b d b d 

Whence, 2±A = ^±A (or 3). 
b c 

Let the pupil prove 4 in like manner, and obtain 5 from 3 and 4. 
Ex.1. Solve f^"^o^g"^^-; = of "'i^f"'"?' 

4ar«+6a;2 6ar«-f2a;2 



By addition and subtraction. 



6Z-.2 4x4-2 



Dividing by 2£!, 2x+^^3£il 

^ ^ 2' 3a:-l 2a: + l 

Whence, 5a^»-8z-4 = 

X = 2, - |. Ans. 
The factor 2 x^ also gives the roots x = 0, 0. Ans, 
Check. For x = 2. 

2x»-f Sx" +3x^1 ^ 16 4- 12 + 6 - 1 ^ 33 

2x« + 3x2-3x + l 16 + 12-6 + 1 23 

3x8 + xg + 2x+1 ^ 24 + 4 + 4 + 1 ^33 

3aH»+x2-2x-l 24 + 4-4-1 23 

Let the pupil check the work for x = — | 

Also for X = 



RATIO AND PROPORTION 189 

131. Given some proportion (or equality of several equal 
ratios), aB a : i ssc : (2, a required proportion is often readily 

proved by taking f = 4 = ^ (hence a^hr^ (? = dr\ and 
a 

Bvb%tituting for a and c in the required proportion. 

Ex. Given a : 6 = (? : cZ, 
prove aih = VS^+T? : VJ^ + StP. 

Let ? = ^ = r .*,a = hryC — dr 

a 



Substitute in each ratio the values a = hr^c = dr 
6" h 



I. ? = ^ = r 






•. T = ° ^ « since each of the two fractions equals 

the same expression, viz. r. 

Hence, a given expression may be proved to be identical 
with another expression either (1) by reducing the first 
expression directly to the form of the second ; or (2) by 
reducing both expressions to a common third form. 

132. Composition of Several Equal Ratios. In a series of 
equal ratios^ the sum of all the antecedents is to the sum of all 
the consequents as any one antecedent is to its consequent. 

Given, ? = £ = £ = 2 

' h d f h 

Let each of the equal ratios equal r. 
Then 5 = r,£ = „|=nf = r 

. •. a = 6r, c = dr^ e =fr^ g^hr 



190 DURELL'S ALGEBRA: BOOK TWO 

Adding the last series of equalities, 

a + c + e + g=:(b + d+f+h)r 
. a'\-C'\-e+g _ a 

.*. a + c+e'\-g:b + d+f+h = a:b 



SZSRCI8S 75 

Solve and check : 

^ a^+4a; + 3 ^ 3a:g + lla;-10 
a4^6a;-8 3a:^+9a:+10 

2. 3a:8 + 6a?» + 5a; + ll:3a^+5a?-5a;-ll 

x^-p-q p-^-q + x 
4. Vx + 2:V2x + S = V^^^:^a^-x + S 



5. 



Vi + V6 2 ^ Va:4-1-Va;-1 „ 

— — = = — 6. — z=; = o — a? 

Vi-V6 1 VJ+l+Va;-l 
a;-V36:^ V3-1 



^ 5;>-V4a;-3jp3 ^ 3p-Va;4-jP^ 
3j[? + V4a; — 3j92 ^+Va;+j[?2 

If a : 6 = <? : (i, prove : 



10. a2 + 62:_^ = o2+d2._^ 

a + 6 <? + a 

11. ^/a^ + ab + l^:^/c^ + cd + d? 



= Va2- oJ + 62 . Vc2 - c(i + cP 



RATIO AND PROPORTION 191 

If a, J, c, d are in continued proportion, prove : 

12. a:b-hd=^c^: M-^<P 13. a: c = a^ -[' ab: b^ + bc 

14. a^ + b^ + c^ : ab + be -{- cd = ab + be -^ cd : b^ + (^ +cP 

15. If (p 4- 9) (r - «) + (j + r) (« - 1?) + ^J - r« = 0, 
prove that p : q = r : s. 

16. Given (2 a + 2 6 + 3 (? + 3 d)(2 a- 2 J- 3c+ 3d) 
= (2 a - 2 6 4- 3 c - 3 d)(2 a -h 2 6 - 3 c- 3 d), prove a : 6 
= c : d, 

17. Find two numbers, such that if 3 is added to each, 
they shall be in the ratio of 1 to 2. But if 1 is subtracted 
from each, they shall be in the ratio 1 to 3. 

18. Separate 52 into two parts, such that if each part 
is diminished by 6, the results shall be in the ratio of 
7 to 3. 

19. Find two numbers, such that their sum, difference, 
and the sum of their squares are in the ratio 5 : 3 : 17. 

20. Given x + y :x — y =^ a -{- b: a — b^ and a^ + y^ 
= a2j2(^2 ^ J2)^ find ^ and y. 

21. U-^ = -l- = ^—,findx + y + z. 

a— b — e e— a 

22. If four quantities are in proportion and the second 
is a mean proportional between the third and fourth, prove 
that the third will be a mean proportional between the first 
and second. 

23. If -^— = = — ^, prove that each of these 

+ c <? + a a-f-6 

fractions = 1. Also that a = b = e, 

24. Show by the use of a proportion whose terms are 
letters, that if the same number is added to each term, the 
results cannot be in proportion. Also give a numerical 
illustration of this property of proportions. 



192 DURELL'S ALGEBRA : BOOK TWO 

25. Before the adoption of modern sanitary methods by 
armies, the ratio of the number of men who died from 
wounds received in battle to the number who died of 
disease was as 1 : 2. If under improved sanitation the 
above ratio was changed to 2 : 1, find the number of lives 
saved in a war where 64,000 men lost their lives. 

26. If the rear and front wheels of a wagon are respec- 
tively p ft. and q ft. in circumference, how many rotations 
does the rear wheel make while the front wheel rotates r 
times ? 

27. If --2- =—2— =__!!_, show that ©-0^4-^ = 0. 

+ c c + a a — b 

28. lip + iq-^riq + r^p+qiq^ prove that g is a 
mean proportional between p and r. 

29. If a, by c, d are in continued proportion, prove that 

ac ^ a^ + e^ ^ a^+2b^+S(^ ^a^ 
bd b^ + d^ 62 + 2^2 + 3^2 62' 

30. Make up and work an example similar to Ex. 9. 
To Ex. 18. 

Meaning of the Ratio Forms -, % ^^^' 
a i) 

133. The Meaning of - has been made clear in Exs. 33- 

35, p. 15. The same result has also been made evident in 
the process of constructing certain graphs. 

In general. 

Zero divided by any number (except zero) gives zero. 

134. Infinity is a number (or quantity) greater than any 
assignable (or definitely expressible) number. 

The symbol for infinity is oo. 

The symbol = means "approaches." 



RATIO AND PROPORTION 
a 



193 



135.. The Meaning of g is made clear most readily by the 
use of graphs. 



Ex. Graph y= — 

X 



X 


y 


4 


\ 


2 


\ 


1 


1 


i 


2 


\ 


4 


t 


8 



etc. 



-2 


-i 


-1 


-1 


-I 


-2 


-i 


-4 


-J 


-8 



etc. 







































r 

































































































r 


























^ 
























S 






A 






^~ 


J 


^— . 


^ 
























5^ 


^ 
























\ 
























c\ 






















f 


opj 


























F' 













1 



Graphing y = - for the positive values of ar, we obtain the branch 

X 

ABC m the above diagram. 

From the diagram it is evident that as the value of x hecomes 
smaller, that of y ( or - ] increases, and as a: = 0, y f or - J = oo. 

If y = - is graphed for the negative values of or, in like manner as 



r = 0, y f or V=-oo. 



Hence, in general, 

A% the value of the denominator of any fraction approaches 
zero {the value of the numerator being finite^, the value of 
the fraction approaches infinity (or negative infinity'). 

This statement is often abbreviated into the form - = oo* 



194 DURELL'S ALGEBRA: BOOK TWO 

136. The Meaning of ^ is best shown algebraically. 

a? — 1 

Ex. Find the value of r- when a? = 1. 

x — 1 

If we substitute the given value of x directly in the given fraction, 

^eobtam __ = __ = - 

If, however, we simplify the fraction before substituting and then 
substitute 1 for x, we obtain 

X— 1 X— 1 

Hence, in this case - represents the value 2. 

Show, in a similar manner, that when a; = 1, the value 

of ^— T- is 3. Also that the value of — ^ is 4. 
a;--l a?— 1 

Hence, the value of - varies with circumstances, or as it 
is usually expressed, 

- 18 indeterminate. 

137. The Meaning of • is best shown algebraically. 

Ex. Find the value of -+■ -^ — -, when a: = 1. 

a; — 1 a;^ — 1 

By direct substitution, -^ — — r- == ;: -^ ;: = S" (s^® ^^- i^^)* 

X— Ix^ — lOO*' 

But if the given expression is simplified before the substitution 
for X is made, 

we have -il H- — ^- = ^ X ?^ll = X + 1 = 1 + 1 = 2. 

X-1 X2-1 X-1 1 

Hence, in this case ^ stands for the value 2. 



RATIO AND PROPORTION 196 

Show, in like manner, that ao may stand for 3, 4, or any number. 
We express the result arrived at as follows : 

S w indeterminate. 
The above results for the meaning of -, ^, -, ^ might 

be obtained by purely logical methods, but the thorough 
discussion of these methods lies beyond the scope of this 
book. 

EXERQSE 76 

2 2 2 

1. Graph y = - and thus find the value of - and -- • 

X ^ 

2. Graph y = — ^— • In this process, what special ratio 

X — 1 

is evaluated? 

3. Graph y = and make a similar inference. 

2 — X 

By an algebraic process, find the value of 

x^— 4 efi — 1 

4. - — - when x=s2. 6. when a = 1. 

x-2 a-1 

5. — '^-— when a? = 2. 7. -^~ — - when a = 2. 
x-2 a2-a-2 

Find the value of 

8. ^whenrr^O. ^^- a:"^ when a: = 0. 



2 
9. — when a? = 0. 



1 
11. 7^ when a? = 0. 



12. when a? = 0. When a: = 1. 

a:— 1 

13. 6 X - f 14. 3 X - ^ + 1 

Q 

15. 5x h x(x — 1) when a; = 1. Also when a? = ©• 

a? 



196 DURELL'S ALGEBRA: BOOK TWO 

16. axO + 6xO — cxO 

17. ■\/5 when a? = 0. 

18. ^"*" when x^a. 

q I »7-i» 

19. What is the limit of when n is increased 

indefinitely ? n 

20. Find the value which ^ — i— — — — — approaches 
when 6 = 0. When <?= 0. a = 0. 



21. Find the same values for • 

2a 

22. Who invented oo as the sign for infinity? 

23. Make up and work an example similar to Ex. 2. 
To Ex. 8. To Ex. 10. 

24. Make up and work an example similar to Ex. 14. 
To Ex. 18. 

25. How many of the examples in this Exercise can 
you work at sight? 



CHAPTER XV 

THE PROGRESSIONS 

138. A Series is a succession of t^rms formed according 

to some law, as 

1, 4, 9, 16, 25, ... 

1 -x + x2- x» + x*-, ... 

2, 4, 8, 16, 32, ... 

139. Utility in an Algebraic Treatment of Series. 

Ex. If a car going down an inclined plane travels in 
successive seconds, 2 ft., 6 ft., 10 ft., 14 ft., etc., how far 
will it go in 30 seconds ? 

The direct method of solution would be to set down the 30 num- 
bers involved and add them. But by investigating the laws of the 
series involved and expressing these as formulas, it will be found 
(see Art. 142) that this long addition can be converted into two 
stiOTt multiplications and much labor can thus be saved. 

The algebraic study of the laws of series will enable us to save 
labor in various ways, and to obtain other important results. 

Arithmetical Progression 

140. An Arithmetical Progression is a series each term of 
which is formed by adding a constant quantity, called the 
difference^ to the preceding term. 

Thus, 1, 4, 7, 10, 13, ••• is an arithmetical progression (denoted 
by A. P.) in which the difference is 3. 

Given an arithmetical progression, to determine the 
difference : from any term Bubtract the preceding term. 

Thus, in the A. P., f , - J, - 3, 

the difference =-}-!=- J. 

197 



198 DURELL'S ALGEBRA: BOOK TWO 

141. Quantities and Symbolfl. In an A. P. we are con- 
cerned with five quantities : 

1. The first term^ denoted by a. 

2. The common difference^ denoted by d. 

3. The last term^ denoted by I. 

4. The numier ofterms^ denoted by n. 

5. The sum of the terms^ denoted by s. 

142. Two Fundamental Formulas. Since in an A. P. 
each term is formed by adding the common difference, d, 
to the preceding term, the general form of an A. P. is 

a, a + dj a+2dj a+3(i, ••• 

Hence, the coefficient of (2 in each term is one less than 
the number of the term. 

Thus, the 7th term is a + 6 (2, 

12th term is a + 11 d^ 

nth term is a + (n — l^d. 
Hence, Z = a + (w — l)df (1) 

Also, 
« = a-f (a + (r) + (a + 2rf)+ ... +(Z-i)+r . (2) 

Writing the terms of this series in reverse order, 

« = Z + G-rf) + G-2(f)+ ... +(a + (r) + « • • (3) 
Adding (2) and (8), 

2«=(a4-0 + (a + + (« + 0+ - +(a+0 + («+0 
= n(a + 1) 

•••« = |(« + (4) 

If we substitute for I in (4) from (1), 
« = ^[2aH-(n-l)d] (6) 



ARITHMETICAL PROGRESSION 199 

Hence, combining results, we have the two fundamental 
formulas for I and «, 

I. l = a+(n-iy 

II. . = |(a + 

• -|[2a + (»-l)(r| 

Thus formula I substitutes a multiplication for succes- 
sive additions of the common difference ; and formula II 
substitutes a multiplication for the addition of the succes- 
sive terms. 

Ex. 1. Find the 12th term and the sum of 12 terms of 
the A. P., 6, 3, 1, - 1, - 8, ... 

In this series a = 5, rf = — 2, n = 12 

From I, Z = 5 +(12 - 1)(- 2)= 5 - 22 =- 17 

From II, »=:¥(5-17)=-72. Sum. 

Ex. Find the sum of n terms of the A. P., 
a + 6 a — J a — 86 
2 ' 2 ' 2 ' 

Here a = ^-i-, d= — 6, n = n 

Substituting in the fundamental formula, a = ^ [2 a + (n — l)d] 

» = |[« + ft +(»-!)(-»)] 
= ?[o+(2-n)6]. Sum. 

EXSRCISB 77 

1. Give the value of d in Exs. 2-15. 

2. Find the 8th term in the series 3, 7, 11, ••• 

3. Find the 9th term and the sum of 9 terms in 7, 3, 
— 1, ••• 

4. Find the 20th and 28th terms in 5, J^, J^, ... 



200 DURELL'S ALGEBRA: BOOK TWO 

5. Find the 16th and 25th terms in - 18J, - 9, — 4 J, . - 

6. Find the 7th and 10th tenns and the sum of 10 terms 
in the series f, |, -^j, ••• 

7. Find the 18th term and the sum of 18 terms in the 
series 8, 2.4, 1.8, ••• 

8. Find the 30th term of the series 1, 4, 7, 10, ... by 
successive additions of the common difference. Now 
find the 80th term by use of the formula. About how 
much shorter is the second process than the first ? 

Find the sum of the series : 

9. 3, 8, 13, ... to 8 terms. 

10. 8, — 3, — 9, ... to 9 terms. 

11. 2 J, 3|, 5, ... to 14 terms. 

12. — ^, J, |, ... to 88 terms. 

13. — |, — |, — \l^ ... to 56 terms. 

14. 5 V2 - 2V3, 4V2 - 3V3, ... to 11 terms. 

15. 3 a , 2 a, a + -, ... to 12 terms. 

a a 

16. 1, 4, 7, 10, ... to n terms. 

17. 6, 3, 0, — 3, — 6, ... to ^ terms. 

18. 5, 3, 1, — 1, ... to w terms. 

19. 2 a; — y, a; + y, 3 y, ... to r terms. 

20. Find the sum of the first 30 odd numbers by writing 
them down and adding them. Now find their sum by use 
of the formula. Compare the amount of work in the two 
processes. 

21. How many strokes does a clock make in striking 
each hour of the day ? 



ARITHMETICAL PROGRESSION 201 

22. If a man saves f 100 in his 20th year, $150 the next 
year, and 1200 the next, and so on through his 60th year, 
how much will he save in all ? 

23. If a body falls 16.1 ft. in one second, 8 times as far 
in the next second, 5 times as far in the third second, and 
so on, how far will it fall in 6 seconds ? In 15 seconds ? 

24. If the velocity of a falling body at the end of 1 second 
is 32.2 ft. per second, at the end of the next second is 64.4 
ft., at the end of the third second is 96.6 ft., what is it at 
the end of 10 seconds ? 

25. A body rolling down an inclined plane goes 6 ft. in 
the first second, three times as far in the next second, 5 
times as far in the third second, and so on. How far will 
it go in 10 seconds ? 

26. State in general language the first of the formulas 
obtained in Art. 142. 

Sua. " The last term equals the first term increased by," etc. 

27. State the second formula of Art. 142 in general 
language. 

28. State the third formula of Art. 142. 

29. Make up and work an example similar to Ex. 9. 
To Ex. 16. Ex. 22. Ex. 25. 

30. How many examples in Exercise 35 (p. 96) can 
you now work at sight ? 

143. Oiven Any Three of the Five Quantities, a, d, I, n, s, to 
find the Other Two. 

The method in general is as follows : 

If the three known quantities are found in one of the formulas 
of Art. 142, BvbBtitute the three given values in the formula 
and solve the resulting equation. 



202 DURELL'S ALGEBRA: BOOK TWO 

The revnaining unknottm quantity may then be found by 
use of one of the other formuiai of Art. 142. 

If the three given quantitieB do not occur in one of the 
formulae of Art. 142, mbititute in two of these formulas and 
solve by elimination. 

Ex. 1. Given ? = 13, « = 49, n = 7, find a and d. 

Since the letters /, s, n, and a all occur in the formula « = - (a + /), 

substitute the values of /, «, and n in this formula. 
Hence, 49 = }(a + 13) 

98 = 7a + 91 .-.0=1. Ans, 

From l = a-\-(n- l)d 

13 = 1 + (7 - l)d .-.(/ = 2. Ans. 

Ex. 2. Given d = 2, Z = 21, « = 121, find a, n 

Since d, I, and s do not occur in one formula, we 
Substitute for </, /, s in Formulas I and II, 

21 = a+(n-l)2 (1) 

^g^^ n(a + 21) ^2^ 

.-. a + 2 Fi = 23 (3) 

an + 21 n = 242 (4) 

Substitute for a in (4) from (3), 
n(23-2fi)+21n = 242 

Whence, ^^^^'\Ans. 

Hence, from (3), a = 1. J 

EXERCISE 78 
Find the first term and the sum of the series when 
1. (i = 8, Z = 40, n=13 2. (i = |, Z=18J, n=33 
Find the first term and the common difference when 
3. «=275, 7 = 45, w=ll 4. « = 4, ? = -10, n=:8 

5. « = -246i, ? = -34|, w = 17 

6. « = 9, Z = 2f , n = 9 

7. «=-i^,Z=- 4,^ = 47 



ARITHMETICAL PROGRESSION 203 

Find n and d when 

8. a= - 5, Z = 15, « = 105 

9. a=19, Z = -21, «=-21 

10. a = J;j,/ = -f,« = -2J 

11. a=-8J, Z = 9i, « = 48 
Find a and n when 

12. «=10, (i = 3, Z = 8 13. « = 10, i = -3, Z=-4 

14. Z = - 8, d = - 3, « = - 8 

15. ? = -|,d = -^,ir=-ijj^ 

How many consecutive terms must be taken in the series 

16. 1, 1^, 2, ••• to make the sum 45? 

17. |, ^, J, ... to make the sum — 1? 

18. f , |, 1, ••• to make the sum 4.5 ? 

19. A body rolling down an inclined plane goes 6 ft. 
the first second, 18 ft. the next second, 30 ft. the third 
second, and so on. In how many seconds will it have 
traveled 486 ft.? 

20. Make up and work an example similar to Ex. 1. 
To Ex. 12. Ex. 19. 

21. How many examples in Exercise 36 (p. 99) can 
you now work at sight? 

144. Arithmetical Means. 

Ex. Insert 9 arithmetical means between 1 and 5. 
We have given a = 1, / = 5, n = 11. Hence, we find rf = J. 
The required means are therefore 11, 1|, 2|, ••• Ans. 
In case only a single arithmetical mean is to be inserted 
between two quantities, a and 6, this one mean is found 

most readily by use of the formula ^ "^ . For if x de- 
notes the required mean, the A. P. is a, a;, b. 



204 DURELL'S ALGEBRA : BOOK TWO 

Hence, x— a = h — x 



EXERCISE 79 

Insert 

1. Four arithmetical means between 7 and — 3. 

2. Seven arithmetical means between 4 and 6. 

3« Thirteen arithmetical means between | and — |. 

4. Fifteen arithmetical means between — 4J and 9. 

5. The arithmetical mean between 2^ and — 6J. 

6. The arithmetical mean between x + 1 and a; — 1. 

7. Find the A.M. between ? and -. and 



b a x+y x—y 

8. If the height of Bunker Hill Monument is 221 ft, 
of the Washington Monument 655 ft., and the length of 
the Olympic 882 ft., by how much does the middle one of 
these numbers differ from the arithmetical mean between 
the other two ? 

9. Make up and work a similar example concerning 
12 J mi., 49 mi., and 100 mi., which are the lengths of the 
Simplon Tunnel, the Panama Canal, and the Suez Canal 
respectively. 

10. Rome was founded 758 B. C. and fell 476 A. D. 
How far is the latter number from being an arithmetical 
mean between the former and the number of the year in 
which Columbus discovered America? 

11. Ether boils at a temperature of 96° F., alcohol at 
167% and water at 212°. How far is 167° from being an 
arithmetical mean bet\yeen the other two temperatures? 



ARITHMETICAL PROGRESSION 205 

12. Show that if twice one number equals the sum of 
two other numbers, the three numbers may be arranged 
as an A. P. 

13. State the formula a? = ^^ (of Art. 144) in gen- 
eral language. 

14. Work at sight such examples on pp. 110-111 as the 
teacher may indicate. 

145. MisceUaneouB Examples. 

Ex. 1. The 7th term of an A. P. Ls 5, and the 14th 
term is — 9. Find the first term. 

By the use of Formula I (Art 142), 

the 7th term is a + 6 cf, and the 14th term is a + 13 e?. 

.-. a + 6rf = 5 (1) 

a + 13rf=:-.9 (2) 

Subtracting (1) from (2), 7 rf = - 14 

rf = -2 
Substitute for d in (1), a - 12 = 5 

a = 17. Ans, 

Ex. 2. The sum of five numbers in A. P. is 15, and the 
sum of the 1st and 4th numbers is 9. Find the numbers. 

Denote the numbers by 

a;-2y, x-y, ar, x -V y, x+2y 

Add, 5a: = 15 (1) 

Also, (x-2y) + (x+y)=9 

.•.2a:-y = 9 (2) 

From (1) a: = 3 hence, from (2), y = — 3 

Hence, the numbers are 9, 6, 3, 0, — 3. Ans. 

Similarly, in dealing with four unknown quantities in A. P., we 
denote them by 

ar-3y, x - y^ ar + y, ar + 3y 



206 DURELL'S ALGEBRA: BOOK TWO 

EXERCISE 80 

Find the first two terms of the series wherein 

1. The 4th term is 11 and the 10th is 23. 

2. The 6th term is - 3 and the 12th is - 12. 

3. The 7th term is — ^ and the 16th is 2^. 

4. The 5th term is c — 3 6 and the 11th is 3 6 — 5 c. 

5. Find the sum of the first n odd numbers. State 
the result obtained as a rule in general language. 

6. Set down the first 20 odd numbers and find their 
sum by addition. Now find their sum by the formula re- 
sult obtained in the preceding example. Compare the 
amount of work in the two processes. 

7. Find the sum of the first n numbers divisible by 7. 

8. Make up and work an example similar to Ex. 6, 
but showing the utility of the result obtained in Ex. 7. 

9. Which term in the series 1^, IJ, IJ, ••• is 18? 

10. The first term of an arithmetical progression is 8 ; 
the 3d term is to the 7th as the 8th is to the 10th. 
Find the series. 

11. Find four numbers in A. P., such that the sum of 
the first two is 1, and the sum of the last two is — 19. 

12. Find four numbers in A. P. whose sum is 16 and 
whose product is 105. 

13. A man travels 2-^ mi. the first day, 2f the second, 3 
the third, and so on ; at the end of his journey he finds 
that if he had traveled 6 J mi. every day he would have 
required the same time. How many days was he 
walking ? 



ARITHMETICAL PROGRESSION 207 

14. The sum of 10 numbers in an A. P. is 145, and the 
sum of the fourth and ninth terms is 5 times the third 
term. Find the series. 

15. If the 11th term is 7 and the 21st term is 8§, find 
the 41st term of the same A. P. 

16. In an A. P. of 21 terms the sum of the last three 
terms is 23, and the sum of the middle three is 5. Find 
the series. 

17. Required five numbers in A. P., such that the sum 
of the first, third, and fourth terms shall be 8, and the 
product of the second and fifth shall be — 54. 

18. The sum of five numbers in A. P. is 40, and the sum 
of their squares is 410. Find the numbers. 

19. The 14th term of an A. P. is 38 ; the 90th term is 
152 ; and the last term is 218. Find the number of terms. 

20. How many numbers of two figures are there divisi- 
ble by 3? By 7? How many numbers of three figures 
are divisible by 6? By 9? 

21. How many numbers of four figures are there divisible 
by 11 ? Find the sum of all the numbers of three figures 
divisible by 7. 

22. If a car starts at the top of a hill and runs down an 
inclined track 2 ft. the first second, 6 ft. the next second, 
10 ft. the next, etc., and reaches the bottom in 12 seconds, 
how long is the track ? 

23. Sulphur fuses at a temperature of 239° F., tin at 442^ 
and lead at 617®. By how much does 442° differ from the 
arithmetical mean between the other two temperatures? 

24. Copper fuses at a temperature of 2200° F., gold at 
2518°, and iron at 2800°. Treat these temperatures in a 
way similar to that used in the preceding example. 



208 DURELL'S ALGEBRA : BOOK TWO 

25. The heights of Mt. Washington, Pike's Peak, Mt. 
McKinley, and Mt. Everest are respectively 6290 ft., 
14,147 ft., 20,464 ft., and 29,002 ft. Find the difference 
between each of these numbers and the corresponding term 
in an A. P. whose first term is 6290 ft. and common differ- 
ence 7500 ft. 

26. Using the data in Ex. 25, find the difference be- 
tween each number and the corresponding term in an A. P. 
whose second term is 14,200 ft. and common difference 
7600 ft. 

27. If a body falls 16^^ ft. in the first second; three 
times this distance in the next second ; five times in the 
third, and so on, how far will it fall in the 30th second? 
How far will it have fallen during the 30 seconds? In 
how many seconds will it have fallen 6433 J ft. ? 

28. If a, 6, c,rf, are in A. P., prove: (1^ t\rdta+d=b + c; 
(2) that ak, bk, ck^ dk^ are also in A. P. ; and (3) that a + A:, 
6 + i, t?-|-i, d'\-k are in A. P. State this problem with- 
out the use of the symbols, a, (, c^ d^k\ that is, in general 
language. 

29. Make up and work an example similar to Ex 1. 
To Ex. 7. Ex. 22. 

30. Practice oral work with fractions as in Exercise 30 
(p. 79). 

Geometrical Progression- 

146. A Oeometrical Progression is a series each term of 
which is formed by multiplying the preceding term by a 
constant quantity called the ratio. 

Thus, 1, 3, 9, 27, 81, ••• is a geometrical progression (or G.F.) in 
which the ratio is 3. 



GEOMETRICAL PROGRESSION 209 

Given a geometrical progression, to determine the ratio : 

divide any term hy the preceding term. 

Thu8, iD the G. P., - 3, f, - }, ••• 

the ratio = — j > or — - 

— o 2 

147. QuantitieB and Symbols. The symbols a, 7, n, « have 
the same meaning as in A. P. Besides these, r is used to 
denote the ratio. 

148. Two Fundamental Formnlas. Since in a G. P. each 
term is formed by multiplying the preceding term by the 
common ratio, r, the general form of a G. P. is 

a, ar, ai^^ ar^, ar^^ ••• 

Hence, the exponent of r in each term is one less than 
the number of the term. 

Thus, the 10th term is ar^ 

15th term is ar^^ 
nth term, or Z = af^~^ . . (1) 

In deriving a formula for the sum, we know, also, 

B = a-{-ar'\-ar^-\ \-ar''-'^ .... (2) 

Multiply (2) by r, 

r8=:ar + a7^'\-ar^+"'-\-ar'^~^ + ar^ . . (3) 

Subtract (2) from (3), 

r« — « = ar^ — a 

ar"* — a . ^ 



r— 1 
Multiply (1) by r, rl = ar^ 
Substitute rl for ar" in (4), 



(4) 



«=7ff (') 



210 DURBLL'S ALGEBRA: BOOK TWO 

Hence, collecting the results obtained in (1), (4), (5), 
we have the two fundamental formulas for I and s: 
I. l=.ar^-^ 
ar*^ a 



11. « = 



« = 



r-1 



r-1 

Ex. 1. Find the 8th term and sum of 8 terms of the 

G". P., 1, 3, 9, 27, ••• 

In this case, a = l, r=3, n = 8 

From I, / = 1 X 37 = 2187 

From II, « = ?Ji|l?Lzil = 3280. Sum. • 

Ex. 2. Find the 10th term and the sum of 10 terms of 

theG. P., 4, -2, 1, -i, ... 

Here a = 4, r = — J, n = 10 

Hence, Z= 4(- *)• = -^ = - tJt 

s^i-iX-Th)-^^^^. Sum. 
- J - 1 128 

EXERCISE 81 
Give the value of r in Exs. 2-15. 

1. Find the 6th term in the series 2, 6, 18, ... 

2. Find the 7th term in 3, 6, 12, ... 

3. Find the 6th and the sum of 6 terms in 45, — 15, 5, ••• 

4. Find the 6th and the sum of 5 terms in 81, — 54, ••• 

5. Find the 7th and the sum of 7 terms in IJ^, —|, ..• 

6. Find the 9th term in the series 2, 2V2, 4, ... 

7. 15th term of ti t^^ li^ -• 

8. nth term of », ^, ^-^ ■^, ... 

9 f f 



GEOMETRICAL PROGRESSION 211 

Find the sum of the series 
9. 3, — 6, 12, ... to 6 terms. 

10. 27, - 18, 12, ... to 7 terms. 

11. - f, 1 J, - 2, ... to 9 terms. 
^* h "" h aV ••' to 8 terms. 

13. -— , 1, V8, ... to 8 terms. 

V3 

14. V2 - 1, 1, V2 + 1, ... to 6 terms. 

15. 1, 2, 4, 8, ... to n terms. 

16. The following is a series of specific gravities : cork, 
.25; oak wood, .75; aluminum, 2.5; iron, 7.5; platinum, 
21.5. By how much does each term of this series differ 
from the corresponding term in a G. P. whose first term 
is .26 and whose ratio is 3 ? 

(What is meant by specific gravity?) 

17. If the average age of parents is taken as 30 years, 
find the total number of a person's ancestors in a period 
of 600 years. 

18. The population of the United States in the year 
1900 was 76,300,000. If this should increase 50% every 
25 years, what would the population be in the year 2000 ? 

19. If a man saves $ 300 each year for 10 years, what 
is the amount of his savings in 5 years at compound inter- 
est at 5 per cent ? In 10 years ? 

20. A ship was built at a cost of f 70,000. Her owners 
at the end of each year deducted 10% from her value as 
estimated at the beginning of the year. What is her esti- 
mated value at the end of 10 years? 

21. A grain of wheat, when planted, produced a stalk 
on which were 30 other grains. The next year each of 



212 DURELL'S ALGEBRA: BOOK TWO 

the grains was planted and produced similar stalks. If 
this process were continued, at the end of 10 years bow 
many bushels would be produced in the last crop if 
1 quart contains 2000 grains ? 

22. Make up and work a similar example concerning 
the amount of corn produced from one grain, using prob- 
able numbers. 

23. If 32 nails are used in shoeing a horse, make up 
and work an example concerning a man who paid a black- 
smith for shoeing a horse at the rate of ^^ for the first 
nail driven, J^ for the second nail. Iff for the third, etc. 

State in general language : 

24. The first formula obtained in Art. 148. 

25. The second formula. The third formula. 

26. Make up and work an example similar to Ex. 4. 
To Ex. 15. Ex. 20. 

27. Practice the oral solution of simple equations as in 
Exercise 19 (p. 48). 

149. Given Three of the Five Quantities, a, I, n, s, r, to deter- 
mine the Other Two. 

Use the same general method a% that given in Art. 143 
(p. 201), /or A. P. 

Ex. 1. Given a = - 2, n = 7, Z = - 128 ; find r, «. 

From I, -128=-2r« 

Hence, r« = 64 r = ± 2 

From n, if r = + 2, « = '^{- Vipj{- '^) = _ 256 + 2 = - 254 

Kr = -2, ,^(-2)(-128)-(^2)^256 + 2^_3g 

Hence, tbere are two sets of answers ; viz. 

r = + 2, 5 = - 254.1 . 
r = -2, . = -86. ^'"- 



GEOMETRICAL PROGRESSION 213 



Ex. 2. Given a = |, r = — J, « = ^^ ; find Z, n. 
The most convenient method of s 
Substituting in the formula, s = - 



The most convenient method of solution is to find, first /, then n. 

rl — a 

^ = riil^^ whence, / = --!-. Ans. 
162 - J - 1 324 

Using I = ar--\ - ,h = i( - i)""' 

Whence, - ib x ♦ = (— \Y'\ and n^^Q, Ans. 

Let the pupil check these results. 

EXERCISE 81 
Find the first term and the sum when 
1. 71=6, r = 3, Z = 486 2. n = 8, r = - 2, Z = - 640 

3. « = 8,r = -f Z = -f|$ 

4. w = 7, r= JV6, Z = 8 
Find the ratio when 

5. a = -2, Z = 2048, w = 6 

6. a=s9, Z = J^, « = 23^ 

7. a = 2|f,Z = -||,7i = 6 

8. a = -16i,Z = -3^,* = -123^ 
Find the number of terms when 

9. a=2, r=2, «=:62 11. a==|, Z=^^, r=J 

10. a=4, r=-^, ««2| 12. a = 3, Z= -96, «= -68 

13. a = 18, r=-f, « = 12| 

How many consecutive terms must be taken from the 

series 

14. 2, 4, 8, ••• to make the sum 62 ? 

15. |, J, J, ••• to make the sum |^ ? 

16. 5 J, - 8, 12, ... to make - 22 J ? 

17. Make up and work an example similar to Ex. 1. 
To Ex. 13. 



214 DURELL'S ALGEBIU : BOOK TWO 

18. How many examples in Exercise 40 (p. 108) can 
you now work at sight ? 

150. Geametrie Heaiis. 

Ex. Insert 5 geometrical means between 3 and ^^. 

We have given a = 3, / = ^^ n = 7, to find r. 

Solving by Art 149, r = i 

Hence, the reqiiired geometrical means are 
1. h h *. A- ^w- 

In case only one geometrical mean is to be inserted be- 
tween two quantities, a and 5, this one mean is found 
most readily by using the formula Voi. For if x repre- 
sents the geometrical mean between a and &, the series 
will be 



a, X, b 



X h 



Hence, ? = - .*. a? = ai, aj = Vai 

a X 

Insert 

1. Three geometrical means between 8 and \. 

2. Three geometrical means between \ and ^. 

3. Six geometrical means between ^ and — ^. 

4. Four geometrical means between — J and 3584. 

5. Six geometrical means between 56 and — ^* 
Find the geometrical mean between 

6. 4J and | 10. .7 and .343 

7. 3|and6f ^ .5 and .125 

8. 28 a^a? and 63 aar/ ^ ^ , ^^^ 

r- 8/-5 12. .005 and .125 

9. ^ and 2^ ^ ^ 
c^^/y xV? 13. 5V2 + 1 and 5 V2 - 1 

/^ Q 

14. Insert 6 geometrical means between -— and —^^ 
^ 16 Va 



GEOMETRICAL PROGRESSION 216 

8 ti^ 

15. Insert 7 geometrical means between -- and —- • 

16. Is a mean proportional between two numbers the 
same as the geometric mean between the numbers ? 

17. State the formula x = Va6 (of Art. 150) in general 
language. 

18. Make up and work an example similar to Ex. 2. 
To Ex. 12. 

19. How many examples in Exercise 31 (p. 84} can 
you now work at sight ? 

161. Limit of the Sum of an Infinite Decreasing Oeometrical 
Progression. 

If a line AB 

A ? 



V y V -v^ 

is of unit length, and one half of it (AC) is taken, and 
then one half of the remainder (CD)^ and one half of the 
remainder, and so on, the sum of the parts taken will be 

This is an infinite decreasing G. P. in which r = ^. 
The sum of all these parts must be less than 1, but must 
approach closer and closer to 1 as a limit, the greater the 
number of parts taken. This illustrates the meaning of 
the limit of an infinite decreasing G. P. 

In general, to find the limit of an infinite decreasing 
G. P. we have the formula 

»=-^ Ill 

l — r 



216 DURELL'S ALGEBRA: BOOK TWO 

For formula II of Art. 148 may be written, 8 = ^""^ » 

1 — r 

Then, as the number of terms increases, 
I approaches indefinitely to 
.•. rl approaches indefinitely to 
.". a — rl approaches indefinitely to a — = a 

' ^ "" ^ approaches indefinitely to ^ 



1-r ^^ ' 1-r 

1 — r 
Ex. Find the sum of 9, — 3, 1, — J, ... to infinity. 
Here a = 9, r = ~ J 

9 9 27 e 



l-(-i) 1 + J 4 
152. Eepeating Decimals. By the use of Art. 151, the 
value of repeating decimals may be determined. 
Ex. 1. Find the value of .373737 ... 

.373737 ... = .37 + .0037 + .000037 + ... 
Here a = .37, r = .01 

- .- .37 _.37^3_7. ^^^ 



1 - .01 .99 

Ex. 2. Find the value of 3.1186186 ... 
Setting aside 3.1, and treating the remaining terms as a G. P., 
a = .0186, r = .001 
.0186 ^ .0186 ^ 186 ^ 62 
'*' * 1 - .001 .999 9990 3330 
.-. 3.1186186 ... = 33V + Tfiirr = 3*. Ans. 

EXERCISE 84 

Find the sum to infinity of the series 

1. 2,§,|,... 4. l|,li,|, ... 

2. 2, -1,},... 5. 4, -2^,1^,... 

3. - 9, 6, - 4, ... 6. ^, ^, gJy, ... 



GEOMETRICAL PROGRESSION 217 

7. 2H,-H,li,... 9. _J_, 1, _1_, ... 

& 6,3V2,3, ... V2-1 V2 + 1 

10. iV2 + JV3 + JV2, ... 

12. Give the ratio in the G. P. in each of the following: 
(1).333... (2) .272727... (3) .356356 (4) .79127912 
(5) .5333 ... (6) In Exs. 13-21. 

Find the values of 

13. .63 14. .417 15. 5.846 

16. 3.52424 ... 19. 1.02727 ... 

17. 1.4037037 ... 20. 1.027027 ... 

18. 3.215454 ... 21. .30102102 ... 

22. Find the first term in an infinite decreasing geomet- 
rical progression whose sum is f and whose ratio is — J. 

23. If the velocity of a sled at the foot of a hill is 60 
ft. per second and this velocity should be diminished by 
one third each second as the sled moves out on the hori- 
zontal, how far would the sled move before coming to 
rest? 

24. Make up and solve a similar example concerning a 
car which first ran down an inclined track and then out 
on a horizontal track. 

25. If a ball, dropped from a height of 80 ft., rebounded 
40 ft., and on striking the ground again rebounded 20 ft., 
and so on, how far would it travel before coming to rest ? 

26. Make up and work an example similar to Ex. 25. 

27. State the formula 8 == ^ in general language. 

1 — r 



218 DURELL'S ALGEBRA: BOOK TWO 

28. Make up and work an example simple to Ex. 8. 
To Ex. 16. 

29. Practice oral work with radicals and imaginaries as 
in Exercise 48 (p. 126). 

153. Hi8odlaii60iiB ProUems. 

Ex. Find four numbers in G. P. such that the sum of 
the first and fourth is 56, and of the second and third 
is 24. 

Denote the required numbers by a, ar, ar^, ar*. 

Then a + ar» = 56 

Or a(l + r») = 66 (1) 

ar(l + r) = 24 (2) 

Divide (1) by (2). L^L±^=1. 
r o 

Hence 3-3r + 3r* = 7r 

3ra-10r = -3 

r = 3, or J 

And a = 2, or 54 



54.1 A 



Or 54, 18, 6, 



£ZEfiaS£ 85 

Find the first two terms of the series in which 

1. The 3d term is 2, and the 5th is 18. 

2. The 4th term is | and the 9th is 48. 

3. The 6th term is 6 and the 11th is ^. 

Determine the nature, whether arithmetic or geometric, 
of each of the following series : 

5. iJ.Ji- 8. 3^,41,7^,... 



GEOMETRICAL PROGRESSION 219 

10. Divide 65 into 8 parts in geometrical progression, 
such that the sum of the first and third is 3^ times the 
second part. 

11. There are 3 numbers in G. P. whose sum is 49. 
The sum of the first and second of these numbers is to the 
sum of the first and third as 3 to 5. Find the numbers. 

12. The sum of three numbers in G. P. is 21, and the 
sum of their reciprocals is -^. . Find the numbers. 

13. Find four numbers in G. P., such that the sum of 
the first and third is 10, and of the second and fourth 30. 

14. Three numbers whose sum is 24 are in A. P., but if 
3, 4, and 7 are added to them respectively, these sums will 
be in G. P. Find the numbers. 

15. The sum of $225 was divided among four persons 
in such a manner that the shares were in G. P., and the 
difference between the greatest and least was to the differ- 
ence between the means as 7 is to 2. Find each share. 

16. Find the sum of —n , V2, — = , ... ad in- 

finUum. ^2-1 V2 + 1 

17. There are four numbers, the first three of which 
are in G. P., and the last three are in A. P.; the sum of 
the first and last is 14, and of the means is 12. Find the 
numbers. 

18. If the series f, ^, ••• is arithmetical, find the 102d 
term ; if geometrical, find the sum to infinity. 

19. Insert between 2 and 9 two numbers, such that the 
first three of the four may be in A. P., and the last three 
in G. P. 

20. Prove that the series V2-1, 3V2-4, 2(5V2-7), 
... is geometrical; that its ratio is 2 — V2; and that its 
sum to infinity is unity. 



220 DURELL'S ALGEBRA: BOOK TWO 

21. If the areas of Rhode Island, New Jersey, New 
York, and Texas are respectively 1250, 7815, 49,170, and 
265,780 sq. mi., how far are these numbers from forming 
a G. P. of which the second term is 7815 and the ratio 6 ? 

22. On p. 173 a table gives the amount of f 1 in differ- 
ent periods of time at simple interest and also at compound 
•interest. Which of these three series of numbers forms an 
A. P. and which a G. P. ? 

23. If an air pump at each stroke removes ^ of the air 
in a receiver, what fraction of the air is left at the end 
of 10 strokes? . 

24. If the amount of air in a receiver is indicated by 
the height of a mercury column in a tube attached to the 
receiver, and this height is 30 in. at the start, what will 
the height of the mercury be at the end of the 10 strokes? 

25. There were 2,500,000,000,000 tons of coal in the 
United States in the year 1910, and 3,000,000,000 tons 
were consumed between the year 1900 and 1910. If the 
consumption of coal should double every decade, tell to 
the nearest decade how long the coal in the United States 
would last. 

26. Work again Exercise 41 (p. 109), or similar examples 
suggested by the teacher or pupils. 



CHAPTER XVI 

BINOMIAL THEOREM 

164. The Binomial Formula. The results obtained by 
iuspection in Art. 39 (p. 82) may be combined in a 
formula as follows : 

(x + ay = 2:~ + rur-^a + ^^^/^^^ oT^a^ 

We shall now; give a proof of this formula for all posi- 
tive integral values of n. 

155. Proof of the Binomial Formula for Positive Integral 
Values of n. This proof may be conveniently divided into 
three parts. 

I. By actual multiplication it is found that, for any definite value 
of n, as n = 4, 

(x + ay = x^ ^ i 3*a -^ QxW -^ ixa^ + a^ 
That is, the binomial formula is true when n = 4. 

II. We shall now prove the general principle that if the binomial 
formula is true for any power, as the Arth, it is true for the next 
higher power, the A; + 1 power. 

We write out the formula for the ^th power and multiply both 
sides by X + a. 

221 



222 DURELL'S ALGEBRA: BOOK TWO 



1 X ^ 

(X + a)»+i = a;*+» + (t + l)j:*a + [^^^ + *]^'«* 

^L 1x2x3 ^ 1x2 J 
or, (x+o)*+i 

This is the result which would be obtained by expanding (x+a)*+^ 
according to the formula. 

Hence, we have proved that if the binomial formula is true for any 
power, as the ifcth, it is true for the next higher power, the X: + 1 power. 

III. But by actual multiplication (in I) the binomial formula was 
shown to be true for (x + a)^ or the 4th power. Hence, by the gen- 
eral principle just proved (in II), the formula must be true for the 
next higher power, the 5th. In like manner, it must be true for the 
6th, etc., to the nth power. 

The method of proof used in this Article is called mathematical 
induction. 

156. When a is negative, «», o^, etc., are negative ; hence, 

rx - ay = a:» - nx-'^a + ^^J'-'P x-'^a^ 

1x2 

^n(n-l)(n--2)^,_3^3 (1) 

1x2x3 

This formula may be proved by changing a into — a in 
the proof given in Art. 155. 

EXERCISE 86 

1. Write out the formula for (a; + a)^ 

2. For (a: + a)«+^ 3. For Qx + ay-\ 



BINOMIAL THEOREM 223 

4. How many terms are there in the expansion of 
(x + ay? Oi(x + ay? (x + a)^? (x+a)-? (a^+a)-'? 

Before the papil attempts the proof of the following laws, each 
law shoald be illustrated numerically till its meaning is thoroughly 
understood. 

5. By mathematical induction prove that 

1 + 2 + 3+ ... + w = ^(w+l). 
Sua. (1) We have 1 + 2 + 3 = J (3 + 1), or 6. 

(2) If 1+2 + 3+ ... +it = |(ifc + l), 
adding ^ + 1 to each member, 

1+2 + 3+ ... + ib +(ifc+l)= |(X:+1)+ it +1 

2 

(3) Hence, etc ' 

By mathematical induction, prove that 

6. The sum of the first n even numbers equals n(n + 1). 

7. The sum of the first n odd numbers equals w*. 

8. 12 + 22 + 3a+ ... +w2 = ^w(w+l)(2n + l). 

9. 2^ + 42+62+ ... + (2n)2 = fw(w + l)(2w+l). 

10. 18+28+38+ ... +W8 = jw2(w+1)2 

= (1 + 2 + 3+ ... +n)2. 

U. a" — 6" is always divisible by a — 6 when n is an 
integer. 

157. Key ITnmber and rth Term. In memorizing the bi- 
nomial formula, it is helpful to observe that a certain 
number may be regarded as governing the formation of 
each term of the formula. This number is one less than 
the number of the term. 



224 DURELL'S ALGEBRA: BOOK TWO 

Thus, for the third term we have ^^!^ ^ }^ af^-^a^ in 

1x2 

which are two factors in the numerator of the coefficient; 

two factors in the denominator ; the exponent of x is w — 2, 

and that of a is 2. Hence, we regard 2 as the key number 

of the term. 

The number 8 occurs in a similar way in the formation 
of the fourth term ; 4, in the fifth term, and so on. 

For the rth term, the key number would be r — 1. 
Hence, 

rth term = n(n - 1) ... (n - r + 2) ^_,^,^,_, 



158. Examples. 

Ex.1. Expand (f-^J. 

-k-iDV'hHm'-'r-Hfivr 

Ex. 2. Find the sixth term of (^ ^Y . 

The key number for the sixth term is 5. Hence, we 
obtain 

Sixth term = ^ >< » x 7 x 6 x 5 /.N -»/ _ 2 4^4)* 
1x2x3x4x5 VL>/ V 3 ^ / 

12* 36 27 ^ 



Ex. 3. Find the term in f a;^ _ Ji j which contains x^. 
We must first find the number of the term and then the term itself. 



BINOMIAL THEOREM 226 

The rth term of (x^ - 2 x"J)" = (coeff.) (a«)"-'+i(-. 2 x'i)^h 
For the required term, the x's collected must = x*'. 

Hence, (x2)i2-r(a:-i)r-i = a^u 

24- 2r-tLpi = i2 

Whence, r = 5 

5th term = ^] ^ l^ \^ ^/ (:p')"-^( - 2 x"*)* 

= 5280x". Ans, 

EXERCISE 87 

1. Change each of the given expressions in Exs. 6-15 
to a form in which it can be most readily expanded by 
the binomial formula. 

Expand : 

2. (2a-xy 4. (3rr*-2ya)* 6. (a?"* + Viy 

3. (i + lj 5. cxi^2xy ^- (^-^^J 

\ ^ y 14. (a?-rr + 2)8 

10. (2\^+v^)* 15. (2-3a; + a?)8 

U. /'22:. y Y 1«- (2a^ + 2:-3)* 

\Vy 2Vi/ 17. (a2 + 2 aa; - a?)* 

Find the 

18. Sixth term of (a - 2 2^)^. 

19. Eighth term of (1 -H a:Vy)^. 

20. Find the seventh and eleventh terms of 

(a;^ — y Va:)^*. 

21. Find the sixth and ninth terms of Q^ a% — 2-v^)^. 



226 DURELL'S ALGEBRA: BOOK TWO 

Find the ratio of 

22. The third to the fifth term in the expansion of 



23. The tenth and twelfth terms of (x^ + ^^) 



8 

24. Find the middle term of (8 a* - xVa)^. 
Write the formula for 

25. The r + 1st term of (x + a)\ 

26. The r — Ist term. For the r + 8d term. 

27. The rLh term of (x + ay^\ 

28. Term containing a:* in ( a: J • 

29. Term containing u^ in f a:^ — - j . 

30. Term containing a:^ in f | -H V? j • 

31. Term not coQtaining a? in f a;^ J • 

32. Term containing x in (yV^-f-V^) • 

33. By use of the hinomial formula find the value of 
(1.1)12 tQ three decimal places. 

Suo. Expand (1 + .l)n. 

Find the value of 

34. C1.2)M 35. (1.8)8 36. (2.2)8 

37. Find the coefficient of a:^ in Ta; - - Y*. 



BINOMIAL THEOREM 227 

38. In the shortest way find the 98th term of 

/ 1 \ioo 

39. Expand (x + a)"+' to 4 terms. 

40. Expand (x+ a)*"^ to 6 terms. 

41. Expand (1 — 1)* by the binomial theorem. 

42. Prove that in the binomial formula the sum of the 
coefficients of the odd terms equals the sum of the coeffi- 
cients of the even terms. 

43. Prove that the sum of the coefficients of the terms 
in the expansion of (a + by^ is 2^^. That the sum of the 
coefficients in the expansion of (a -H J)* is 2*. 

44. Who discovered the binomial theorem and when? 
(See p. 265.) Find out all you can about this man. 

45. State the advantages or utilities in the binomial 
theorem. 

46. Make up and work three examples similar to such 
of the above as the teacher may indicate. 

47. Practice oral work with exponents as in Exercise 39 
(p. 104). 



CHAPTER XVII 
INEQUALITIES; VARIATION 

Inequalities 

159. The Signs of Inequality are >, which is read ''is 
greater than "; and <, which is read "is less than." 

Thus, a>b means that a is greater than b. 
c<b means that c is less than 6. 

Observe that both signs of inequality are written with 
the opening toward the greater quantity. 

160. An Inequality is a statement in symbols that one 
algebraic expression represents a greater or less number 
than another ; as, a? -H y < «* + **• 

Remember that any positive number is greater than any negative 
number, and that of two negative numbers the smaller as to its nu- 
merical size is the greater in relative value. Thus, 2 > — 5, and 
-2>-3. 

The first member of an inequality is the expression on 
the left of the sign of inequality; the second member is the 
expression on the right of this sign. 

161. Inequalities of the Same Kind. Two inequalities are 
said to be of the same kind^ or to subsist in the same sense, 
when the greater member occupies the same relative posi- 
tion in each inequality ; that is, is the left-hand member in 
each, or the right-hand member. Hence, in inequalities of 
the same kind the signs of inequality point in the same 
direction. 



INEQUALITIES 229 

Thus, z>2x- 



o 

but a < 6 ' 

2a<6-| 



are of the same kind; 
are of opposite kinds. 



162. Properties of Inequalities. The following primary 
properties of inequalities are recognized as true: 

(1) Adding and subtracting quantities. An inequality will 
he unchanged in kind if the same quantity is added to or 
subtracted from each member. Hence, 

(2) Terms transposed. A term may be transposed from 
one member of an inequality to the other^ provided its sigri is 
changed. 

(3} Signs ohanged. 7%e signs of all the terms of an in- 
equality may be changed^ provided the sign of the inequality 
is reversed. 

(4) Positive multiplier. An inequality will be unchanged 
in kind if all its terms are multiplied or divided by the same 
positive number. 

(5) Raised to a power. An inequality will be unchanged 
in kind if both members are positive and both are raised to 
the same power. 

( 6) Equalities combined with inequalities. If the members 
of an inequality are subtracted from equals^ the result will be 
an inequality of the opposite kind. If the members of an 
inequality are divided into equals^ the result will be an ine- 
quality of the opposite kind. 

(7) Inequalities combined. If the corresponding members 
of two inequalities of the same kind are added^ or multiplied^ 
the resulting inequality will be of the same kind. But if the 
members of an inequality are subtracted from^ or divided by., 



230 DURBLL'S ALGEBRA: BOOK TWO 

the eorretponding members of another inequality of the same 
kind, the resulting inequality mil not always he of the same 
kind. 

The following are numerical illustrations of the above 
principles : 

(5) 7> 5 

.•.7«> 6* 
or 49 > 25 

(6) 



(7) 

Adding, 



21>15 Subtracting, 9<13 

163. A Conditional Inequality is an inequality which is 
true only for certain special values of the letter or letters 
involved. 

Thus, (3 — xY > (x — 4)' is a conditional inequality, since it may 
be proved that it is true only when x > 3J. 

An unconditional or absolute inequality is an inequality 
which is true for all possible values of the letter or letters 
involved. 

Thus, a^ + 5^ > 2 a& is an unconditional inequality, since it may be 
proved to be true for all possible values of a and h (the value zero not 
being considered in this case). 

Hence, the conditional inequality corresponds to the conditioDal 
equation, and the unconditional inequality to the identity. As with 
the equality sign, so with the signs > and <, the particular sense in 
which each is used is, for the present, to be determiued by the context. 



(1) 


7>5 






3 = 3 






10>8 




(2) 


12>7- 


-8 


.•.12 + 3>7 




(3) 


-7>- 


10 




.•.10> 


7 


or 


7< 


10 


(4) 


7>5 
3 = 3 





20 = 20 
7> 5 


13<15 

17>15 
8> 2 


25>17 

17>15 
8> 2 



INEQUALITIES 



231 



164. Solution of Cronditionallnequalities. 

Ex. 1. For what value ofa?i8 8x — 4>1 — 2a?? 

Transposing terms, 3 x + 2 x > 5 

5 a: > 6 ,\x>l, Ang. 



T 


^ t 


-t 


J 


^^ t 


^ 4- 


^ 5 


%-t 


MS^ ^ 


-It 


t^ 


t % 


3 -^ 


7 ^ 


-L it 


4: 



This process may be illustrated graphically as in the diagram. 

Graphing p = 3 x — 4, we obtain line AB. Also by graphing 
^ = 1 — 2 X, we obtain line CD. These lines intersect at the point F, 
whose abscissa is 1. To the right of F (where x > 1), p > 9, that is, 
3x-4>l-2x. 

Ex. 2. Given that x is an integer, determine its value 

from the following inequalities: 

f4a;-7<2a:-f 3 

32;4-l>13-ic 
Transposing terms, 

f2x<10 

I4x>12 
Dividing by coefficient of x in each inequality, 

[Itl •••^=*- ^"*- 

Let the pupil illustrate this solution graphically. 



232 DURELL'S ALGEBRA : BOOK TWO 

165. Proof of TTnoonditional Inequalities. 

Ex. 1. Prove that the sum of the squares of any two 
unequal quantities is greater than twice their product. 

Let a be the greater of two quantities, and b the less. 
Then, a-b>0 

.-. (a-6)2>0 
.-. a* - 2 aft + 6« > 

a^ + b*>2ab 

Ex. 2. Prove (a + 6) (6 + c) (a + (?) > 8 abe. 

The left-hand member when expanded becomes 

a(b^ + c«) + 6(a« + c«)+ c(a« + &=) + 2 abc 

But from Ex. 1, a(b^ + c^)>a(2bc) (1) 

6(a« + c2) > 6(2 ac) (2) 

c(aa + 62)>c(2a6) (3) 

Also, 2 abc ==2 abc (4) 

Adding (1), (2), (3), (4), 

(a + b)(b + <?)(«+ c)> 8 «^ 

EXERCISE 88 

Reduce: 

a. (3-a;)2>(a;-4)2 J 

6 a! + l a;+3 
3. 7aa; + 5>3aa; + 5J ' a;_2a;— 4 

4 a;— 3 a; 3a;-h 8 ajf£ _J_+£_ 

3 2 21 ' a-a: 26-a; 

4(x+3) 8a; + 87 7 a;- 29 
9 18 6a;-12 

Find the limits of a; : 

9. 3a; + l>2a: + 7- ^ 60>^^>60 

2a;-l<a! + 6 5 

10. 3(x-4) + 2>4(x-3) ioo>:r + ^±l>90 

2(a:+l)<4(a;-l) + 3 ^ ^ 2 ^ 



INEQUALITIES 233 

Solve the following: 

13. x — y>5 14. 3a;-4y>6 

a; + y=:12 4a: + 5y=80 

15. One fifth of a certain number plus its sixth is 
greater than 6, while its third minus its eighth is less than 4. 
Find the number. 

16. A certain integer decreased by | of itself is greater 
than J of the number increased by 5J ; but if J of itself is 
added to the number, the sum is less than 20. Find the 
number. 

If a, 6, and c are positive and unequal, prove : 

17. 8a2 + }2>2a(a + 6) ^ ^-f *>2 

18. a8-J8>3a26-3a62 ^^ , ^^^ .^ 

21. a-\-b>2wao 

19. a^ + b^>aH-\-ab^ 22. a^-\'b^+c^>ab-\'ac-\'bc 

23. 6aJc<a(62-|.a6 + e?2) + <62 + 6<? + a*) 

24. ab(^a + J) + ac(a + c) + *<?(* + c) < 2(a8 + J^ + c^) 

25. a^-\'b^+(^>Sabc 

26. A baseball team has won 22 games out of 35 games 
played. What is the least number of games which the 
team must win in succession, in order that the average of 
games won may exceed .75 ? 

27. Make up and work a similar example concerning 
games won by a basketball team. 

28. A boy has worked correctly 13 examples out of 18. 
What is the least number of examples which he must work 
correctly in succession, in order to bring his average above 
90 % ? Above 80 % ? 

29. Who invented the signs > and < to represent in- 
equality ? What other signs were invented for this pur- 
pose ? Which set of signs do you consider superior and why ? 



234 DURELL'S ALGEBRA: BOOK TWO 

Variation 

166. As stated in Art. 24 (p. 67), a YariaUe is a quan* 
tity which has an indefinite number of diiferent values. 

A constant is a quantity which has a single fixed value. 

167. Selation of Variables; Variations. One variable 
(called the function, see Art. 24) may depend on another 
variable for its value in a definite manner. 

Thus, if a man is hired to work for a certain sum per 
day, the number of dollars he will receive as wages will 
vary as the number of days he works. 

Thus, if z = number of dollars in his wages, 
t = number of days he works, 
xccL (The symbol « reads " varies as.") 

This expression is called a variation. 

This variation may also be expressed in the form 

a; = mf , or - = m 
t 

where m denotes the number of dollars in one day's wages. 

Thus, if the ratio of two variables is always constant, 
their relation may be expressed in any one of three ways : 

(1) As a ratio. (2) As an equation. (3) As a 
variation. 

168. I. Simple Birect Variations. The case considered 

in Art. 167, viz.: ^^.^ o..«. *«./ 
' a: QC y, or a? = my 

is called a direct variation. 

II. Inverse Variations. If x varies inversely as y (that 
is, as X increases, y decreases, and vice versa)^ then x and 

- have a constant ratio, 

y 

1 m 

xcc-^ or a: = — . 

y y 

Ims IS called an inverse variation. 



VARIATION 235 

Thus, the number of days required to do a given piece of 
work varies inversely as the number of workmen employed. 
Also, in triangles of a given area, the altitude varies 
inversely as the length of the base. 

III. Joint Variation. If x varies as the product of two 
or more other variables, as of y and z^ then x and yz have 
a constant ratio, and 

xocyz, or x = myz 
This is called sl joint variation, 

IV. Direct and Inverse Variation, x may also vary di- 
rectly as one variable, as y, and inversely as another, as z. 
Then 

V my 

a:«^, or x = — - 

z z 

Thus, the number of days it takes to reap a giren field varies 
directly as the number of acres in the field and inversely as the 
number of laborers. 

169. Fundamental Property of Variations. A variation 
may be converted into an equation by the use of a coefficient 
which is afterward to be determined; also the properties of 
variations may be derived and problems solved by the use of 
the properties of equations. . 

170. Elementary Properties of Variations. 

I. li xocy a,nd yccz^ then XQcz. 
For X = my, y = nz 

.'. X = mnz 

.'. XQCZ 

In like manner, let the pupil show that 

II. It xccz, yccz^ then x±yccz and Vxyccz. 

III. Itxccz B,nd yxu^ then xyocuz, 

IV. It xccy^ then a;* oc y*. 



236 DURELL'S ALGEBRA: BOOK TWO 

171. Examples. 

Ex. 1. li X varies inversely as y*, and re =4 when 
J = 1, find X when y = 2. 

Since x«— , we have x =^ (1) 

y* y* 

Substitute a: = 4, y = 1, in (1), 4 = m. 
Substitute for m its value in (1), 

'=? (^> 

Let y = 2 in (2), then x = 1. Ans. 

Ex. 2. The area of a circle varies as the square of its 
diameter. Find the diameter of a circle whose area shall 
be equivalent to the sum of the areas of two circles whose 
diameters are 6 and 8 inches respectively. 

Let A denote the area of a circle, and D the diameter. 
Then AxD^ 

And A = mD^ 

Denote the areas of the two given circles by *4' and -4" - 
Then .4' = fw x 6* = 36w 

A" = Tnx 82 = 64m 
Adding, A' + A"j or X = lOOwi 
Hence, since A = mD^j and also 100 m, 
mZ)2 = 100m 
2)2 = 100 
D = 10 
Thus, the required diameter is 10. Ans, 

EXERCISE 89 

1. If a; X y, and x = 8 when y = 6, find y when x=S. 

2. If a: + 1 X y — 5, and x=2 when y = 6, find x when 
y = 7. 

3. If a? X y8, and a: = 4 when y = 2, find y when x = 32. 

4. If a? X y^ + 8, and x = f VS when y = 1, find y when 
a:=3. 



VARIATION 237 

5. If X varies inversely as y, and equals 2 when y is 4, 
find y when a: = 5. 

6. If X varies jointly as y and 2, and is 6 when y = 3 
and 2 = 2, find a; when y = 5 and 2 = 7. 

7. a; varies directly as y and inversely as 2, and = 10 
when y = 15 and 2 = 6. Find y when a: = 16 and 2=2. 

8. If a; varies inversely as y and directly as 2, and 
a; = 2 when y = J and 2 = — 3, find the value of x when 
y = 2 and 2 = — J. 

9. The distance («) passed over by a body falling from 
a state of rest is found by experiment to vary as the 
square of the number of seconds (Q. Express this law 
as a variation. 

If m is found to be 16.1 ft., express the law as an equa- 
tion. 

10. The number of seconds {€) required by a pendulum 
to make a complete oscillation is found by experiment to 
vary as the square root of the length (Z) of the pendulum. 
Express this law as a variation. 

If m is then found to equal 2 tt -h v^, express the law 
as an equation. 

11. The number of vibrations (JP) made by a wire of 
given length (V) stretched by a weight (w) is found by 
experiment to vary directly as the square root of w, and 
inversely as I, N is also found to vary inversely as the 
diameter (rf) of the wire, and inversely as the square root 
of the specific gravity («) of the material composing the 
wire. Express this law as a variation. 

If m is found to equal VI-htt, express the law as an 
equation. 



238 DURELL'S ALGEBRA : BOOK TWO 

12. Assuming that the velocity of a body falling from 
a state of rest varies directly as the time, and knowing 
that the velocity is 160 ft. per second after 5 seconds of 
falling, find the velocity at the end of 8 seconds. 

13. Assume the law that the time required by a pendu- 
lum to make one vibration varies as the square root of the 
length of the pendulum. If a pendulum 100 centimeters 
long vibrates once in one second, find the time of vibra- 
tion of a pendulum 36 centimeters long. 

14. The volume of a cone of revolution, of which the 
altitude is 7, and the radius of the base is 3, is 66. Find 
the volume of a cone of revolution of altitude 6 and radius 
6. 

Note. The volume of a cone of revolution (or cylinder of revolu- 
tion) varies jointly as the altitude and the square of the radius of the 
base. 

15. Find the altitude of a cone of revolution, of which 
the radius of the base is 7, and the volume is equivalent to 
two cones with altitudes 5 and 11 and radii 2 and 4 re- 
spectively. 

16. Assume that the illumination from a source of light 
varies inversely as the square of the distance. A book is 
now held 18 in. from a lamp. How much farther away 
must it be moved in order to receive J as much light? 
Interpret the two results. 

17. Make up and work two examples similar to such of 
the above as the teacher may indicate. 



CHAPTER XVIII 
LOGARITHMS 

172. The Logarithm of a number is the exponent of that 
power of another number, taken as the base, which equals 
the given number. 

Thus, 1000 = 10». Hence, log 1000 = 3, 10 being taken as the base. 

Again, if 8 is taken as the base, 4 = 8'. Hence, log 4 = }. 

If 5 is taken as the base, log 125 = 3, log ^ = - 2, etc. 

The base is sometimes stated as above; but when desir- 
able, it is indicated by writing it as a small subscript to 
the word log. 

Thus, the above expressions might be written, 
logio 1000 = 3; log34 = |; log, 125 = 3; logj,A = -2; etc. 

In general, by the definition of a logarithm, 
number = (base)^^«*'"*'°», 
or -Z\r= jB*. Hence, log^N^^ I 

173. Uses of Logarithms. One of the principal uses of 
logarithms is to simplify numerical work. For instance, by 
logarithms the numerical work of multiplying two numbers 
is converted into the simpler work of adding the logarithms 
of these numbers. 

To illustrate this principle, we may take the simple case 
of multiplying two numbers which are exact powers of 10, 
as 1000 and 100. Thus, 

1000 = 10« 
100 = IQg 
Hence, 1000 x 100 = 10* = 100,000, 

the multiplication being performed by the addition of exponents. 

239 



240 DURELL'S ALGEBRA : BOOK TWO 

Similarly, if 384 = 10«-««»+ 

and 25 = 10i«w»^ 

To multiply 384 by 25, 
Add the exponents of 102-6m»+ and 10i-»t»4+^ thus obtaining 

10S.982Z7+. 

Then get from a table of logarithms the value of 10»-»w+, viz. 9600. 

In like manner, by the use of logarithms, the process of 
dividing one number by another is converted into the 
simpler process of subtracting one exponent, or log, from 
another. The process of involution, also, is converted 
into the simpler process of multiplication ; and the extrac- 
tion of a root into the simpler process of division. 

We can save labor still further, through the use of 
logarithms, by committing to memory the logs of numbers 
that are frequently used, as 

2, 3, .-. 9, TT, Vn^, -, V2, V3, etc. 

IT 

By the use of the 9lide nde^ the practical use of loga- 
rithms is reduced to sliding one rod along another and 
reading off the number at one end of a rod. 

It will be a useful exercise to teach the class the use of the slide 
rule in connection with the study of this chapter. 

174. Systems of Logarithms. Any positive number, ex- 
cept unity, may be made the base of a system of logarithms. 
Two principal systems are in use : 

1. The Common (or Decimal) or Briggsian System, in 

which the base is 10. This system is used almost exclu- 
sively for numerical computations. 

2. The BTatural or BTapierian System, in which the base 
is 2.7182818"*". This system is generally used in algebraic 
processes, as in demonstrating the properties of algebraic 
expressions. 



LOGARITHMS 241 

EXSRCISE 90 

1. Give the value of each of the following : logg 9, 
logg 27, log4 64, log4 3^^, logs i, logg ^, log^o ^, log^o .01, 
logio .001. 

2. Also of loga 32, loga ^, logg ^ J^, log^ 8, logg 16. 

3. Simplify logg 4 + logg 9 + log^o .1 - logg f 

4. Write out the value of each power of 2 up to 2*^ in 
the form of a table. 

Thus, 21 = 2, 2« = 4, 2» = 8, etc. 

5. By means of this table, multiply 32 by 8, perform- 
ing the multiplication by the addition of exponents. 

6. In like manner, convert each of the following mul- 
tiplications into an addition : 82 x 16, 64 x 32, 1024 
X 16, 612 X 64. 

7. Convert each of the following divisions into a 
subtraction : 1024 -h 16, 512 -t- 64, 32,768 -h 1024. 

8. Convert each of the following involutions into a 
multiplication : (32)8, (64)2, ^32)4. 

9. Convert each of the following root extractions into 
a division: VM, -y/Vm, </^m6. 

10. Make up two examples like those in Ex. 6. In 
Ex. 8. In Ex. 9. 

11. Construct a table of powers of 3 and make up 
similar examples concerning it. 

12. How many of the above examples can you work at 
sight ? 

175. Characteristic and Mantissa. If a given number, as 
384, is not an exact power of the base, its logarithm, as 
2.58433"*", consists of two parts: the whole number, called 
the characteristic^ and the decimal part, called the mantissa. 



242 DURELLS ALGEBRA: BOOK TWO 

To obtain a rule for determining the characteristic of a 
given number (the base being 10), we have : 

10,000 = 10*, hence log 10,000 = 4 ; 

1000 = 10«, hence log 1000 = 3 ; 

100 = 102, hence log 100 = 2 ; 

10 = 101, hence log 10 = 1. 

Hence, any number between 1000 and 10,000 has a 
logarithm between 8 and 4 ; that is, 3 plus a fraction. 

But every integral number between 1000 and 10,000 
contains four digits. Hence, every integral number con- 
taining /owr figures has 8 for a characteristic. 

Similarly, every number between 100 and 1000, and 
therefore containing three figures to the left of the decimal 
point, has 2 for a characteristic. 

A number between 10 and 100 (i.e., a number contain- 
ing two integral figures) has 1 for a characteristic. 

Every number between 1 and 10 (that is, every number 
containing one integral figure) has for a characteristic. 

Hence, the characteristic of an integral or mixed number 
is one less than the number of figures to the left of the deci- 
mal point. 

176. Characteristic of a Decimal Fraction. 

1 = 100. ... logl = 0; 

•1 = ^=10-^ .-. log.l = -l; 

•'' = T^=ii^ = l«"- ••• log .01 = ^2; 

•''' = l4 = i^ = l«"'- ••• log .001= -3, etc. 
Hence, the logarithm of any number between .1 and 1 
(as of .4, for instance) will lie between - 1 and 0, and 
hence will consist of - 1 plus a positive fraction. 



LOGARITHMS 243 

The logarithm of every number between .01 and .1 (as of 
.0372, for instance) will be between — 2 and— 1, and hence 
will consist of —2 plus a positive fraction ; and so on. 

Hence, the characteristic of a decimal fraction is negative^ 
and is numerically one more than the number of zeros be- 
tween the decimal point and the first significant figure. 

There are two ways in common use for writing the 
characteristic of a decimal fraction. 

Thus, (1) log .0384 = 2.58433, the minus sign being placed over 
the characteristic 2, to show that it alone is negative, the mantissa 
being positive. 

Or (2) 10 is added to and subtracted from the log, giving 
log .0384 = 8.58433 - 10. 

In practice, the following rule is used for determining 
the characteristic of the logarithm of a decimal fraction : 

Take one more than the number of zeros between the deci- 
mal point and the first significant fi^ure^ subtract it from 10, 
and annex — 10 after the mantissa, 

EXERCISE 91 
Give the characteristic of 

.08267 11. 7 

1.0042 12. 6267.3 

7.92631 13. .000227 

.007 14. 100.58 

.0000625 15. 23.7621 

to the left of the decimal point 
(or how many zeros immediately to the right) are there 
in a number, the characteristic of whose logarithm is 3 ? 
2? 5? 1? 0? 4? 8-10? 7-10? 9-10? 

17. Can you make up a rule for fixing the decimal point 
in the number which corresponds to a given logarithm ? 



1. 


452 


6. 


2. 


16,730 


7. 


3. 


767.5 


8. 


4. 


64.56 


9. 


5. 


9.22678 


10. 


16. 


How many 


figures 



244 DURELL'S ALGEBRA: BOOK TWO 

18. If log 632 = 2.8007, express 632 as a power of 10. 

19. If 267 = 102-^, what is the log of 267 ? 

20. If a number lies between 9000 and 20,000, what 
will its characteristic be ? 

21. If a number lies between 10,000 and 100,000, 
between what two numbers must its logarithm lie ? 

177. Mantissas of numbers are computed by methods, 
usually algebraic, which lie outside the scope of this book. 
After being computed, the mantissas are arranged in 
tables, from which they are taken when needed. In this 
connection, it is important to note that 

The portion of the decimal point in a number affect% only the 
characteristic^ not the mantissa^ of the logarithm of the number. 
Thus, if log 6754 = 3.82956 

log 67.54 = log ^ = log 15!:^ = log 10i.8»6a = 1.82956. 

In general, log 6754 = 3.82956 
log 675.4 = 2.829.56 
log 67.54 = 1.82956 
log 6.754 = 0.82956 
log .6754 = 9.82956 - 10 
log .06754 = 8.82956 - 10, etc. 

178. Direct Use of a Table of Logarithms ; that is, given a 
number^ to find its logarithm from a table. From the follow- 
ing small table of logarithms, the student may learn 
enough of their use to understand their algebraic proper- 
ties. The thorough use of logarithms for purposes of 
computation is usually taken up in connection with the 
study of trigonometry. 

In the table (see pages 246, 247), the left-hand column is a column 
of numbers, and is headed N. 

The mantissa of each of these numbers is in the next column op- 
posite. In the top row of each page are the figures 0, 1, 2, 3, 4, 5, 6, 
7, 8, 9. 



LOGARITHMS 245 

To obtain the mantissa for a number of three figureSi as 364, we 
take 36 in the first column, and look along the row beginning with 36 till 
we come to the column headed 4. The mantissa thus obtained is .5611. 

If the number whose mantissa is sought contains four or five figures, 

Obtain from the table the mantissa for the first three figures^ and also 
that for the next higher number^ and subtract; 

Multiply the difference between the two mantissas by the fourth (or 
fourth and fifth) figure expressed as a decimal; 

And ADD the result to the mantissa for the first three figures. 

Thus, to find the mantissa for 167.49, 

Mantissa for 168 = .2253 

Mantissa for 167 = .2227 

Difference = .0026 

Since an increase of 1 in the number (from 167 to 168) makes an 
increase of .0026 in the mantissa, an increase of .49 of 1 in the number 
will make an increase of .49 of .0026 in the mantissa. 
But .0026 X .49 = .001274 or .0013 - . 
Hence, .2227 

13 

Mantissa for 167.49 = .2240 

Hence, to obtain the logarithm of a given number, 
Determine the characteristic hy Art. 175 or Art. 176; 
Neglect the decimal pointy and obtain from the table 
(pp. 246, 247) the mantissa for the given figures. 
Exs. Log. 62.6 = 1.7210. Log. .00094 = 6.9731 ~ 10. 
Log. 167.49 = 2.2240. Log. .042308 = 8.6264 - 10. 



EXERCISE 92 

Find the logarithms of the following numbers ; 



1. 


37 


6. 


175 


11. 


.0758 


16. 


.7788 


2. 


85 


7. 


32.9 


12. 


5780 


17. 


.04275 


3. 


6 


8. 


4.75 


13. 


.00217 


18. 


234.76 


4. 


90 


9. 


.08 


14. 


63.21 


19. 


5.6107 


5. 


300 


10. 


1.02 


15. 


3.002 


20. 


7781.4 



246 



DUBELL'S ALGEBRA: BOOK TWO 



N. 





1 


S 


3 


4 


5 


6 


7 


8 


9 


10 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


11 


414 


453 


492 


531 


569 


607 


646 


682 


719 


755 


IS 


702 


828 


864 


899 


934 


969 


1004 


1038 


1072 


1106 


13 


1139 


1173 


1206 


1239 


1271 


1303 


335 


367 


399 


430 


14 


461 


492 


523 


563 


584 


614 


644 


073 


703 


732 


15 


1761 


1790 


1818 


1847 


1875 


1903 


1931 


1959 


1987 


2014 


16 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


279 


17 


304 


330 


356 


380 


405 


430 


455 


480 


604 


629 


18 


558 


577 


601 


625 


648 


672 


696 


718 


742 


765 


19 


788 


810 


833 


866 


878 


900 


923 


945 


967 


989 


SO 


3010 


3032 


3054 


3076 


3096 


3118 


3139 


3160 


3181 


3201 


21 


222 


243 


263 


284 


804 


324 


345 


365 


385 


404 


S8 


424 


444 


464 


483 


602 


522 


541 


560 


679 


598 


23 


617 


636 


655 


674 


692 


711 


729 


747 


766 


784 


24 


802 


820 


838 


856 


874 


892 


909 


927 


945 


962 


25 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4116 


4133 


26 


4150 


4166 


183 


200 


216 


232 


249 


265 


281 


298 


27 


314 


330 


346 


362 


378 


893 


409 


426 


440 


466 


28 


472 


487 


502 


618 


538 


548 


564 


679 


594 


609 


29 


624 


639 


654 


669 


683 


698 


713 


728 


742 


767 


30 


4771 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


31 


914 


928 


942 


956 


969 


983 


997 


6011 


5024 


5038 


32 


5051 


5066 


5079 


5092 


5105 


5119 


5132 


145 


169 


172 


33 


185 


198 


211 


224 


237 


260 


263 


276 


289 


302 


34 


315 


328 


340 


363 


366 


378 


391 


403 


416 


428 


35 


5441 


5453 


5465 


6478 


5490 


6502 


5514 


5627 


5539 


5551 


36 


563 


575 


587 


599 


611 


623 


636 


647 


658 


670 


37 


682 


694 


705 


717 


729 


740 


762 


763 


775 


786 


38 


798 


809 


821 


832 


843 


865 


866 


877 


888 


899 


39 


911 


922 


933 


944 


966 


966 


977 


988 


999 


6010 


40 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 


41 


128 


138 


149 


160 


170 


180 


191 


201 


212 


222 


42 


232 


243 


253 


263 


274 


284 


294 


304 


314 


326 


43 


335 


345 


365 


365 


376 


385 


896 


405 


416 


425 


44 


435 


444 


454 


464 


474 


484 


493 


508 


518 


622 


45 


6532 


6542 


6551 


6561 


6671 


6580 


6590 


6599 


6609 


6618 


46 


628 


637 


646 


im 


665 


675 


684 


698 


702 


712 


47 


721 


730 


739 


749 


768 


767 


776 


785 


794 


808 


48 


812 


821 


830 


839 


848 


867 


866 


875 


884 


893 


49 


902 


911 


920 


928 


937 


946 


965 


964 


972 


981 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7069 


7067 


51 


7076 


7084 


093 


101 


110 


118 


126 


136 


143 


162 


52 


160 


168 


177 


185 


193 


202 


210 


218 


226 


235 


53 


243 


251 


259 


267 


276 


284 


292 


300 


308 


316 


54 


324 


332 


340 


348 


356 


364 


872 


380 


888 


396 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 



LOGARITHMS 



247 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


55 


7404 


7412 


7419 


7427 


7436 


7443 


7461 


7469 


7466 


7474 


6« 


482 


490 


497 


606 


613 


620 


628 


636 


643 


651 


57 


569 


666 


674 


682 


689 


697 


604 


612 


619 


627 


58 


634 


642 


649 


667 


664 


672 


679 


686 


694 


701 


69 


709 


716 


723 


731 


738 


746 


752 


760 


767 


774 


60 


7782 


7789 


7796 


7803 


7810 


7818 


7826 


7832 


7839 


7846 


61 


853 


860 


868 


876 


882 


889 


896 


903 


910 


917 


62 


924 


931 


938 


946 


962 


969 


966 


973 


980 


987 


63 


993 


8000 


8007 


8014 


8021 


8028 


8036 


8041 


8048 


8055 


64 


8062 


069 


076 


082 


089 


096 


102 


109 


116 


122 


65 


8129 


8136 


8142 


8149 


8166 


8162 


8169 


8176 


8182 


8189 


66 


195 


202 


209 


216 


222 


228 


236 


241 


248 


264 


67 


261 


267 


274 


280 


287 


293 


299 


306 


312 


319 


68 


325 


331 


338 


844 


361 


367 


363 


370 


376 


382 


69 


388 


396 


401 


407 


414 


420 


426 


432 


439 


446 


70 


8461 


8457 


8463 


8470 


8476 


8482 


8488 


8494 


8500 


8606 


71 


613 


519 


626 


631 


637 


643 


649 


555 


661 


667 


72 


673 


679 


686 


501 


697 


603 


600 


615 


621 


627 


73 


633 


639 


645 


651 


667 


663 


669 


675 


681 


686 


74 


692 


698 


704 


710 


716 


722 


727 


733 


739 


746 


75 


8761 


8756 


8762 


8768 


8774 


8779 


8786 


8791 


8797 


8802 


76 


808 


814 


820 


826 


831 


837 


842 


848 


854 


859 


77 


866 


871 


876 


882 


887 


893 


899 


904 


910 


916 


78 


921 


927 


932 


938 


943 


949 


964 


960 


966 


971 


79 


976 


982 


987 


993 


998 


9004 


9009 


9016 


9020 


9026 


80 


9031 


9a36 


9042 


9047 


9063 


9058 


9063 


9069 


9074 


9079 


81 


085 


090 


096 


101 


106 


112 


117 


122 


128 


133 


82 


138 


143 


149 


164 


169 


166 


170 


175 


180 


186 


83 


191 


196 


201 


206 


212 


217 


222 


227 


232 


238 


84 


243 


248 


263 


258 


263 


269 


274 


279 


284 


289 


85 


9294 


9299 


9304 


9309 


9316 


9320 


93-25 


9330 


9336 


9340 


86 


345 


360 


356 


360 


366 


370 


376 


380 


386 


390 


87 


395 


400 


406 


410 


415 


420 


426 


430 


436 


440 


88 


445 


460 


466 


460 


465 


469 


474 


479 


484 


489 


89 


494 


499 


604 


609 


613 


618 


623 


628 


633 


638 


90 


9642 


9647 


9662 


9657 


9662 


9666 


9671 


9676 


9681 


9686 


91 


690 


696 


600 


605 


609 


614 


619 


624 


628 


633 


92 


638 


643 


647 


652 


657 


661 


666 


671 


676 


680 


93 


685 


689 


694 


699 


703 


708 


713 


717 


722 


727 


94 


731 


736 


741 


745 


760 


754 


769 


763 


768 


773 


95 


9777 


9782 


9786 


9791 


9796 


9800 


9805 


9809 


9814 


9818 


96 


823 


827 


832 


836 


841 


846 


860 


864 


869 


863 


97 


868 


872 


877 


881 


886 


890 


894 


899 


903 


908 


98 


912 


917 


921 


926 


930 


934 


939 


943 


948 


952 


99 


966 


961 


966 


969 


974 


978 


983 


987 


991 


996 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 



248 DURELL'S ALGEBRA: BOOK TWO 

179. Inyene Use of a Table of Logarithms ; that is, given 
a logarithm^ to find the number corresponding to this loga- 
rithm^ termed antilogarithm : 

From the table^ find the figures corresponding to the man- 
tissa of the given logarithm ; 

Use the characteristic of the given logarithm to fix the 
decimal point of the figures obtained. 

Ex. Find the antilogarithm of 1.5658. 

The figures corresponding to the mantissa, .5658, are 368. 
Since the characteristic is 1, there are 2 figures at the left of the 
decimal point. 

Hence, antilog 1.5658 = 36.8 

In case the given mantissa does not occur in the table, 
obtain from the table the next lower mantissa with the corre- 
sponding three figures of the antilogarithm ; 

Subtract the tabular mantissa from the given mantissa; 

Divide this difference by the difference between the tabular 
mantissa and the next higher mantissa in the table ; 

Annex the quotient to the three figures of the antilogarithm 
obtained from the table. 

Ex. Find antilog 2.4237. 

.4237 does not occur in the table, and the next lower mantissa is 
.4232. The difference between .4232 and .4249 is .0017. 
Hence, we have antilog 2.4237 = 265.29 

4232 



17)5.00(.29 

If a difference of 17 in the last two figures of the mantissa makes 
a difference of 1 in the third figure of the antilog, a difference of 5 in 
the mantissa will make a difference of ^ of 1 or .29 with respect to 
the third figure of the antilog. 



13. 


0.4183 




14. 


1.4900 




IS. 


3.8500 




16. 


1.8904 




17. 


2.4527 




18. 


9.6402 - 


-10 



LOGARITHMS 249 

EXERCISE 93 

Find the numbers corresponding to the following 
logarithms : 

1. 1.6335 7. 0.6117 

2. 2.8865 8. 9.7973-10 

3. 2.3729 9. 7.9047-10 

4. 0.6776 10. 8.6314-10 

5. 3.9243 11. 7.7007-10 

6. 1.8476 12. 6.1004-10 

19. Write log 17 = 1.2304 as a number equal to a power 
of 10. 

20. Make up and work a similar example for yourself. 
180. Properties of Logarithms. It has been shown 

(Arts. 85, 51, 62, 53) that 

a« X a** = a"*"*"*, 
when m and n are commensurable. By the use of suc- 
cessive approximations approaching as closely as we please 
to limits, the same law may be shown to hold when m and 
n are incommensurable. It then follows that 

(1) log ah = log a + log h (3) log a^=p log a 

(2) log g)= log a -log J (4) logV^ = l2S^ 

Proof : 

Let a = 10"». .-. log a =m, 

b = 10». .'. log b = n.. 

ab = 10*»+». .•. log aft = w + n = log a + log J . . . . (1) 

« = 10«-n. ... log (^^ = m - n = log a - log 6 . . . . (2) 

aP = 10»«. .«. log a' = p7n = ploga (3) 

V^=107. ...iogV« = - = ^^^ (4) 

P P 

The same properties may be proved in like manner for a system of 

logarithms with any other base than 10. 



250 DURELL'S ALGEBRA: BOOK TWO 

181. Properties Utilized for Purposes of Computation. 

I. To Hultiply Vnmbers, 

Add their logarithms^ and find the antilogarithm of the 
sum. This will he the product of the numbers. 

II. To Divide One Vnmber by Another, 

Subtract the logarithm of the divisor from the logarithm of 
the dividend^ and obtain the antilogarithm of the difference. 

III. To Raise a N'limber to a Required Power, 

Multiply the logarithm of the number by the index of the 
power. Find the antilogarithm of the product. 

IV. To Extract a Required Root of a n'limber, 

Divide the logarithm of the number by the index of the re- 
quired root. Find the antilogarithm of the quotient. 

Ex. 1. Multiply 527 by .083 by the use of logs. 

log 527 = 2.7218 

log .083 = 8.9191 - 10 

antilog 1.6409 = 53.7+, Product. 
The following form is the arrangement of work used by many 
practical computers. It has the advantages of brevity and of 
showing all the steps in a complex logarithmic computation. 
527 log 2.7218 
.083 lo g 8.9191 - 10 
Product 53.7 log 1.6409 

Observe that " 527 log 2.7218 " is read « 527, its log is 2.7218." 

Ex. 2. Compute the amount of $1 at 6 % for 20 years 
at compound interest. 

The amount of $ 1 at 6 % for 20 years = (1.06)». 
1.06 log 0.0253 
20 



^fw.»3.21 log 0.5060 
Computing (1.06) *> by direct multiplication, will make clear the 
amount of labor sometimes saved by the use of logarithms. 



LOGARITHMS 251 

Ex. 3. Extract approximately the 7th root of 15. 

15 log 1.1761, } log 0.1680 
Root 1.47 log 0.1680 

182. Cologarithm. In operations involving division, it is 
usual, instead of subtracting the logarithm of the divisor, to 
add its cologarithm. The cologarithm of a number is ob- 
tained by subtracting the logarithm of the number from 
10 — 10. Adding the colog gives the same result as sub- 
tracting the log itself from the logarithm of the dividend. 

The use of the cologarithm saves figures, and gives a 
more compact and orderly statement of the work. 

The cologarithm may be taken directly from the table 
by use of the following rule : 

Subtract each figure of the given logarithm from 9, except 
the last significant figure^ which subtract from 10. 

Ex. 1. Find colog of 36.4. 

log 36.4 = 1.5611 
colog 36.4 = 8.4389 - 10 

Ex. 2. Compute by use of logarithms — f • . 

2V576x3.78 
8.4 log 0.9243 
32.4 log 1.5105 
2 log 0.3010 colog 9.6990 - 10 
576 log 2.7604 i log 1.3802 i colog 8.0198 - 10 
3.78 log 0.5775 colog 9.4225 - 10 
Ans. 1.5 log 0.1761 

SXERCISS 94 
Find, by use of logarithms, the approximate values of 
1. 75 X 1.4 4. 831 X .25 336.8 

2 98x35 ' ^^^* 

3. 15.1 X .005 ' 13.4 ' .0049 



262 DURBLL'S ALGEBRA: BOOK TWO 

-78.9 10. .48-1- (-1.79) 3.51 x 67 

* 98.7 97.7 

^ 42.316 „ 1.78 X 19 „ 12.9 



.06214 28.7 4.7x9.1 

14. 47.1 X 3.66 X .0079 

15. .0631 X 7.208 X. 51272 

16. 4.77x(-.71)-i-(.83) 
623 X 249 
767x396 

18. (2.3)» 83. Vl9 28. \/.00429 

19. (1.032)» 24. </SM 29. (2.91)5 

20. (3.57)* 25. </TM 30. -v^I^ 

21. (.96)T 26. Vm 31. VSO'x^''^^ 

22. (.796)« 27. <^ 32. </T9-i-Vi6 

33. \/:005 X -5^0766 35. </Jx</^ 

34. 2^x7* 36. </2xV3x^v^ 

37. -(3.12)« + ^(-42.8)* 



39. 



38. \/'.000479-j--v':0668 
il 529 ^ ^/37.56 x 26.6 



'67x518 ^22.7x16.78 

By the use of logarithms : 

41. Find the amount of $1250 at 6 per cent compound 
interest for 12 years. Also make the computation with- 
out the use of logarithms. What fraction of the work is 
saved by the use of logarithms ? 

42. Find the amount of $26 at 6 per cent compound in- 
terest for 600 years. 

43. Find the amount of $ 300 at 6 per cent for 60 years, 
interest being compounded semiannually. 



LOGARITHMS 253 

44. Find the amount of $300 at 6 per cent for 50 years, 
interest being compounded quarterly. 

45. Find the radius of a circle whose area is 100 sq. yd. 

46. Find the radius of a sphere whose volume is 20 cu. ft. 
(User=|7ri28.) 

47. A given parallelogram is 12.7 ft. long and 8.9 ft. 
high. Find the side of a square whose area is equal to 
that of the parallelogram. 

48. Compute Vl5 to three decimal places without the 
use of logarithms. Now obtain the same result by the use 
of logarithms. Compare the amount of work in the two 
processes. 

49. Find log -5/10 x -y/TOQ without the use of tables. 

50. By use of logarithms, find the value of V6^ — c^ 
when h = 276.5, c = 172.4. 

51. How many years will it take a sum of money to 
double itself at 5 ^ compound interest ? At 7 per cent ? 

52. If the area of a lot is 401.8 sq. ft. and the length is 
62.37 ft., find the width. 

53. The diameter of a spherical balloon which is to lift 
a given weight is calculated by the formula 



'>-<-. 



W 



.6236(^ - a) 

where D = diameter of the balloon in feet. 

A = weight in pounds of a cubic foot of air. 

(} = weight in pounds of a cubic foot of the gas 

in balloon. 
W= weight to be raised (including weight of 

balloon materials). 

If ^ = .08072, a = .0056, W= 1250 lb., find 2). 



254 DURELL'S ALGEBRA: BOOK TWO 

54. Also in Ex. 53, if ^ = .08072, & = .0056, D = 35.5, 
find W. 

55. In warming a building by hot-water pipes, the 
required length of pipes 4 in. in diameter is determined 
by the formula 

^^(P^K|llO ^.0046(7 

where L = length of pipes in feet. 

P= temperature (F.) of the pipes. 
T— temperature required in the building. 
t = temperature of the external air. 
(7= number of cubic feet of space to be warmed 
per minute. 

Find L when P= 120° F., t = 40.5° F., T^ 61.5° F., 
and (7= 35.6 x lO^. 

56. Make up and work three examples similar to such 
of the above as the teacher may indicate. 



CHAPTER XIX 
HISTORY OF ELEMENTARY ALGEBRA 

183. Epochs in the Development of Algebra. Some knowl- 
edge of the origin and development of the symbols and 
processes of algebra is important to a thorough under- 
standing of the subject. 

The oldest known mathematical writing is a papyrus roll, 
now in the British Museum, entitled " Directions for 
Attaining to the Knowledge of All Dark Things." It was 
written by a scribe named Ahmes (a'mes) at least as early 
as 1700 B.C., and is a copy, the writer says, of a more ancient 
work, dating, say, 3000 B.C., or several centuries before the 
time of Moses. This papyrus roll contains, among other 
things, the beginnings of algebra as a science. Taking 
the epoch indicated by this work as the first, the principal 
epochs in the development of algebra are as follows: 

1. Egyptian: 3000 B.C.-1500 B.C. 

2. Greek (at Alexandria) ; 200 A.D.^00 A.D. Principal 
writer, Diophantus (di 6 fan'tus). 

3. Hindoo (in India): 500 A.D.-1200 A.D. 

4. Arab : 800 A.D.-1200 A.D. 

5. European : 1200 A.D.-. Leonardo of Pisa, an Italian, 
published in 1202 a.d. a work on the Arabic arithmetic 
which contained also an account of the science of algebra 
as it then existed among the Arabs. From Italy the 
knowledge of algebra spread to France, Germany, and 
England, where its subsequent development took place. 

265 



256 DURELL'S ALGEBRA : BOOK TWO 

We will consider briefly the history of 
I. Algebraic Symbols. 
II. Ideas of Algebraic Quantity. 
III. Algebraic Processes. 

I. History of Algebraic Symbols 

184. Symbol for the Unknown Quantity. 

1. Egyptians (1700 B.C.) : used the word hau (expressed, 
of course, in hieroglyphics), meaning "heap." 

2. Diophantua (Alexandria, 350 A.D.?): 9', or 9°'; plural, 
99. 

3. Hindoos (500 AD.-1200 a.d.): Sanscrit word for 
" color," or first letters of words for colors (as blue, yellow, 
white, etc.). 

4. Arabs (800 A.D.-1200 a.d.): Arabic word for 
" thing " or " root " (the term root^ as still used in algebra, 
originates here). 

5. ItaUans (1500 a.d.): Radix, R, RJ. 

6. Bombelli (Italy, 1572 A.D.) : O'- • 

7. Stifel (stee'fel) (Germany, 1544): A, B, (7, — 

8. Stevinus (sta vee'nus) (Holland, 1586): ®. 

9. Vieta (ve a'ta) (France, 1591) : vowels Ay JS, 7, 0, U. 
10. Descartes (dakart') (France, 1637): a:, y, «, etc. 

185. Symbols for Powers (of x at first) ; Exponents. 

1. Diophantus: Bwafii^^ or S" (for square of the un- 
known quantity); >«;/8o9, or /c" (for its cube). 

2. Hindoos: initial letters of Sanscrit words for "square" 
and "cube." 

3. Italians (1500 A.D.): " census " or " zensus " or "2 " 
(for a?) ; " cubus " or " c " (for a^). 

4. Bombelli (1579): \^, ^, ^, for (a:, 2^^ a?^. 

5. Stevinus (1586): ®, (2), (3), (for x, a?, a?). 



HISTORY OP ELEMENTARY ALGEBRA 257 

6. Vieta (1591) : A^ A quadratus^ A cuius (for a:, a?, ofi^. 

7. Harriot (England, 1631) : a, aa, aaa. 

8. Herigone (er'igon) (France, 1634): a, a2, a3. 

9. Descartes (France, 1637) : a;, a:^, ofi. 

Wallis (England, 1659) first justified the use of frac- 
tional and negative exponents, though fractional exponents 
had been suggested earlier by Oresme (o rem) (1350), 
and negative exponents by Choquet (shoka) (c. 1500). 

Newton (England, 1676) first used a general exponent, 
as in a;*, where n denotes any exponent, integral or frac- 
tional, positive or negative. 

186. SymbolB for Known Quantities. 

1. Diophantus: fiovaBe; (i.e. monads), or /a®. 

2. Regiomontanus (re ji 5 mon ta'nus) (Germany, 1430) : 
letters of the alphabet. 

3. Italians: df, from dragma. 

4. Bombelli: O. 

5. Stevinus: ©. 

6. Vieta: consonants, -B, O^ D^ F^ »•• 

7. Descartes : a, 6, c, d. 

Descartes possibly used the last letters of the alphabet, x, y, z, to 
denote unknown quantities because these letters are less used and less 
familiar than afb,c,d, •••, which he accordingly used to denote known 
numbers. 

187. Addition Sign. The following symbols were used: 

1. Egyptians: pair of legs walking forward (to the 
left), _A. 

2. Diophantus: juxtaposition (thus, a6, meant a + 6). 

3. Hindoos: juxtaposition (survives in Arabic arithme- 
tic, as in 2|, which means 2 + |). 

4. Italians: plus^ then p (or e, or <^). 

5. Germans (1489): + +, +. 



258 DURBLL'S ALGEBRA : BOOK TWO 

188. SnUnetion Sign. 

1. Egyptians: pair of legs walking backward (to the 
right), A_; or a flight of arrows. 

2. Diophantus: ^ (Greek letter ^fr inverted). 

3. Hindoos: a dot over the subtracted quantity (thus, 
mh meant m — n). 

4 Italians: minus^ then Morm or de. 
5. Germans (1489): horizontal dash, — . 

The signs + and — were first printed in Johann Widman's Mer- 
cantile Arithmetic (1489). These signs probably originated in 
Grerman warehouses, where they were used to indicate excess or 
deficiency in the weight of bales and chests of goods. Stifel (1544) 
was the first to use them systematically to indicate the operations of 
addition and subtraction. 

189. Hnltiplication Sign. Multiplication at first was 
usually expressed in general language. But 

1. Hindoos indicated multiplication by the syllable bhcL, 
from bharita^ meaning " product,'* written after the factors. 

2. Oughtred (ot'red) and Harriot (England, 1631) in- 
vented the present symbol, x. 

3. Descartes (1637) used a dot between the factors 
(thus, a -b). 

190. Division Sign. 

1. Hindoos indicated division by placing the divisor 
under the dividend (no line between) Thus, 5 meant 
c-i- d. 

2. Arabs, by a straight line. Thus, a — 6 or a | 6, or -. 

3. Italians expressed the operation in general language. 

4. Oughtred, by a dot between the dividend and 
divisor. 

5. Pell (England, 1630), by -^. 



HISTORY OP ELEMENTARY ALGEBRA 259 

191. Equality SigiL 

1. Egyptians: Z □ (also other more complicated sym- 
bols to indicate diflferent kinds of equality). 

2. Diophantus: general language or the symbol, \. 

3. Hindoos: by placing one side of an equation imme- 
diately under the other side. 

4. Italians: ce ova; that is, the initial letters of cequalia 
(equal). This symbol was afterward modified into the 
form, 30, and was much used, even by Descartes, long after 
the invention of the present symbol by Recorde. 

5. Recorde (England, 1540): =. 

He says that he selected this symbol to denote equality because 
^Hhan two equal straight lines no two things can be more equal." 

192. Other SymbolB used in Elementary Algebra. 

Inequality Signs, > <, were invented by Harriot (1631). 

Oughtred, at the same time, proposed "H, JU as signs of in- 
equality, but those suggested by Harriot were manifestly superior. 

Parenthesis, ( ), was invented by Girard (1629). 

The Vinculum had been previously suggested by Vieta 
(1591). 

Eadical Sign. The Hindoos used the initial syllable of 
the word for square root, Ka^ from Karania^ to indicate 
square root. 

Rudolph (Germany, 1525) suggested the symbol used 
at present, V (S^^ script form, of the initial letter, r, of 
the word radix^ or root), to indicate square root, /w to de- 
note the 4th root, and A^ to denote cube root. 

Girard (1633) denoted the 2d, 3d, 4th, etc., roots, as at 
present, by ^, ■^, ^, etc. 

The sign for Infinity, oo, was invented by Wallis 
(1649). 



260 DURELL'S ALGEBRA : BOOK TWO 

193. Other Algebraic Symbols have been invented in recent 
times, but these do not belong to elementary algebra. 

Other kinds of algebra have also been invented, employ- 
ing other systems of the symbols. 

194. General IllastratioiL of the Evolntioii of Algebraic Sym- 
bols. The following illustration will serve to show the 
principal steps in the evolution of the symbols of algebra: 

At the time of Diophantus the numbers 1, 2, 3, 4, ••• were denoted 
by letters of the Greek alphabet^ with a dash over the letters used ; 

as, a,Ayi- 

In the algebra of Diophantus, the coefficient occupies the last place 
in a term instead of the first as at present. 

Beginning with Diophantus, the algebraic expression, 
a? + 6 a; — 4, was expressed in symbols as follows: 

S^a^Se^i fi^'B (Diophantus, 350 A.D.). 

lzp.5 Em A (Italy, 1500 a.d.). 

1^ + 5 N- 4 (Germany, 1575). 

lii; p.5^; ?».4vo; (Bombelli, 1579). 

1® + 5® - 4© (Stevinus, 1586). 

1 ^2 + 5 ^ - 4© ( Vieta, 1591). 

1 aa + 5 a - 4 (Harriot, 1631). 

1 a2 + 5 al - 4 (Herigone, 1634). 

a? + 5x-4: (Descartes, 1637). 

195. Three Stages in the Development of Algebraic Symbols. 

1. Algebra without SymbolB (called Rhetorical Algebra). 
In this primitive stage, algebraic quantities and operations 
were expressed altogether in words, without the use of 
symbols. The Egyptian algebra and the earliest Hindoo, 
Arabian, and Italian algebras were of this sort. 

2. Algebra in which the Symbols are Abbreviated Words 
(called Syncopated Algebra). For instance, jt? is used for 
plus. The algebra of Diophantus was mainly of this 



HISTORY OP ELEMENTARY ALGEBRA 261 

sort. European algebra did not get beyond this stage 
till about 1600 a.d. 

3. Symbolic Algebra. In its final or completed state, 
algebra has a system of notation or symbols of its own, 
independent of ordinary language. Its operations are 
performed according to certain laws or rules, "inde- 
pendent of, and distinct from, the laws of grammatical 
construction." 

Thus, to express addition in the three stages we have 
plus^ p^ + ; to express subtraction, minus^ w, — ; to ex- 
press equality, cBqualu^ cb, = . 

Along with the development of algebraic symbolism, 
there was a corresponding development of ideas of alge- 
braic quantity and of algebraic processes. 

II. History of Algebraic Quantity 

196. The Kinds of Quantity considered in algebra are 
positive and negative ; particular (or numerical) and 
general ; integral and fractional ; rational and irrational; 
commensurable and incommensurable ; constant and vari- 
able ; real and imaginary. 

197. Ahmes (1700 B.C.) in his treatise m^q^ particular^ 
positive quantity, both integral and fractional (his frac- 
tions, however, are usually limited to those which have a 
unity for a numerator). That is, his algebra treats of 
quantities like 8 and |^, but not like — 3, or — |, or V2, 
or —a. 

198. Diophantus (850 a.d.) used negative quantity, but 
only in a limited way ; that is, in connection with a 
larger positive quantity. Thus, he used 7 — 5, but not 
5 — 7, or — 2. He did not use, nor apparently conceive 
of, negative quantity having an independent existence. 



262 DURELL'S ALGEBRA : BOOK TWO 

199. The Hindoos (500 A.D.-1200 a.d.) had a distinct 
idea of independent or absoltUe negative quantity, and used 
the minus sign both as a quality sigQ and a sign of opera- 
tion. They explained independent negative quantity 
much as it is explained to-day by the illustration of debts 
as compared with assets, and by the opposition in direc- 
tion of two lines. 

Pythagoras (Greece, 520 B.C.) discovered irrational 
quantity, but the Hindoos were the first to use this in 
algebra. 

200. The Arabs avoided the use of negative quantity as 
far as possible. This led them to make much use of the 
process of transposition in order to get rid of negative 
terms in an equation. Their name for algebra was "al 
gebr we'l mukabala," which means "transposition and 
reduction." 

The Arabs used surd quantities freely. 

201. In Europe the free use of absolute negative 
quantity was restored. 

Vieta (1591) was principally instrumental in bringing 
into use general algebraic quantity (known quantities 
denoted by letters and not figures). 

Cardan (Italy, 1545) first discussed imaginary quanti- 
ties, which he termed " sophistic " quantities. 

Euler (oi'ler) (Germany, 1707-1783) and Gauss (gous) 
(Germany, 1777-1855) first put the use of imaginary quan- 
tities on a scientific basis. The symbol i for V— 1 was 
suggested by Gauss. 

Descartes (1637) introduced the systematic use of 
variable quantity as distinguished from constant quantity. 
This led to the construction of graphs in connection with 
algebraic equations. 



HISTORY OP ELEMENTARY ALGEBRA 263 

III. History of Algebraic Processes 

202. Solution of Equations. Ahmes solved many simple 
equations of the first degree^ of which the following is an 
example : 

"Heap its seventh its whole equals nineteen. Find 
heap." In modern symbols this is, 

Given - -|- a; = 19 ; find x. 

The correct answer, 16J, was given by Ahmes. 
Hero (Alexandria, 120 B.C.) solved what is in effect the 
quadratic equation^ 

where d is unknown, and s is known. 

Diophantos solved simple equations of one unknown 
quantity, and simultaneous equations of two and three un- 
known quantities. He solved quadratic equations much 
as is done at present, completing the square by the 
method given in Art. 99. However, in order to avoid 
the use of negative quantity as far as possible, he made 
three classes of quadratic equations, thus, 

Iax^ -h 6a; = c, 
aofi + c =bxy 
aa? =sbx+ c. 

In solving quadratic equations, he rejected negative 
and irrational answers. 

He also solved equations of the form ax"^ = 6a;*. 

He was the first to investigate indeterminate equations^ 
and solved many such equations of the first degree with 
two or three unknown quantities, and some of the second 
degree. 



264 DURELL'S ALGEBRA : BOOK TWO 

The Hindoos first invented a general method of solving a 
quadratic equation (now known as the Hindoo method, 
see Art. 106). They also solved particular cases of 
higher degrees, and gave a general method of solving 
indeterminate equations of the first degree. 

The Arabs took a step backward, for in order to avoid 
the use of negative terms, they made six cases of quadratic 
equations ; viz. : 

03^ = 6a;, aa? + 6a: = c, 

aa? = (?, aofi + £? = 6a;, 

6a; = (?, aa? =hx-\-c. 

Accordingly, they had no general method of solving a 
quadratic equation. 

The Arabs, however, solved equations of the form 
aa;2p -|_ j^p = (?, and obtained a geometrical solution of 
cubic equations of the form a? +px + q=^Q. 

In Italy, Tartaglia (tar tal'ya) (1500-1559) discovered 
the general solution of the cubic equation, now known as 
Cardan's solution. Ferrari (f er ra're), a pupil of Cardan, 
discovered the solution of equations of the fourth degree. 

Vieta discovered many of the elementary properties of 
an equation of any degree; as, for instance, that the num- 
ber of the roots of an equation equals the degree of the 
equation. 

203. Other Processes. Methods for the addition^ subtrac- 
tion, and multiplication of polynomial expressions were 
given by Diophantus. 

Transposition was first used by Diophantus, though, as a 
process, it was first brought into prominence by the 
Arabs. The word algebra is an Arabic word and means 
"transposition" (jil meaning "the," and ^e6r meaning 
"transposition"). 



HISTORY OF ELEMENTARY ALGEBRA 266 

The Greeks and Romans had a very limited knowledge 
of fractions. The Hindoos seem to have been the first to 
reduce fractions to a common denominator. 

The square and cube root of polynomial expressions were 
extracted by the Hindoos. 

The methods of using radicals, including the extrac- 
tion of the square root of binomial surds and the rational- 
izing of the denominators of fractions, were also invented 
by the Hindoos. 

The methods of using fractional and negative exponents 
were determined by Wallis (1659) and Sir Isaac Newton. 

The three progressions were first used by Pythagoras 
(pi thag'6 ras) (569 B.C.-500 B.C.). 

Permntations and combinations were investigated by Pascal 
(pas'kal) and Fermat (fer ma) (France, 1654). 

The binomial theorem was discovered by Newton (1655), 
and, as one of the most notable of his many discoveries, 
is said to have been engraved on his monument in West- 
minster Abbey. 

Graphs of the kind treated in this book were first in- 
vented by Descartes (France, 1637). 

Logarithms were invented by Lord Napier (Scotland, 
1614) after a laborious search for means to diminish the 
work involved in numerical computations, and were im- 
proved by Briggs (England, 1617). 

The fundamental laws of algebra (the Associative, Com- 
mutative, and Distributive Laws ; see Arts. 204-205) were 
first clearly formulated by Peacock and Gregory (Eng- 
land, 1830-1845), though, of course, the existence of these 
laws had been implicitly assumed from the beginnings of 
the science. 



APPENDIX 
Fundamental Laws of Algebra 

201. The following Laws of Algebra have been used in 
the preceding pages without formal statement : 

A. The Commutative Law (or Law of Order). 

1. For addition^ a + 6 = 6 -f a. 

2. For multiplication^ ah = ha, 

3. For division^ a-hhxc=axc-i-h. 

B. The Associative Law (or first Law of Grouping). 

1. For addition^ a 4- J 4- ^ = « 4- (J 4- c) = (a + 5) -f {?. 

2. For multiplication^ ahc = a(hc') = (aJ)c. 

C. The Distributive Law (or Second Law of Grouping). 

1. For multiplication^ a(h + c')^ah-{- ac. Hence, 

inversely, ab-^-ac^ a(h 4- c). 

2. Jbrdivmow, ^^ = - + -. 

a a a 

Who first formulated the laws of algebra? (See p. 
265.) 

205. TTtility of the Laws of Algebra. The laws stated 
in Art. 204 are methods adopted for arranging and 
grouping algebraic symbols so as to decrease the amount 
of work and to increase the importance of the results 
attained . 



CUBE ROOT 267 

Thus, in the following example we are able to eliminate the 
parenthesis by use of the Distributive Law and to collect terms by 
use of the Commutative and Associative Laws. 

Ex. 6(ar + y)+3(j:-y + 2)+2(x+2y-2). 

= 6a: + 6y + 3ar-3y + 32 + 2j: + 4y-2z 
= 6a:4-3a: + 2x + 6y-3y + 4^ + 32-22 
= lla: + 7y4-z. Arts. 

The use of these laws enables us to diminish the 23 symbols used 
in the first expression to the 8 symbols used in the last expression. 

It should be noted that by changing the laws stated in Art. 204, 
kinds of algebra different from that presented in this book, and 
adapted to other uses, may be devised. Thus, in a certain important 
kind of algebra ab = — ba, not ba. 

Even in arithmetic the Commutative Law holds only in a limited 
way. For, while 5 x 7 = 7 x 5, 57 does not equal 75. 

Cube Root 
2M. Cube Root of FolynomialB. A general method for 
determining the cube root of any polynomial which is a 
perfect cube may be found by studying the relation be- 
tween the terms of a binomial — or, in general, of a poly- 
nomial — and the terms of its cube (as a + b^ and its cube, 
cfi + S a^b + S al^ -{- b^}. This relation stated in the inverse 
form gives the method for extracting the cube root. 

The essence of this method consists in writing cfi + SaH 
-h 3 a62 4- 68 in the form o^ + 6(3 «« + 3 aJ 4- 6^). 
Ex. Extract the cube root of afi-^-Sx^^da^-^-Bx— I. 

\x^-\- x-1 Root 
a:«+3a:6-5a;« + 3j:- 1 

x^ 

3(x2)^=3x* 



3(a:2) j:+ x^ = +3a:»+ 



Sx^-\-^x*-^x^ 



Complete divisor = 3 a;* + 3 a;« + x^ 

3(a:2+a:)2=3a:*+6a:«+3a:3" 
3(x3+a:)(- l) + (-l)2= -3a:g-3ar + l 



Complete divisor = 3a:* + 6j:*-3a:-f-l 



-3x*-6a:«+3x-l 
-3x*-6x«+3x-l 



Let the pupil state this process as a formal rule. 



268 DUBELL'S ALGEBRA : BOOK TWO 

BZESCISB M 
Find the cube root of 
X. efi + 6ah; + 12aa? + Safi 2. 27 -27 a+9a'- cfi 

3. a«-3a»-3<i* + lla8+6a»-12a-8 

4. 12a:*-36a:+64a^-6a^-8 + 117a«-144a:6 

5. 95a8+72a*-72aa + 15a» + 15a + a«-l 

6. 114«*-171a?-27-135x+.8a^-60a;» + 55a:8 

X or or 

a '62fi 15s^ 45a^ 27a: 27 lOafi 
• "^ y "^ 2ya 4y4"^2/ 8/ f 

207. Cube Boot of Arifhmetical Vumbers. The same 
general method as that used in Art. 206 can be used to 
extract the cube root of arithmetical numbers. The 
process is slightly different from the algebraic process, 
owing to the fact that all the numbers which compose a 
given cube are united or fused into a single number. 

Thus, (42)« =(40 + 2)« = 40« + 3 X 402 X 2 + 3 X 40 X 22 + 2« 
= 64000 + 9600 + 480 + 8 
= 74088 

Reversing this process, we obtain a method of extracting the cube 
root of a number. 

Ex. 1. Extract the cube root of 74088. 

^4088 [42 Root 
40« = 64 
Trial divisor, 3 x 402 = 4300 

3 X 40 X 2 = 240 

22 = 4 

Complete divisor, _ 5944 



10088 



10088 



CUBE ROOT 



269 



Ex. 2. 



Extract the cube root of ^^ to 4 decimal places. 
^ = .416666666666+ 



.416666666+ | .7469+ R oot 
343 



3 X (70)« = 14700 

3 X (70 X 4) = 840 

4«= 16 



15556 

3 X (740)« = 1642800 

3 X (740x6)= 13320 

62= 36 



1656156 
3 X (7460)2 3, 166954800 



73666 



62224 



11442666 



1505730666 
1502593200 



3137466 



The first three figures of the root are found directly. The last 
figure is then found by division of the remainder, using three times 
the square of the root already found as a divisor. The number of 
figures of the root that may thus be found by division is two less 
than the number of figures already found. 

Let the pupil state the above process as a rule. 

EZ£RCISE 96 

Find the cube root of 

1. 3375 4. 43,614,208 7. 344,324.701729 

2. 753,571 5. 32,891,033,664 8. .000127263527 

3. 1,906,624 6. 520,688,691.125 9. .991026973 
Find to three decimal places the cube root of 

10. 75 12. 5.6 14. 7^ 16. ^ 18. 1^ 

11. 6 13. 3| 15. 19J 17. ^^j 19. 8^ 
Compute the value of 

V3V10-2VT0 



20. 



1. V5 + 2^/5 



21. 



22. 



V3V.8-2VT:936 



270 DURELL'S ALGEBRA : BOOK TWO 

Visualize the following objects by the aid of cube root : 

23. 150,000,000 cu. yd. of earth. 

24. 40,000,000,000 feet of lumber. 

25. 60,000,000 tons of iron (taking 480 lb. as the weight 
of one cubic foot of iron). 

26. Make up and work an example similar to Ex. 12. 
To Ex. 23. Ex. 25. 

EXERCISE 97 

General Review 

1. Factor: 

(1) a* + 4 (2) x*-6a:V + y* 

(3) m^ - 2 mn + n2 + 5m - 5 n 

(4) aa - n« - m2 - 2ah + 2m6 + &* 

(5) x{x + \){x -f 2) -3 a; - 3 

(6) ar* + 2x«- 13x2 -14x4- 24 

2. Simplify ra^-^^mn+^n^ ^ 8mi-_18n_a ^ 2rn±%n ^ 

^ ^ 2m -3n 3m8-24n« m^^^n' 

3. Solve -^ L = _J_, _^ + J^=. ^ 



pJtq p-q p-\-q p -^ q p-q p-q 

4. Factor : 

(1) a:* -9 (4) ar^" - r ' 

(2) a:* +27 (5) a*- 8ft-i 

(3) 4x-y2 (6) 25n«-y-a 

5. Factor: 

(1) 3x- 8x^-35 (4) aM-Sa* + 5x^-15 

(2) lOx*- 19x^-56 (6) 60-7V3a-6a 

(3) 12x* + 5x*-72 (6) 15x-2V^-24y 

6. (a-6)-i+(a + 6)-> 

- (a + 6)(a - 6)-i - (a - h)(a + 6)-i 
l_(a2 4.62)(a + 6)-2 

8. Solve v^x^nr+ 2 VT^Ti -1 = 0. 



GENERAL REVIEW 271 

9. By letting a, 6, c, etc., have special values, convert ax^ + bxy 
-}- cy^ + dx + ey -\- f = into 

(1) a homogeneous equation of the second degree. 

(2) a symmetrical equation of the second degree. 

(3) a homogeneous symmetrical equation of the second degree. 

Solve: 

10. ^^ =3-^^ 
X- 1 3a: 

11. x^ + ^xy + 2y2 = U, 4x2- 2xy-Sy^ + 9 = 

12. 1 + 1=2 !*• V'xTy + Vx-y = 4 

X + y = 10 

13. x + y + V^ = 14: 16. Vxy-Vx^=ll 
y/xy{x + y) = 40 Vx — y Vxy = 60 

17. Write the equations whose roots are 1 ± t. Also J ± } t. 

18. Find, by inspection, the sum of the roots of 3 x^ — 2 x + 1 = 0. 
Find also the product of the roots. Verify your result by solving 
the given equation. About how much shorter is the first process 
than the second ? 

19. What must be added to each of the terms of a^ : &^ to make 
the resulting ratio equal to a : 6 ? 

20. If a box car 36' x 8i' x 8' has a capacity of 60,000 lb., by 
how much must the length be increased to make the capacity 
100,000 lb.? 

21. The rates of two boys traveling on bicycles are as p to q. 
If the first boy rides a miles in a given time, how far does the other 
boy travel in the same time ? 

22. For what value of x will the ratio x^ — x + lix^+x+l be 
equal to 3 : 7 ? 

23. If g = £ = £ = g, prove g = 5a -f- 3c - 26 

b d f q^ q 5b + dd-2f 

24. lia:b = c:df show that ab + cd is a mean proportional between 
a2 + c2 and b^ + d^ 



272 DURELL'S ALGEBRA : BOOK TWO 

25. Find the sum of 30 terms of the A. P. 3, 5, 7, ••• by the ad- 
dition of snccessive terms. Now find this sum by the use of one of 
the formulas of Art. 142. Compare the amount of work in the two 
processes. 

26. Prove that the differences between the squares of successive 
integers form an A. P. 

27. Prove that equimultiples of the terms of an A. P. form 
another A. P. 

28. Obtain a formula for the nth term of the A. P. 9, 7, 5, ••• . 
Also for the n + 2d term. 

29. If the hours of the day were numbered from 1 to 24, how 
many times would a clock strike in striking the hours during one day? 

30. If each stroke of an air pump removes } of the air in a receiver, 
what fraction of the air will be left in the receiver after 10 strokes ? 

31. Find a G. P. in which the sum of the first two terms is 2} and 
the sum to infinity is 4|. 

32. Find the 7th term of f 2 -x^ ^ J by finding all the terms 

up to the 7th. Now find the 7th term by the method of Art. 158. 
Compare the amount of work in the two processes. 



33. Expand ( Vx + 1 - Vx - 1)*. 

34. Find the two middle terms of ( 2v^ ) • 

\ 2a:V 

35. In the equation 10 x^ - (ifc + 19) a: + ifc = 0, what value must 
k have in order that the roots be reciprocals of each other? Find 
these roots. 

36. Find n when the coefficients of the 4th and 6th terms in the 
expansion of (1 4 a:)* are equal. 

(2 X* 3 \ * 
— TT-) f fi^d the coefficient of x*. 
o 2 x/ 



38. Solve Vx+T + Vx - 2 = V2x + 3. 

39. Find the sum of all positive integers of three digits which are 
divisible by 9. 

40. Extract the square root of 4x" + 9x-» + 28 - 24x « - 16x«. 

41. Solve x«-fx-4ax + 3a2-5a-2 = 0. 



GENERAL REVIEW 273 

42. Simplify: (1) ?_±.2V-J.. (2) (2 + 3vC-l)(3 - 4V3T). 

43. The sum of 5 terms of an A . P. is — 5, and the 6th term is 
— 13. What is the common difference? 

44. What is the ratio of the mean proportional between a and b to 
the mean proportional between a and c ? 

45. Form an equation whose roots are =— 

46. A boat crew, rowing at half their usual speed, row 3 mi down- 
stream and back again in 2 hr. 40 min. At full speed, they can 
go over the same course in 1 hr. 4 min. Find, in miles per hour, the 
rate of the crew and of the current. 

47. Simplify 8"* + 25* - (i)-« + 13° - (^j)"* + 1"*. 

48. Show that the sum of the squares of the roots of the equation 
x2 _ 5 X + 2 = is 21. 

49. The mean annual rainfalls at Phoenix (Ariz.), Denver, 
Chicago, and New Orleans are 7.9 in., 14 in., 34 in., and 57.4 in. re- 
spectively. By how much do these numbers differ from the corre- 
sponding terms in a G. P. whose first term is 7.9 in., and whose ratio 
is2? 

50. Solve in the shortest way — i— + — ^ + — ^r + -^ = 0. 

51. If a:b = c:d=e:f, show that a« + c« + e» : 68+ rf» + /« = 
ace : bdf. 

52. Without extracting roots, determine which is greater, vY or Vs. 

53. Free the following from radicals, and find the value of x when 
5 = 0: ____^^^ 

54. By use of the binomial theorem, find the ratio of the 5th 
term to the 7th term in the expansion of (1 — V2 xy. 

Simplify : 

1-ahi . (1-faM)-^ gg a"26-}-5a-^\/6-66 

hiVi-ab' a-ib ' a-26 + 3a-iV6-54 

57. (.09)"*. Also (- .064)*. 



274 DURELL'S ALGEBRA : BOOK TWO 

58. Which term of the series j, f , f , etc., is 8 ? 

59. Solve x« - 3 X - 6Vj:2-3j:-3 + 2 = 0. 

GO. Show that the roots of the equation x^ + ox — 1 = are real 
and unequal for any real value of a. 



61.Solve(x+iy = 4-H(l-i)(l+l). 



62. Find four numbers in A. P., such that the sum of the first 
and third shall be 18, and the sum of the second and fourth shall be 
30. 

63. Find the G. P. whose sum to infinity is 4 and whose second 
term is }. 



64. Solve —^-fl+—V+- + - = 0. 
in-\' n \ mnl m n 



65. Find the sum to infinity of — 3 + j — ^^^ .... 

66. Given K = vR^, and C = 2 ir/2, eliminate R and find K in 
terms of C. 

67. Given 5 = wRL and T = irR(R + X), eliminate R and find 
T in terms of S and L, 

68. Simphfy ^^— ^ ^. 

69. A man sold a horse for % 96 and, in doing so, gained as many 
per cent as the horse cost him dollars. What did the horse cost him ? 

70. Solve X + y + Vx + y = 20, xy = 63. 

71. Given / = distance iu feet between two adjacent supports of a 
trolley wire, 

s = sag of the wire in feet, 
t = tension of the wire in pounds, 
w = weight of wire in pounds, 
/' = actual length of wire between two adjacent supports, 

and (a)*=^... and (6) ^' = '+|f» 

(1) Find the value of / in equation (a). Also in equation (h). 

(2) Eliminate / between the two equations. 

72. Expand (2 V3 - jV^)* and simplify. 

73. The sum of the first seven terms of a G. P. is 635, and the 
ratio is 2. What is the fourth term? 



GENERAL REVIEW 275 

74. Divide 2 x^y^ - 5 x^y-^ + 7 x^y-^ - 5 x- + 2 ar^y by a:»y-» - 
x^-2 4 ary-i. 

75. Find the value of x* - 6 x + 14, if a: = 3 + V^. 

76. If a and p are the roots of the equation jox^ + ^x + r = 0, find 
the values of a + )3, a — )3, and a)3 in terms of p, 7, and r. 

77. A man finds that it takes him 2 hours less to walk 24 miles, 
if he increases his speed 1 mile per hour. What is his usual rate? 

78. If a:b = b:c, prove that a + J:6 + c = ft'; ac^. 

79. Insert four geometric means between 160 and 5. 

80. How many terms of the A. P. 42, 39, 36, ••• must be taken to 
make 315? 

81. Express the repeating decimal .3232 ••• as a fraction. 

82. Solve (x-2 + })-» = 27. 

83. Solve 9 x2 + 25 y2 = 148, 5 xy = 8. 

84. The first term of a geometrical series is 2 and the sum of the 
fourth term and three times the second term is equal to four times 
the third term. Find the series. 

85. Solve (xi - x)(x + 2)= 0. 

86. Solve ^ - 2^ = 81, X - « = 2. 

y X 

87. If --^— = — L = «£__, show that x - y + z = 0. 

b + c c + a a — b 

88. Solve x(x - y) = 0, x^ + 2 xy + y^ _ 9 ^y the factorial method 
as far as possible. 

89. By the factorial method, solve 

(y + 3x - 7)y = 0, (y + X - 3)(y + 2x - 4)= 0. 

90. Solvexl + 3x-i = 4. 

91. Solve 2 x^ - 5 nx = 3 n', both by factoring and by completing 
the square. 

92. Solve xV - lOxy + 24 = 0, X + y = 5. 

93. The sum of the first seven terms of an A. P. is 98, and the 
product of the first and seventh terms is 115. Find the common 
difference. 

94. Solve x^ + xy + y« = 133, x - Vxy + y = 7. 



276 DURELL'S ALGEBRA : BOOK TWO 

95. Find the sum of the odd integers between and 200. How 
many of these are not divisible by 3? 

96. Find the values of x and y which will satisfy the following : 

y X 

97. In an A. P., given a = J, Z = — 2}, a = — 4, find n and d. 

98. Extract the square root of 

(1)7 + 4V3; (2) 3 + V6; (3) 2a + 2y/a^--bK 

99. Plot the graphs of the following system of equations: 
x2 -f y2 — 4^ 3 X — 2.y = 6. From the graphs find the approximate 
values of x and y that satisfy both equations. 

100. Solve ^^ "^ = 1. Determine the value of x (1) when a=:c; 

cx-^d 

(2) when b = d', (3) when a = c and h = d. 

101. Solve Vi + V3 - V3x + ar» = V3. 

102. What is meant by an extraneous root of an equation ? Give 
an example of an extraneous root. 

103. What two numbers, whose difference is h, are to each other 
as a : b'i 

104. What distance is passed over by a ball which is thrown 60 
feet vertically upward and at every fall rebounds | the distance from 
which it fell ? 

105. Solve a^» + y2 + 2(ar + y) = 12, xy -(x + y)= 2. 

106. What number added to both numerator and denominator of 

- , and subtracted from both numerator and denominator of - , will 
b d 

make the results equal ? 

107. Show that — ^ ^ 3[— + — i^= x* + 2 . 

ari - 1 xi + 1 x^ - I a?i + 1 

108. Solve x^ - 2 V7a: + 2 = 0. 
x-y . 1 



109. Simplify '^^^^'w 

\5x+7yJ 



110. Ua-.b = b:c = c:d, show that ( + c is a mean proportional 
between a + b and e + d. 



GENERAL REVIEW 



277 



111. If a, bf c, and d are in A. P., show that a + d :=b + c, 

112. Solve (a: + H.ar-i)(a;-l+x-i)=5i. 

113. The following table gives the normal or average height of a 
boy and girl at different ages : 



Age in years . 


3 


6 


9 


12 


16 


18 


21 


Height of boy . 
Height of girl . 


2' 11" 
2' 11" 


3' 8" 
8' 7" 


4' 2" 
4' 2" 


4' 7i" 
4' 9" 


5' 2}" 
6' li" 


6' 6}" 
6' 31" 


6' 8V' 
5'3f" 



Graph the above facts as two graphs on one diagram. 
From these graphs, determine, as accurately as you can, the normal 
height of a boy and of a girl at 10 years of age. At 14 years. 

114. Insert between 1 and 21 a series of arithmetical means such 
that the sum of the last three shall be equal to 48. 

115. Find the coefficient of ar» in the expansion of f a:^ — 2- J . 

116. Prove that either root of the quadratic equation a:^ — g = is 
a mean proportional between the roots oi x^ + px + q = 0. 

117. Simplify (VaT^ + Vi"^^)« +(Vir}r6 --Va^^y. 

ft 

118. Solve X + V = 6 — xvy x + y = — 

119. Find the sum of n terms of the series 



<'-'H^.-si<$-iy 



120. The formula used for determining the elevation of the outer 
rail of a railroad track on a curve is as follows : E = , where 

E =z elevation of outer rail in inches, 

B = width of track in feet, 

R = radius of the curve in feet, 

V = maximum speed in miles per hour of a train taking the curve. 

Find E when B = 4 f t. 8J in., R = 425 ft., F = 20 mi. per hour. 
Also when F = 60 mi. per hour. 

121. Solve the formula for F. From this result determine the 
maximum speed at which a train can take the track when ^ = 5 in. 



MATERIAL FOR EXAMPLES 

Formulas 

Formulas used in the following subjects may be made the 
basis for numerous examples. 

I. Arithmetic 

P-ir 4(l+r)"-l] 
t — prt ^ ;; 



o = p + prt 



II. Geometry 



K = WV3 T = 27rR{R+H) 

K = 7rRL 8 = "^^ 
8 = 47riP 180 

K = Vsis - a) (* - b) (* - c) 

V = ^H{B + b + VBb) 

III. Physics 



8 = f flT ,. 

* = r< + i^ ^ 

278 



<=V- 



MATERIAL FOR EXAMPLES 279 

2a g + 8 

e=— 1 = 1 + 1 

bh^m f P T^ 

H = .u(?m 

g ' 4n^Pw 

C = ^ "^ 

R C = f(F-32) 

IV. Engineering 

H. P. = ^ ^^ (horse-power in an engine) 

s = -^ and ^' = ^ + ^ (sag in a suspended wire) 
or 3/ 

jB = .,-, (elevation of outer rail on a curve) 
oK 

B XD^ 

W = = — k (weight a beam will support) 

Ij 

L = ^^ :^ — (length of hot-water pipe to heat 

a house) 

D^PL 
T = --- (tractive force of a locomotive) 
yy 



G = — — (no. gal. water delivered by a pipe) 

Ij 

D = y (diameter of a pump to raise a given 

amount of water) 

v ^ 

D = \ fi<}oa(A_n\ (diameter of balloon to raise a given 
weight) 



280 



DURBLL'S ALGEBRA: BOOK TWO 



Impohtant Numerical Facts 

Areas 

8q. Mi. 

Rhode Island 1250 

New Jersey 7815 

New York 49,170 

Texas 265,780 

United States 3,025,600 

North America 6,446,000 

Land surface of earth 51,238,800 

Great Britain and Ireland 121,371 

France 207,054 

Europe 3,555,000 



Astronomical Facts 



Planet 


Diameter 


Distance from Sun 


Time of Revolu- 


Synodic 
Period 
in Days 


in Miles 


in Million Miles 


tion about Sun 


Mercury 


3030 


36 


88 da. 


116 


Venus 


7700 


67.2 


225 da. 


584 


Earth 


7918 


92.8 


365 da. 




Mars 


4230 


141.5 


687 da. 


780 


Jupiter 


86,500 


483.3 


11.86 yr. 


399 


Satum 


73,000 


886 


29.5 yr. 


378 


Uranus 


31,900 


1781 


84 yr. 


369 


Neptune 


34,800 


2791 


165 yr. 


367 



Sun's diameter 866,400 mi. 

Moon's diameter 2162 nu. 

Moon's distance 238,850 mi. 

Distance of nearest fixed star, 21 millions of millions of 
iniles (or 3.6 light years). 



MATERIAL FOR EXAMPLES 



281 



Dates (a. d. unless otherwise stated) 



Rome founded . . 753 b. c. 
Battle of Marathon 490 b. c. 
Fall of Jerusalem . . 70 
Fall of Rome .... 476 
Battle of Hastings . . 1066 
Printing with movable 

type 1438 

Fall of Constantinople 1453 
Discovery of America 1492 
Jamestown foimded. . 1607 



Declaration of Indepen- 
dence 1776 

Washington inaugurated 1789 
Battle of Waterloo . . 1815 
Telegraph invented . . 1844 
First transatlantic cable 

message 1858 

Telephone invented . . 1876 
Battle of Manila Bay . 1898 



Distances 



From New York to Miles 

Boston 234 

Buffalo 440 

Chicago 912 

Denver 1930 

San Francisco .... 3250 



From New York to Miles 

Philadelphia 90 

Washington 228 

New Orleans 1372 

Havana 1410 

London 3375 



San Francisco to Manila 4850 

New York to San Francisco via Panama 5240 

London to Bombay via Suez 6332 



Heights of Mountains 





Feet 


Mt. Washington . 


. 6290 


Pike's Peak . . . 


. 14,147 


Mt. McKinley . . 


. 20,464 


Mt. Everest . . . 


. 29,002 



Fed 
Mt. Mitchell .... 6711 
Mt. Whitney. . . . 14,501 

Mt. Blanc 15,744 

Acongua 23,802 



282 



DURELL'S ALOEBRA: BOOK TWO 



Heights (or lengths) of Sthuctdbes 



Feet 
Bunker Hill Monument. 221 
Washington Monument 555 
Singer Building (N.Y.). 612 
Metropolitan Life 

Building 700 

Eiffel Tower 984 



Olympic .... 


. 882 ft. 


Deepest shaft . . 


. 5000ft. 


Deepest boring . 


. 6673 ft. 


Simplon Tunnel . 


. 12imi. 


Panama Canal . 


. 49 mi. 


Suez Canal . . . 


. 100 mi. 



Lengths of Rivers 

Miles Miles 

Hudson 280 Mississippi 3160 

Ohio 950 Rhme 850 

Colorado 1360 Amazon 3300 

Missouri 3100 Nile 3400 



Rainfall (mean annual) 

Inches Inches 

Phoenix (Ariz.) ... 7.9 New York 44.8 

Dienver 14 New Orleans .... 57.4 

Chicago 34 Cherrapongee (Asia) . 610 

Records (year 1910) 

100-yard dash 9f sec. 

Quarter-mile run 47 sec. 

Mile run 4 m. 15f sec. 

Mile walk 6 m. 29^ sec. 

Rimning high jump 6 ft. 5f in. 

Running broad jump 24 ft. 7i in. 

Pole vault 12 ft. lOjinu 



MATERIAL FOR EXAMPLES 283 

100-yard swim 55f sec. 

l-ndle swim 23 m. IQ^ sec 

100-yard skate 9^ sec. 

l-mile skate 2 m. 36 sec. 

1 mile on bicycle 1 m. 5 sec. 

1 mile in automobile 27J sec. 

1 mile by running horse 1 m. 35f sec. 

1 mile by trotting horse in race . 2:03 J m. 

Throw of baseball 426 ft. 6 in. 

Drop kick of football 189 ft. 11 in. 

Transatlantic voyage (from N. Y.) 4 da. 14 h. 38 m. 
Typewriting from printed copy. . 123 words in one minute 
Typewriting from new material . 6,136 words in one hour 

Shorthand 187 words in one minute 

Cost 1 lb. radium $2,500,000 

Com crop per acre 255f bu. 

Milk from 1 cow (1 year) .... 27,4321b. 
Butter from cow (1 year) .... 1164.6 lb. 

Resources (crops, etc., year 1910) 

(All these figures are approximate estimates.) 

Coal lands in U. S 400,000 sq. mi. 

Coal in U. S 2,500,000,000,000 tons 

Iron ore in U. S 15,000,000,000 tons 

Water-power of Niagara 7,000,000 H. P. 

Natural water-power in U. S. . . 75,000,000 H. P. 

Possible water-power in U. S. (de- 
veloped by storage dams, etc.) . 200,000,000 H. P. 

Reclaimable swamp lands in U. S. . 80,000,000 acres 

Lands in U. S. reclaimable by irri- 
gation 100,000,000 acres 



284 DURELL'S ALGEBRA: BOOK TWO 

National forest reserves of U. S. . 168,000,000 acres 

Com crop of U. S 3,000,000,000 bu. 

Wheat crop of U. S 700,000,000 bu. 

Cotton crop of U. S 13,000,000 bales 

Temperatures (Fahrenheit) 



Nonnal temperature of the human body 98.7^ 

Ether boils at 96° Temperature of arc light 5400° 

Alcohol boils at 173° (approx.) 

Water boils at 212° Average change of temperature 

Sulphur fuses at 238° below earth's surface 1° per 

Tin fuses at 442° 62 ft. (increase) 

Lead fuses at 617° above earth's surface 1° per 

Iron fuses at 2800° (approx.) 183 ft. (decrease) 

Velocities 

Wind 18 mi. per hr. (av.) 

Sensation along a nerve .... 120 ft per sec. (av.) 

Sound in the air 1090 ft. per sec. (av.) 

Rifle bullet 2500 ft. per sec. (av.) 

Message in submarine cable . . 2480 mi. per sec. 

Light 186,000 mi. per sec. (approx.) 

Weights 

Boy 12 years old 75 lb. (av.) 

Man 30 years old 150 lb. (av.) 

Horse 1000 lb. (av.) 

Elephant 2^ tons (av.) 

Whale 60 tons (approx.) 

1 cu. ft. of air Ij oz. (approx.) 

1 cu. ft. of water 62.5 lb. 



MATERIAL FOR EXAMPLES 285 

Specific Gravities 

Air ^oT Stone (average) . . . 2.5 

Cork 24 Aluminum 2.6 

Maple wood 75 Glass 2.6-3.3 

Alcohol 79 Iron (cast) 7.4 

Ice 92 Iron (wrought) .... 7.8 

Sea water 1.03 Lead 11.3 

Water 1 Gold 19.3 

Clay 1.2 Platinum 21.5 

Miscellaneous 

Heart beats per minute — Frog 10 

Man 72 

Bird 120 

Smallest length visible to unaided eye. . . -sh^ inch 
Smallest length visible by aid of microscope 1^5,000 inch 
Accuracy of work in a machine shop . . . io\o inch 
Accuracy in most refined measurements . . lo.OoVoOO inch 
Dimensions of double tennis court .... 78 X 36 
Dimensions of single tennis court .... 78 X 27 

Dimensions of football field I6O' X 300' 

Standard width of railroad track 4' 8^'' 

Weights and Measures 

Avoirdupois weight, 1 ton = 2000 lb.; 1 lb. = 16 oz. = 7000 gr. 
Troy weight, 1 lb. = 12 oz. = 5760 gr.; 1 oz. = 20 pwt. = 

480 gr. 
Long measure, 1 mi. = 1760 yd. = 5280 ft. = 63,360 in. 
Square measure, 1 A. = 160 sq. rd. = 43,560 sq. ft.; 1 sq.yd. 

= 9 sq. ft. = 9 X 144 sq. in. 



286 DURBLL'S ALGEBRA: BOOK TWO 

Cubic meaaure, 1 cu. yd. = 27 cu. ft. = 27 X 1728 cu. in. 

Dry measure, 1 bu. = 4 pk. = 32 qt. = 64 pt. 

Liquid measure, 1 gal. = 4 qt. = 8 pt.; 1 pt. = 16 liquid oz. 

Paper meajture, 1 ream = 20 quires = 480 sheets. 

Metric system, 1 meter = 39.37 in.; 1 kilometer = .621 mi. 
1 liter = 1.057 liquid qt.; 1 kilogram = 
2.2046 lb. 
1 hectare = 2.471 A. 

1 kilometer =*= 10 hectometers = 100 decame- 
ters = 1000 meters = 10,000 decimeters = 
100,000 centimeters = 1,000,000 millimeters. 

At the option of the teacher, the pupil may insert on the 
blank pages at the end of the book other important formulas 
or numerical facts, particularly those which are important 
in the locality in which the pupil lives. 



INDEX 



Abbreviated division . 

multiplication . . . 
Abscissa of a point . . 
Absolute term . . . 
Addition in a proportion 
Affected quadratic equa- 
tion • . . . 126, 
Ahmes . . . 255,261, 
Algebra, derivation of . . 

Alternation 

Antecedents . . . 181, 
Arabs . 255,256,258,262, 
Areas, important 
Arithmetic formulas 
Arithmetical means 

progression . . 
Astronomical facts . 
Axes 



Binomial surd 

theorem . . 

BombeUi . . 



256, 



Cardan 262, 

Characteristic . . 241, 

Choquet 

Coefficient of a radical 
Commensurable ratio . . 
Common difference . . . 
Comparison, elimination by 
Completing the square 
Complex fraction . . . 

number 

Compound ratio .... 
Consequents . . . 181, 

Constant 

Continued proportion . . 



19 

17 

68 

146 

188 

128 
263 
262 
188 
183 
264 
280 
278 
203 
197 
498 
68 

114 
221 
257 

264 
242 
257 
106 
182 
198 

49 
128 

37 
118 
182 
183 
234 
183 



287 



Codordinates of a point . 69 

Cube root 267 

of numbers .... 268 

Dates, important . . . 281 
Degree of a radical . . . 107 
Descartes . 256, 257, 262, 265 
Detached coefficients . 11, 13 
Diophantus 256-261, 263, 264 

Discriminant 174 

Distances, important . . 281 
Distributive law .... 266 

Division 13 

Duplicate ratio .... 182 

Egyptians . . 255-259, 261 
Elimination ..... 49 
Engineering formulas . . 279 

Entire surd 107 

Equation 39 

conditional 39 

fractional 39 

linear 71 

literal 39 

numerical 39 

Equivalent equations . . 43 

Euler 262 

Evolution 81, 86 

Exponent, fractional . 92, 93 

law of 81 

negative 92, 95 

Extraneous root .... 44 
Extremes of a proportion 183 

Factor theorem ... 26, 27 
Factorial method of solu- 
tion .... 133, 157 



288 



INDEX 



Factoring .... 22, 178 
Formula method of solution 139 
Formulas, important . . 278 

Fractions 32 

Functions .... 67, 234 

Gauss 262 

Geometrical progression . 208 
Geometry formulas . . . 278 
Germans . . 256, 257, 259 

Girard 259 

Graphs .... 67, 164, 265 

of quadratic equations . 164 

Gregory 265 

Harriot ... 257, 258, 259 

Heights of mountains . . 281 

of structures .... 282 

Herigone 257 

Hero 263 

Highest common factor . 30 
Hindoo method .... 138 
Hindoos 255, 256, 258, 259, etc. 
History of algebra . . . 255 
Homogeneous equation . 147 

Imaginary number . 86, 118 
Important numerical facts 280 

Index law 81,86 

Indicated roots .... 106 

Inequality 228 

conditional 230 

of same kind .... 228 

signs of 228 

unconditional . ' . . . 230 

Infinity 192 

Inverse ratio 181 

variation • 234 

Inversion 188 

Involution 81 

Italians 256,257,258,259,260 



Joint variation 
Key number . 



235 
223 



Laws of algebra . . 265, 266 

Lengths of rivers . . . 282 

Limit 215 

Linear equations ... 71 

Literal quadratic equations 131 
Logarithm .... 239, 265 

table of 246 

Lowest common multiple . 30 

Mantissa .... 241, 244 

Mean proportional . . . 183 

Means of a proportion . . 183 

Measures 284 

Members of an equation . 39 

of an inequality . . . 228 

Multiplication .... 11 

Napier 265 

Newton .... 257, 265 

Numerical facts, important 280 

Ordinate of a point ... 68 

Oresme 257 

Origin 68 

Oughtred .... 258, 259 

Peacock 265 

PeU 258 

Physics formulas . . . 278 

Principal root .... 86 

Progressions 265 

Proportion 182 

Pure quadratic equations 

126, 127 

Pythagoras 265 

Quadrants 69 

Quadratic equation . . . 126 

properties of .... 174 

Quadratic surd .... 114 

Radical equations . • . 137 

Radicals 106 

addition of Ill 

division of Ill 

multiplication of . . . Ill 



INDEX 



289 



Radioand 106 

Rainfall statistics . . . 282 

Ratio 181,209 

Rational root 106 

Rationalizing a denomina- 
tor 113 

Real number ... 86, 118 

Recorde 259 

Records, athletic, etc. . . 282 
Regiomontanus .... 257 
Repeating decimals . . . 216 

Resources 283 

Rhetorical algebra . . . 260 

Root 39,86 

^ principal 279 

Series 197 

Similar radicals .... 107 
Simplification of a radical 107 
Simultaneous equations . 49 

quadratic 146 

Specific gravities . . . 285 
Square root . . 88, 114, 116 
Stevinus .... 256,257 

Stifel 256 

Substitution, elimination by 49 
Subtraction in a proportion 188 



Surd 

Symbolic algebra 
Symmetrical equation 
Syncopated algebra 



106 
261 
151 
260 

264 
284 
183 
181 
183 
39 
406 



Tartaglia .... 
Temi)eratures . . 
Terms of a proportion 

of a ratio . . . 
Third proportional . 
Transposition . . 
Triplicate ratio . . 

Utility of algebraic pro- 
cesses ... 67, 126, 
146, 181, 197, 239, 266 

Variable 234 

Variation 234 

Velocities 284 

Vieta 256, 257, 260, 262, 264 

Wallis 257,265 

Weights .... 284,285 

Widman 258 

Written problems ... 57 



Zero exponent 



95 



DURELL'S ALGEBRA: BOOK TWO 





ANSWERS 




SXSRCISS 1 


1. 


25 4. 7 7. 6 


10. 2 13. 1 16. 


2. 


86 S. 33 8. 90 


11. 6 14. 17. 21 


s. 


28 6. i 9. 18 


12. 20} 16. 13 18. 10 


19. 


(a + 6)« = a«4-&^ + 2a6 


24. 65; 73 in. or 6 ft. 1 in. 


80. 


4(a2-96)<(7o + 6»)2 


26. 164.6592 


21. 


182 ; 3 ; .8464 


26. 325.62 ft. 


22. 


1963.5; 143.13915; .02010624 


27. 45°; 982f 


23. 


78.54; 2.010624 






SXSRCISS 2 


1. 


11 8. 


6. 1.4a«x» 7. -4(o-6) 


2. 


-9aj 4. -2a^a 


6. 3(« + y) 8. -4Va-x 


9. 


18. -SVa-a 


10. 


x-j-y 


19. a^c 


11. 


7 09 + 269 


20. (a-4 6)a; 


12. 


llp-2g- 13pg + 25pr 


21. (2o-2c)x 


IS. 


-}x + iiy-V« 


22. (-2a-c-d)y. 


14. 


8.58 a6- 7.35 OC + . 753 &c 


23. 3a -46 


15. 


3o»6» 


24. 7a«-5a24-2a-4 


16. 


-2a« + 2o6 + 96a 


25. 7a;2»_8a;'»-9 


17. 


ll(a:-y) 


26. 14a -46 + 10 c 


27. 


(4rt-106)x 29. 2x2 


-flC-lO 31. 1163 


28. 


8aJ»-5a; 30. -8aj2 + 5x 32. 222° 


Cm 


fright, 1914, Im Charles B. MerrlU Co. 


i 



ii DURELL'S ALGEBRA : BOOK TWO 

SXSRaSS 8 

1. 8a:«+7x+2 8. 2a*-9aj» + llx- 3 

S. 12a!' + xy-20y» 4. 6a«- ITa^a: + 14aa^-8x« 

8. 4y«-18y* + 22yS-7y + 5 

6. 12x*-x»-27a:«-8x+ 10 

7. 6a5» + 9x*y-10a:»y2 + 6a.ay8_43cy44.y6 

8. 16a*-10a:*-17a*-17a;« + 6x + 20 

9. 12a« + 6a»6-4a*6»-o«6« + 2a26*4-a&«-6» 
10. 8a* + 5x*y-3x»y2-xV-4y^ 

11. 2a^-8x* + l 14. J(i8-^6«. 

11. x6-2aj»-39x + 16 16. Aa«-Jx*+A«"-A^+ A 

18. 4a^-9x*-14x»-x2 + 26 16. 2.88 x» + 10.86 x - 19.2 

17. 4.6a«-7.1aa-.4o + .24 

18. 4«^» + 5x»+a-x^+i-x" + x^i 

19. 6a*+i-6x»-6x*-i + 18x*-«-6x'»-» 
29. Ex.7. 

96. 16(x + y)«-12(x + y)a + 6(x + y) 

96. 10(x - y)« - 19(x - y)« - 19(x - y) + 10 

97. x» + y» + «« + 3xy2 + 3x2y 

98. a» + 6» + c»-3a6c 

99. x»-y» + ««-xy«-2x22f + 2y«a 





EXERCISE 4 


1. 4x»-8xa + 2x-l 


11. x2-ix + } 


9. 2x»«8xy + 2y« 


19. .5 0-.46 


8. x«-3xa + 2x-l 


18. 2.5X + 5 


4. x2-2xy«y3 


14. x* + 2x*-i4-3x"-«-2a?'-* 


6. x2_2xy— ya 


16. x» + 2x»»-i 


6. a2 4-06 + 62 


19. 3(x4-y)2-2(x + y)-4 


7. x2-a;-4 


90. 3(a-6)*-4(a-6)«+5 


8. 4x2-6xy + 9ya 


91. a2+62+x2+2a6-ax-6x 


9. -xa-4ax«4a« 


99. x2 + 2xy4-x«+y«+«*+y« 


10. Jx2-4x + J 


93. a2 + 62 + c«— a6— ac-6c 











ANSWERS 




i4. 





28. 


6 


31. 2i 


34. 2 


26. 


-3 


29. 


45 


32. -lOJ 


36. 9 


26. 


6 


30. 





33. 2 


36. 6 


27. 



















SXSRCISS 6 




1. 


l-2a 




3. 


8 6. 


— X 


S. 


-.756 




4. 


-y-1? 6. 


6 -a. 


7. 


a«-6a + 6 






12. x»-10x« + 31x-30 


8. 


Sz + 4y 






13. -2iJ;2 + 17x-30 


9. 


3a;2 + 6x + 3«y + 4y 


14. 362-3x2 


4-5 6X 


10. 


-10a;«+9a; 


+ 4 




16. a2 + 10o- 


■13 


11. 


16x«-19x8 


+ 4 




16. 2a2-aJ2 





lU 



17. 3o2-10a6 4-62 

18. 3x-56x4-21y— 156y-12x2 + 15xy-8y2 

19. 6x» + 2x2y-llxy2_|-l0y8 

20. -18 22. -26 24. 29 

21. 16 23. 54 26. 29x2-10x-60 

26. -4x+16y 28. x«-(-6x2 + 8x - 1) 

27. x«-(5x2 + 3x-l) 29. a-(-6 + c + d) 

30. a-(6 + c4-c?) 

31. (5-<i-c)x + (3-(?)y+(7 + 6 + o)« 

32. (7-8a)x + (-3 + 36)y + ll« 

SXSRCISS 6 

1. 9x5«-30x + 25 6. 12x*-41x2 + 36 

2. 9x2 + 80x + 25 7. 9 x« - 30 x*y + 25 x2y2 

3. 9x2-25 a a2 + 2a6 + 62-c2 

4. 6x2-7x-20 9. l-4x +6x2-4x« + x* 
6. x2 + 3ax-10a2 10. a2-62 + 26c-c2 

11. o* + a2ai2 + x* 

12. o2 + 4x2 + 9y2-4ax + 6ay- 12xy 

13. 9x2»»-25 14. a2»-262 + 6a«+>6»+i + 9a*62'» 

15. l-4x + 10a52-14x8+13x*-6x» + x« 

16. a2 - 2 a6 + 62 _ a.2 . 2 xy - y2 



iv DURELL'S ALGEBRA : BOOK TWO 



17. 16«* + 4a;« + l 


27. 2iB« + 4ya 


18. 12fl:*-28a;«y + 6y* 


is. -12xy 


19. iofl + ix + i 


as. 5-635 


SO. Aa«-266« 


30. 46a«44a6-4a« 


tl. .09aa-.8a6 + .26 6« 


81. 8 a2 + 18 aft -92 6* 


St. 1.44 a:«-. 12 a; + .0026 


82. -.2a2-6c2+2a6+6ac-26c 


as. 2x« + 2y» 


88. 56} 


M. 4xy 


84. 42} ; 00} ; 156} 



£X£RCISS 7 

1. a + y S. a* — a«y4-o^* — ay' + y* 

a. a« + ay + y2 7. 2x-y 

8. a* + a«y + ay + ay« + y* S. 4x2 + 2a:y4-y^ 

4. a^-ay + y* S. 16 x* + 8 x«y+ 4 a;2y2+ 2 jey^+y* 

5. a-y 10. flC2_2a;y -h4ya 

11. a-2x-2y 

la. 4a^ + 8a;y + 4ya + 2a2x4-2a»y + a* 

18. 6*-6V+y« 

14. 9xa + 6aa; + 66x + 4a«4-8a6 + 462 

IB. a*-2aax«+4x« 

SXERCISS 8 
1. x{7^-Sxy + 9r/^) S. (Jx±4y) 

a. «(« - 3 y)2 S. 7(a - b){a -6 + 2) 

8. »(a;±3y) 10. (3x-2y)(9x2-12a;y+4y9+8) 

4. x»(a;+l)« 11. (.3x-.2y)a 

5. a(«±l) 11. «(i«±}) 

S. a2'»(4a»-8 + a2«) 18. (x-3y±}) 

7. (Sa + ihy 14. (x + y + «)* 

IB. 6(a + 6)(«-y)(l + 2a + 2 6) 

16. J^(aJ2-2y2) 81. (.02x±.ly) 

17. (9y-«)(7x-3y) 88. (Jx + iy)^ 

IS. 15(a±2) as. (x-y)(a-6 + c) 

19. (x-y)(6«-6y-l) 84. (4a + 46-3)(3 - 2a- 26) 

80. {x-\-y-ay SB. (x-2y2)2 



ANSWERS V 

S6. 4(2 6-a)» 

27. a(a:8 + l)(«*4-l)(«« + l)(a;4-l)(«-l) 

28. (ofi-S)^ 29. (6a;-3y)(3y-a;) 
80. 1950 32. 8900 34. 1408 
31. 121,600 33. 103,600 35. 780.6 

EXERCISE 9 

1. a(a-l)(a-3) 4. xa(a; i 1) (x? - 12) 

2. (2x-l)(a;-2) 5. (x^ + y«)(x*- xV + !^) 

3. (2x-3)(4x«+ 6x4-9) •• x2(3x-2)(x -3) 

7. (y-.«)(y«6) 

8. (a-x)(a« + ax + x2)(a«+a«x' + x«) 

9. (6a+l)(3a-2) 10. (y-a)(y-.6) 

11. (a + 6 + X4- y)(a24- 2 a6 + ft^ - «« - oy - &a; - &y + «'+ 2 xy+y^) 

12. (2 a^ - 3 6«*») (4 a**» + 6 a^*»68» + 9 6«») 

13. (x + y-2)(x + y-4) 16. x(2x- l)(4xa + 2x + 1) 

14. (2a-.3)(a+.2) 17. x(2x + 6y)(x- 2y) 

15. (8x-2)(x + 2) 18. (a« + 3x«)(a*-3a2x»-|-9x«) 

19. (x-|))(x-g) 

20. (a-2x)(a* + 2a8x + 4a2x2 + 8ax« + 16x*) 

21. (4a + 36)(8a-46) 

22. (2a2+8x4-8y)(4a*-6a2x-6a«y + 9xa + 18xy + 9y3) 

23. (l+Jx)(l-Jx + i«2) 27. (a8+l)(a*+l)(a2+l)(a±l) 

24. (3a±2)(a±l) 28. x*(x + 3y)« 

25. (3 + a2)(9-3aa + a*) 29. (l-x)(l -|-x + xa + x» + x*) 

26. (x-a6)(x-8a6) 30. (x4-3a + 36)(x-3a- 36) 
31. (8a-36-2x)(a-6 + 3x) 

82. (.2x-iy)(.04x2 + Aa;y + iy«) 

EXERCISE 10 

1. (p-(7)(x + y) 6. (4a» + y*±ay) 

2. (x-pig) 7. (x + a-6)(x-a + 6) 

3. (x2±pa; + P«) 8. (x-a)(x + a + 6) 

4. (a--6)(a + y) 9. (2x2±2x + l) 

5. (a''b-^y){a-^h''y) 10. (2xa-.l±3y) 



vi DURELL'S ALGEBRA : BOOK TWO 

M. (a-6)(2a-26-l) If. (l±a + 2a«) 

IS. 2(x-l+2a«){x-l-2a«) M. S(a + 2)(a-l)« 

14. (rf«±ay + ^) ». (6*i6» + l) 

15. S(x-2)(2x-3) 11. (a-6)(a+6+l) 

16. (a-6)(l-a«-a5-6«) ». (2x + 3y + l)(2x-8f - 1) 

17. {a+l)(a-3)(a4-2) 14. (2x- 3f)(2x + 3if + 2) 

la. (8 o» ± 4a + 1) 1«. (a*-3)» 16. 2(a+6)«(2a-6) 

SXSRaSK u 

1. Yes. 7. Cx-l)(x-2)(x-8) 

1. No. Yes. a Cx-l)(x-2)(x-3)(x-4) 

8. No. No. Yes. Yes. Yes. f. (x+l)(x + 2)(x + 3)(x + 5) 

4. (x + 3)(x-3) 10. (x-2)(x-3)(x-4)(x-5) 

5. x(x-l)(x + 2) U. {x-l)(x-2)(2x + 3) 

6. (x-l)(x-2)(x + S) 11. (x + l)(x-2)(Sx-4) 

IS. (x+2)(x-3)(3x + 2) 
14. (a-6)(a«-a-6) 

w. (y-6)(y + e)(y-!r + 8) 

SZBRCISB 11 
1. x(x-2)(x-3) 6. x(3f-4x)(2y +x) 

1. x(x + 2ii)(x - 2 a) 7. (fea db &x + 7^) 

3. 2x(x-2a)« 8. (a + 6)(y-x) 

4. x(x±l)(x«dbx+l) 9. x(x* + l)(X2 + l)(x±l) 

5. (a;a + l-y)(x«-l + y) 10. (a* + x*) (a» - a*x* + x*) 
11. (a±x)(a2±ax + x2)(a*-a2x24-x*)(a«+x3) 

U. (x-l)(x + 4)(x-3) 

18. 10 x(a - 1) (a2 + a + l)((i« 4- a» + 1) 

14. (x-l)(4x2-llx + 4) 15. (9 4-x)(8-x) 

16. (2x2- a - 6)(4x* + 2ax2 + 2 6x» + a^ + 2ab + 6*) 

17. (a + 6-3)(a + 6-l) 

18. (a + 2)(a«-2a6 + 4a*-8a» + 16a3-32a + 64) 

19. (x-.6 + a)(x-6-a) 

10. C2a- 26 + 3x2 +6xy + 3y»)(2a- 26 -3x2 -6xy-3y8) 



ANSWERS VU 

St. (2 62 + l)(6-2 6) 17. (x» + 6)(x + 2)(x-.2) 

tS. (a;-yXx + y-«) S8. (aa + l)(a-l) 

M. (a + lXa^ + a + l) »• (a- l)(a;« + a;-8) 

25. (l±4x + a;«) 80. (3 + x)(a±8) 

M. «a(7aj + 6)(3a;-6) 81. 9a2y2(a.«y)2 

8S. (2a+26-x4-y)(4aa + 8a6 + 4 62 + 2aa; + 26a5-2ay-26y 

+ x»-.2xy+i/«) 

88. (0* 4- « o^ft* + M) (a + &)^(o - by 

84. (a + 6)(x-y)(a + 6 + 3c-y) 

85. (a + 6 + 2x-2y)2 

86. (x + 86)(x-86-6a) 

87. (a;-l)(a-4)(a; + l)(fl;-6) 

88. 27(x + y)«(a;-y)» 

89. (a-6)2(aa + 2a6 + &* + 2a+3) 

40. (p 4-2 + 2a- n)(i) + 2- 2a + n) 

41. (a-2)(2a + l)(a + 6) 46. (a±l)(6±l) 
4S. (aa+4)(a±2)a 47. {afi^a^^b^y 

48. (x + l)(x~2)(a + 8) 48. (x» - y») (x*- + «"F + !/*»») 

44. (6aa±6a6 4-7 6») 49. (x»-»«»)a 

«. (x±l)(x-2)(x + 3) 00. (^a-^b-cy 

51. (a-|-6)(a2 + 63 + l) 

51. (a-x)(3aa-6ax + 3x2 + l) 

68. (x + l)(x*-x« + ai2-x + l)(xi<»-a^ + l) 

64. 29 X 931 
55. x(x -2a)(x* + 2 0x8 + 4a2x2 + 8a8x + 16a*) 
66. x2(7x-3y)(3x + 7y) 57. (a + 2 6)(a- 26)2 

EZ£RCISB 18 

1. 3xy; 36 xV S. babe; SOa^b^c^ 

2. 4a262; I68a«6*c 4. a-3; a2(a ± 3) 

5. 3(x - y) ; 18(x ± y)(x^ + xy + 1/«) 

6. 6(x - 1) ; 72(x + 1) (x - l)^ 

7. a-1; 12(a + l)(a-l)2 

8. 3x(x-l); 18x2(x+l)(x-l)a 



Viii DURELL'S ALGEBRA : BOOK TWO 

9. «; x«(x±l)(x + 2)(a5«4-l) 

10. 1+x; (l-a:)(l + JC)3(l-x4-x«) 

11. 8x; 46a6x2y(x-y)«(x + y)« 
IS. aJ»^(x-y)«;xV(x-if)* 

IS. 9a?(x - f)« ; 108(x + y)«(x - y)» 

14. a — 6; (a-6)(xdby) 

15. (a-6)»; (a-6)2(x±y) 

1«. 6x(a - x) ; 26x»(a + x)(a - x)« 

SXXSaSB 14 

1. f ^3^ ^ 2(o-6) ^ a + 8 

^ 8gy * 4« " 6(a + 6) 2(a-l) 

4a^ 5 y 7 x + 2 ^ 8x-2 

3 8aa*y ' 2 - 3xy * x-2 * x+1 

9 



^^ a» + 4a+16 ^^ 4 + x ^ x- 



11. 
IS. 
13. 

ss. 



y 


f 


2-3xy 


14. 


44-x 
3-x 


16 


/)24.3p4-9 




l>-3 


1ft 


24-x4-y 




2-x + y 


17. 


8-x 



a + 4 3 — X a; — r 

J_ 15 f» + 3pH-9 SO. 6\ 

g + 6 — c 

2a-l 

2a + l "" x-4 

4a-,2+ ^-^ S7. a?'+3xy-2y«-l 

2a+l x+y 



». x-1^ .^-^. S8. ^*+l_ 



x^ + x+l x^+x-l 

24. a-x + l-^=i S9. ^^"^-»«' 

a+x 4 

S5. i|A 3^ ^-1 

,3 3 3^-2x4-5 «+l 

EXERCISE Iff 

1. ?^ 3. «'^ 



10 a + & 

4- 67 x'- 42 ^ g^ 

63x» * a«-4 









ANSWERS 


B «^+^ 


la + 10 




f. 


1)2-21 


■ (a + l)(a 


+ 2)(a 


+ 3) 


(p + 2)(p-3)(p-4) 


6. y" 






10. 


1 


7. 






11. 


66 
1-68 


a. ^-^^ 
r8 + 64 






12. 


2x{x^y) 


13. 1 




16. 


a + 6 — c 


IB «* + ^^* 






17. 


a — b ^c 
20 -7x 
8(1 -a:2) 


4a6 
19 * 


15. 1 


" x»-l 


20. 






25. 


-7 


11. 


12p(p+l) 


22. a 






26. 


a — a6 -1- 6 


23. « 


-6 




27. 


a + l 



IX 



(a-2)(a-3)(a-6) x^-x 

j^ 2 gg + 8 o + 1 
8a 











EXSRCISB 16 








1. 


l + 2a; + 4 
2x 


L*? 




6. 2(x + ?^) 


11. 


1 




2. 


z 






7. />^ + ?^-H 


12. 
13. 


-6a 




3. 


2a2-l 






8. 4 

8a + 8 

9. a-l 


X 




4. 


c(a+6 + < 
c(a — 6 + ( 







14. 


1 




6. 


23cy 


-fy2) 


10. •^ + «' 
a(l-a2) 


15. 


2x 

X2 + 1 












BXERCISB 17 








1. 


6 


8. 


a + 26 14. -J 




20. 21 




2. 


-A 


9. 


6 


15. -H 




21. -J 




3. 


8 


10. 


5 


1«. l^F 








4. 
5. 
6. 


1 

« 
4 


11. 
12. 


ac 
b 


17. «\ 

a + 6 

18. -} 




22. %- 

P 

23. -8 


g + r) 


7. 


5 


13. 


2 


19. 




24. -6 





DURELL'S ALGEBRA: BOOK TWO 
.,6 18. n. .05 

19. — - 



2 St. 392 



4 
St. 1 

S7. 8 80. 5} 88. 



SXSKCISS 18 

1. -8 8. -I 9. 7 10. 6,-3* 

•• -i Y _3 10. -g 11. 3, 2* 

-' 11. M. -j,2* 

8. 1 * c-d If. 4,-2* 18. 6, -2» 

EZBRCI8S SO 

8. - }, } 16. 18, 18 18. c + (2, d 

8. 8,-2 16. 3, 4 17. - 3, 2 

*- 1»2 17.11,7 18. }(o + &).!(«-&) 

•• -^»-^ 18. 2,-1 ». a-6,a-6 

I:-l3 "-^'-^ 30. £^,^-±1 

8.-4,3 «^-^'-^ 

10.^^,4 «•».-! 81. -2, -J 

11. 8, 1 18. a, 6 

11. - V, - V 14 ^'^-^' a'e-ac' ^' ^' ^ 

18. 9^8 ' ab'-a'b' ab'-a'b 84. 6 - 1, a + 1 

14. 12,6 18. 26-a,2a-6 88. (a - 6)», (a + 6)a 
EXERCISE 11 

1. 1,-1 •. i, J 

1. 10,8,6 10. a, -26 

8. A. A ^^- 2a, a- 6, a + 6 

4. 1, 2, - 3 -J 2a 2a 

8. J, -I * l+«^' l-«* 

6. -i, i,2 18. 1,2,3,4 

7. 2,-3 14. a-\-h — e,a — b-\-c,b-\-c-^a 

8. 3, 4, 7 16. 2, - 3, 5, 4 



11. 7, 6 81. i, i 

a a 











ANSWERS XI 


16, 4, 2, 1 

17. h 1, i 










22. 


1 1 
a' b 


i«. i.J.i 










23. 


1,-2, J 


19. —a, a 










24. 


1,1 




2r 

6 + c 






25. 


- 1, J, - 1 








EXSRCISB 23 


1. 11, 12, 18, 


> 14 








7. 


6 mi. per hr. ; 7 mi. per hr. 


2. $6000, $12,000,318,000 




8. 


144 


S. 300,1600, 


4500 








9. 


42,64 


4. 6290 










10. 


$526, $600, $160 


6. 400, 1200, 


3200 








11. 


$2, $7.60 


6. $56.26 










12. 


Iron, 2400 lb.; lead, 2100 lb. 


13. Iron, 5 cu. ft. ; 


lead, 3 


cu. 


ft.; 


; aluminum, 5 cu. ft. 


14. 26;!^, 2^ 










25. 


760 million bu. 


15. 16 










26. 


$1.20, $.82, $.70 


16. 8 










27. 


15^5, 6^4i;^ 


17. f 










28. 


140 lb. 


18. 26 gal. 










29. 


301b. 


19. 4t 










30. 


Agal. 


SO. 6,3,1 










31. 


Oats, 48 bu. ; com, 32 bu. 


11. $8, $60. 










32. 


301b. of 18^; 701b. of 28^ 


22. 12,18 










S3. 


l^gSkhmiYk] 13^gal. cream 


23. 40, 12 










34. 


lida. 35. 24 da. 


24. .06 










36. 


14f min. 



38. 2 hr. 21]^ min. ; 3 hr. 4,<^ min. ; 6 hr. 42f min. 

39. 2:10}f 46. 6 boys, $240 

40. 2 : 43^ 47. Gold, 4} lb. ; silver, 37} lb. 



41. 


6:16^;6:49A 


48. 


Copper, 164 lb. ; tin, 146 lb. 


42. 


23^ mi. 


49. 


i 60. 48 


43. 


4 in., 18 in. 


31. 


324 


44. 


$20,000, $20,000 


62. 


4 mi. per hour 


45. 


$8000 


63. 


$72 



ZU DURELL'S ALGEBRA : BOOK TWO 

14. 85 dimes, 14 qoarterB ^ 6o — b 4a-f 6 

«. 26d«..16|d.. ». ^da. 

W. 20 oxen, 30 da. ^ ^ 

60 «^ a ^L±J 
97. I oz., 3} oz., 6 oz. ' 8 ' ' 8 

EXSKCISB 14 

X. i=A, „ = 4 

to I 

tL* iri?' BL W w ' I 

• J?=_^. ,=_^ "• -P=f C + 32 

2ir* 2J? »» 

7. C = 2vB C 

8b Lr = — 13. g = — ; 8 = «^- 

t « 

14. / = ^;p = -i£:-;p'=^ 
P+P' P'-f P-f 

16. 7282° 17. 1^=320°; C=160° 

16. -40° 18. i^ = -12A°; C=-24A° 

BXSRCISB S6 

8. (1) 6 (2) 10 (3) 13 (4) 3Vl0 

». 42 10. 28 11. 10, 5, V29 

BXSRCISE 17 
1- (1t-1) 3. (2,-1) 6. (3,6) 7. (-2,-3) 

* (1,1) 4. (3,-2) 6. (6,2) 8. (-3,0) 

1«- (1,6), (4, -2), (-3,-3) 

11- (5,0), (3, 4), (-2, 6), (-1,-6) 



ANSWERS XIU 

SX£RaS£ 28 

1. 1 : 48 P.M.; 64 mi. S. 6 p.m.; 106 mi 

3. At end of 3 br.; 12 mi. from A. 

4. Cream : milk = 16 : 8 

6. 4:6 6. 2:6 

BXERCISS S9 

5. -9a;2 + a; + 7 S. 4a2 + ft^ _|. (ja _2a6- 2ac- 6c 
4. (1) (a2±2ax4-4a;«) (2) (^--Y* 



(3) (a + 6)(fl 


5 + 6 -3) (4)(a + 


x-h 6+c)(a+x— 6-c) 


6. 


9. 


-i 


13. 12 




6 ^' + J^ 


10. 


-4 


W. f-i 




2xy 










7. 


11. 


2 


w. ia 




8. l + 2x 


12. 


i 


16. -6,-1 




^^ cq - &r ar- 


-cp 


iQ a4-26 a-26 




aq— bp^ aq- 


^bp 


2 ■ 


' 2 




10 ^ 


1 


20. 3J, - 


-3,-2 




a-{-b-\-d' a 


+ 6 + d 








BXERCiSE 31 






1. A 0*62 


9. 


.0026 


17 »1^ 




S 4a:« 


10. 


.000026 


* 626 6*-*« 




'• 26 


11. 


.000126 


w. « 




S. V«^C8 


12. 


.000000126 


19. ^ 




4. 9a:2nya»-2 


IS. 


32 a«5io 


20. W 




5. ^T^ny^-^ 




243 


21. 12 ; 36 




6. 27a;V 


14. 


nh cfib^^ 


„ 49c2(P 




7. 266 0*6* 


16. 


- 128 a^* 


"• 9 




8. Aa^&* 


16. 


A^^« 


24. 64 




16. 6«+666a; + 156*x2 + 2068a^4-1662x*4-66a* + jB» 




17. 6«-6 6Sa;+15 6*a;2- 


20 6»x8 + 16 62a:*- 


6 6x6 4. a^ 




28. a* + 8 a7a; + 28 083^2 + 66 a«iB» + 70 a*a:* + 56 a«a* 





+ 28 a2a« + 8 (KcT + x» 
29. 69-9 68x + 30 6^x2 - 84 6^x8 + 120 65x* - 126 6*x6 

+ 84 63x6 - 36 62xT + 9 6x8 - a« 



xiv DURELL'S ALGEBRA : BOOK TWO 

80. 6» + 106*a; + 406«a? + 806*B» + 806a5* + 82a* 

81. ^(i« + |<i*64-5a»6« + 20a26« + 40a6* + 3266 
81. aT-7(i« + 21a*-36<i*4-86a»-21a« + 7a-l 

88. p^ - 12i)i0g 4. 60 pV ^ieOjfi^ + 240 p*g* - 192 p^q^ + 64 g« 

86. 81a* -108a^ + 54 35* -12x2 + 1 

86. 81a8-216(i«6 + 216a*62_90a268 + 16 6* 

»7- lb «" + iV a^5 + ^^^ + 80 a86» + 186 a'6* + 248 6* 

88. 128 a" - 224 a" 4- 168 a^ - 70 a^ + y a* - V «* + 1^1 a* - tIt 

8f. 6561 - 8748 x + 5103 «« - 1701 x» + ayijc*-. ijaa* + ffx* 

-A^' + iriira* 

40. xe-3x<^+6x*-7x» + 6x2_3x+l 

41. l+9x + 24a:24.9a:8_24x* + 9x6_a:6 

45. a* + 4xTy^.i0a*ya + i6a*y» + 19x*y* + 16x«y6 + 10xV 

+ 4xy7+y8 
48. 8x«-12x5 + 42x*-37x8+63x3-27x + 27 
44. a^-4x74.2a* + 8x8-6a!*-8x« + 2x2 + 4x + l 

46. 158 a^xw + 18 cw;" + x" ; 153 a^x" - 18 ox" + x" 



1. 4ox» 

2. 3xV 
8. ioxy* 

4. ia^ 

5. 5x*+« 

18. 9, 49, 64, 121, 225 19. 8, 27, 8, 125, 216, 27, 343 

SO. 72 11. 108 22. 420 28. 2376 

EXERCISE 33 

I. 36« + 263-.66 + l ^ 5<j2_«4.? 





EXERCISE 82 






6. 


9a62 


10. 2 0x3 


14. 


3 


7. 


3 0252 


11. 2a«x 


16. 


-2 


8. 


-3a262 


12. iay» 


16. 


2 


9. 


2ab^ 


18. 2a2 

3 6- 


17. 


} 



2. 2p8-36a + 5 



4 3 5 



8. a- 6 + 5 8?E + 8 + ? 

4. 3a« + 4a26-4a62-36« ' a x 

6. 3x»-4x2 + x-2 ^ ^ . ?_« 

6. |xa-xy + 5ya * 2 a x 







ASSWEE& 








^ 


!•• ^H-^-h^ 








ir 


l^i: 


c^iii>^^ 






- 




tt. 




-• 


14. «M 


ift. 


JKMfi 




sr 


.^Sfiv* 


«. 


t.WMtf* 


«. ^•^Ji 


». 


i:r3S, 


0, 


t.WW* 


ST 


*XJl^'r^ 


1«. 2836 


a. 


4a*«S:- 


a. 


'iSJir 




M 


% JOJI*- 


XT- TC.12 


SL 


i'iJift- 


JB. 


J5»-4^K- 


J» 


'X/9t'^j^' 


3*. i#€rrT- 




J&. 


4<^ VC 


^r 


: ^ 


M 


i;.S^J5>- 


• n 


31- «L«»-iL. 




JB 


SM2« 






M. 


;jir> 


• tit 


as. SUTT-iL. 




JB 


s.;**-^- 


1-. 




m 


it.;^^' 


n. 


*S- 2&-4iiiT^ iL. 




>» 


^^^.-y- ; 


L 














Zicvafci 


K « 


( 


m 


^ 




X- %:ii 




: r-^ 




«^ <r 




^ 


tt-^- 






m 


i <;^ 




m. i-wT 




:k 


s»- 






m 


i ' 




4- i^^:"^ 




2r: 


3. 






M 


^ 




S. i -k ^ k t 




Jt 


jr 






it^ 


ir 




«. it^^^ 




3* 


» 






M^ 


■^ / 




T i *^ 




s: 








» 


/ /' 




1. r . . ^ j0^ 




jK 


, 






tr 


X 




9 T'* 




^ 


,„- * 






M 


9^ 




aft f^ 
a Bc-'^ 




^ 


: 






-» 
m 
^ 

m 


•5^ 




a -'.. 




de. 


c 






^ 


^ 





XV 



xvi DURELL'S ALGEBRA : BOOK TWO 



,. ^^ 






8. 3 


X 




: 


10. 1 + 1 


a* 






4a«6V 






a b 


7. \ 






• ^ 


\ 




: 


11. i 


a«y* 


• (=«- 


yy 


IS. ^ 


13. 32 


17. 


A 




SI. 


jij 




SO. 8 X 10-9 


14. 8 


18. 


A 




ss. 


20 




30. 10-9 


16. 4 


19. 


W 


S3. 


a 




SO. io-« 


16. 8 


10. 


-1 




S4. 


i 




S9. 16xlO-« 


SO. £ach = l 




8«. 1,1, 


1,1,27,4 


34. t 


31. 8, 5, 3, 1, 3, 


7 


33. 2 






38. «1; 6J/«^; Uif 


89. Latter 


43. 


-S 


!*♦ 


47. 


^ 




00. (i)-2 


40. ^ 


U. 


a* 
3 




48. 


Sm 

y^ 




01. -1} 
OS. V 


41. A 


40. 


a-a. 






8 














6* 






xy 


46. 


7 6» 




f-Sn 






4S. 4 a 




Sa 


i 


49. 


p^ 












EZBRCISB S6 






1. A 

2. a;-a 




10. 


7y» 






S4. 
SO. 




3. 1 




10. 


a-' 






SO. 


xi 


4. 8 

5. 8a* 




17. 








27. 


-^ 


6. A 
81 

7. 22«or4«» 




18. 
19. 






SO. 
SO. 

30. 


-2J 

Jbifi 


8. 2«'-*^ 

9. a;2y-« 




SO. 


9.J 
4 a* 






31. 


2fl 
IK* 


10. 2a-i6"5a;i 




SI. 


(-!)^ 






3S. 


X0»+aft-2a 


11. i 

la. 1 




ss. 








33. 


a*c2 
bd 


13. -^ 

27 a« 




S3. 


zi^ 






34. 


a^y 


14. xP 






yk 








xi 







se. 


3x 


'2,3a;-*,}x-*, 


3a; 


-^4, 


a-l'^4a-* 


3& 


4 






43. 1 






46. 81 


39. 
40. 
41. 
4S. 


9 
i 
i 
-82 






44. 1 

^^2 

46. 1 

(-3)- 






47. JV 
i8. ¥ 
49. A 










BXSRCISE 


37 




1. 


4x3- 


-3a; + 3- 


ari-|-6x~a; - 


- a-* + a" 


a - 1 + aa + a» 


S. 


Ba- 


13a;*^ 


-12 


a;i-xl-i2 









xvu 



3. 9a-*-16a-8-Kl2a-2-4a-i 

4. 12 o2 - 17 ahi - 10 a*6* + 12 6« 

6. 9x*-27x^+32-16x"+ 17. a + ah^-^h 

, , , i 18. 2x-.3x*-J 

7. xV* + 3-|-4x"^y* . . 

8. 9x»-hl6x-6x-i-8x-» !•• «^-2xiy* 

9. 2x-i4-3 + x + 6x2. ^^ 3xV^ + 2x-4 



10. 

11. 


27a2-30ay-^-|-3|r^ 
3a*-2a* + l 


21. 4x^+3a*-2ax"* 


12. 


x-2 + x-^y + y8 




22. 6*- 3 + 26"* 


13. 


a-i + a'^fti + h 




23. ia*-ia&i+}a*6 


14. 


aji-iciy-i+y-* 




24. 3x-a-3x-Jyi + y 


15. 
16. 


a"*+2a* 

x* + 4x~*4-4x'^ 




26. a*-2o6"i + ^ 
2 






SXSRCISS 38 


1. 


-2Jf,13i 




6. 6a2 + 2x*-3a-2x 


S. 


First, 32 times 




6. x«'-2» 


'a. 


Latter, 227 times 




7. X-" 


8. 
9. 


1 10. 
^-^ 11. 


8 

1 

hi 


12. x-i4-«"M-y^ 

13. x« + x^y"i4-y-* 



XViii DURELL'S ALGEBRA : BOOK TWO 

^*- ^ 17. (x-i±»b 

18. (xt-8)(x*-2) 
Iv. 0*+* 

If. (x*-y*)(aj* + a;V + y^) 

to. a* + aM + a6 + aV + a*62 + 6* 
21. 8f , i ». 2», 2», 2T-+1 SO. 2»"-« 

SI. x«-12xt + 48x"^-64x"» 

SI. a?-8x^ + 24x*-32x* + 16x* 8S. 27isec. 

34. 66} lb. of 18^ coffee ; 83i lb. of 30^ coffee 

SZSKCI8S 40 
1. 2aV%^ 8. -?v^266 "• 12ax»V35? 

S. 2a^^ •• 2^"^ 16. (2x + l)x^ 

5. 2a6c«V65 "' f^^^ 18. ^v^ 

. 8. 2aci^ "• ^^^ ^^^ 

^ ^ ,- IS. 6x»V2a2 19. 24 VS 

7. }a6va 

6(2a-ft)v^^^r6 8S. ^Vi 

21. 2VI6 «4. 1^154 

28. ^^/bi «». f^V2^ 



X 



4x 



26. \V% «?. 1^ SI. ^W^^ 

27. Av^ISO^ SO. — \/36^ ^^ 

88. A^. '^ 3«- TTi^^-^^ 



ln/-r 



7a 

34. 5L±A^(a + 6)Ha-.6) S6. fV^ 

a — b 

87. JL±±^ »+J/(a + &)8(a - 6)«-i S8. 10.3023+; 13.8564+ 

(a -6)2 



ANSWERS 



XIX 



1. 
2. 
S. 
4. 
6. 
6. 
16. 

17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 



1. 
2. 
3. 
4. 
6. 
6. 
7. 

8. 

9. 
11. 
27. 
28. 
29. 
30. 



V76 
\/375 

n 

2a62a;8 



y/ab^ 
y/2bx^ 
y/9^^ 



V2(a-6) 



EXERCISE 42 

7. Vf 



••aS 



10. v^27«2(a-6) 



"•aI 

IS. # 



26. ^/4a^(a;-y) 

"■aS 

28. 



^^^e/126 

29. ^^^3126, ^4 

30. \/S7, \/^ 

31. 1^, </9, n 

32. ^J^, ^'</27 

33. \^, v^, v^ 



14. V (xs - a2)» 
16. V(3a-6 6)(a-6) 

34. VS 37. Jv^ 

36. y/i 38. 2 v^ 

36. 2V2 39. }V} 

40. VE, y/i, V3 

41. \/80, \/2, v^ 

«. n. n n 

43. J\/6, 3, 2V3 



16V3-18V^ 



6bVx 


8.48626+ 

i\/2 

16,32,729, -27V3; J 

14-4V6 

15 V3 - 26 

100A-35VS" 



EXERCISE 43 

12. v^^OO 

13. v^72 

14. VEi 
16. Sa^Vb-Zb^Va 

16. 3V2 

17. 54 + 16\/6 

18. 68\/l0-67 

19. 4 



21. 533 

22. AV6 

23. 2^ 

24. -Vac 



26. ^iv'^^ 

Or 



20. -^ 



26. 2V5-.jV6-J\/7 



31. 109\/2-89\/3 

32. 2a; + 2\/a;2_y2 

33. 6-.2V6 + 2V3-2\/2 



XX DURELL'S ALGEBRA : BOOK TWO 

SXXRCI8B 44 





4V7 
21 


6. 


h</d^7<m 


1. 


12 


s. 


6>/2x 
6x 


7. 


76-22VI0 
13 


s. 


4^ 


8. 
S. 


II4.X-6VX + 2 
7--« 


4. 


a-6 


X + 6-2V3X + 6 
1-x 


s. 


Va* — a6 — Va6 
a-6 


-^ 10. 


6* 


11. 


V3+v^ 






IS. 


4(486 4- T4 Vl6 - 



106^6 - 120\/3) 
49 

14. 




IS. 


5 


6 


IB. 


3V6 

-V3 
2 


IS. .8944+ 
SO. .9449+ 


SS. 7.888+ 

SS. -.18036 


IS. 


SI. .6666+ 


S4. .4031+ 






EXSRCISB 4B 




1. 


2\/2-3 


9. 2V16-3V3 


17. 2-V3 


s. 


V5 + 2V6 


10. JV2 + iV6 


18. 3-V2 


s. 

4. 
B. 

6. 
7. 


3V3-2V2 

V6-V3 
Vli + 2v^ 
3V5 - 2 V7 
2v/6+V6 


11. fV3-jV6 
IS. i+V3 

13. 2-JV3 

14. Vm + n + Vwi 
IB. a + 3Va'2 + l 


19. 1+}V6 

50. 1+v^ 

51. V7-V2 
SS. 3 + 2\/3 

"^ S3. 3V2-V6 
S4. 2V3-\/6 


8. 


3V6-4>/2 


16. VS-l 


SB. 2+V3 






EXERCISE 46 


1. 


6V-1 


3. 8V-2 


5. aV-1-6 


S. 


6V-3 


4. 


6. -(a 4-3 6) V^ 



ANSWERS 



XXI 




16. 3 + V2 

17. -6-5V6 

18. 24 

"6 4-9V2-2V3 
35. 1-V^^ 

a2 + 62 

,^ 24 + 7\/!0 
2 

6 

14 

41. 3_V;^T 

42. v/3-V^=^ 

43. 2V3 + 3\A^ 

44. 2V^^-3\/^^ 

45. 4-3Vir6 ' 

46. 5>/3T + 4V2 

^; ((w;±i6); («±0; (x*+t*)(x*-0 
^Ti; v^rij v^T; -1 

53. 7i-l 56. 2-2V^n; 

54. 6\/31 57. fVin + 4; 



1. 

3. JV2-JVZ^ 

4. (3a2+62)Va2-6'» 



EXBRCISE 47 

5. Sv'^TS 

6. -6V6 

7. vn 

8. 23 



^ 8\/25 
' 16 

10. -tVV^=^ 



xviii DURBLL'S ALGEBRA : BOOK TWO 

^*- ^ 17. (x-i±»b 

18. (x*-3)(x*-2) 

10. 0*+* 

1». (xi-y*)(x* + xV + y^) 

SO. a* + ah^ + a6 + a^6* + ah^ + 6* 

11. 3J, i 19. 2», 2», 27»+i W. 2-"-» 

81. x«-12xi + 48x"i-64x"* 

31. aJ8-8x^ + 24x*-82x* + 16x* 33. 27J sec. 

84. 66} lb. of 18 ^ coffee ; 33J lb. of 30 ^ coffee 

BXBRCISB 40 

1. 2a</2^ 8. ^^V2bb "• 12ax»V5^ 

1. Aa^y/i ^ ,,_ "• «^'^ 

8. 2a</S^ •• ^^'^ 1«. (2x+l)v7 

4. 3 a6c> V6^ ^^' «'^ /iL ". (a + 6) V^:::6 

8. 2a6cW'6^ "' ^^^^ 18. ^f^VS^ 

8. 2aci^ "• 'y^ '''^ 

^ , ^ ,- 18. 6x»v^2 19. 24 V6 

7. Ja6va 

7(x-y) ^« 

11. 2>/l6 «4. |Vl64 

SI. i^VSx M. |^V2^ 



X 



4x 



18. \Vi 19. iv^ 31. ^Wi^ 

IT. A^^iSO^ SO. — \/36^ ^^ 

.• 2 ./To- ^^ »«. -^-vrr^ 

18. -=-v49a l + x 



7a 



83. _«_-V6(^+T) 3». Jv^a& 

(a + 6)' t* 

a — 

ST. aH-5 n+i^^^^^^8(q-5)n-i 38. 10.3923+; 13.8564+ 

(a -6)3 







ANSWERS 


XIX 






EXERCISE 41 






\/76 
^376 


7. V| 


18. # 




10. v^27x2(a-6) 


U. V(a:»-o»)» 


IB. 


Vx*(a;a - a2) 


16. V(3^ 


(,_66)(a-6) 


IT. 
18. 
13. 
SO. 
11. 
S2. 


Va62x8 
2 aft^xs 

V2 6a;2 
V3a262 


16. 4/4aXx-y) 

18. \/lQ, y/n6 

29. ^^3125, '^ 

30. \^, v^x* 

31. </B, </d, n 

32. ^!^, ^,'4/27 

33. </^, ^ ^ 


S4. V5 87. J\^ 

85. y/l 88. 2v^ 

86. 2V2 89. }V} 

40. -J^, </4, V3 

41. '-J^, V2, VU 
«. -n. n ^t 


23. 
S4. 
36. 


V4aa«y*» 
V2(a - 6) 


48. iV5,8, 2V8 






EXERCISE 43 




1. 


16V3-18V2 


12. v^500 


«1. 683 


S. 

3. 

4. 


1^6 

-lOav^ 

19xv^ 


13. v'72 

14. VU 

15. Sa^Vb-Sb^Vi 


M. 2-1^ 


6. 
6. 




bbVx 


16. 3V2 

17. 64 + 16\/6 


c 


7. 
8. 
9. 




8.48626+ 


18. 68V16-67 

19. 4 

62 


25. 1%^6^ 


11. 
17. 
18. 


4V2 ""• "2" 
16,32,729, -27V3, i 
14-4V6 31. 109 V 
15V3-26 32. 2x + 


26. 2\/6-}V6-|V7 
2-89V3 


19. 


2Vx2-.y2 


30. 


100^-35\/6 


33. 6-2V6 + 2V3-2V2 



XX DURELL'S ALGEBRA : BOOK TWO 

BXBRCISE 44 



1. 


4V7 


ft 


b^-7^1^ 


21 




12 


s. 


by/2x 


T. 


76 - 22 VIO 




6x 




13 


3. 


4^ 


8. 


11 + «— 6Va; + 2 






Q 


7 -X 


4. 


Va^-ft^ 


X4-5-2V3X + 6 




a-b 




1-x 


S. 


Va"^ — rtft — Va6 


-^ 10. 


a« + V«*-6* 




a- 6 




6"^ 


11. 


V8+v^ 






14 


4(485 4- 74 VIE - 


- 106^5 - 120V3) 






C 


14. 




13 


Va6 + 6i-Va6 


o+Va2-6a 




& 




b 


IB. 


3V6 


19. .8944+ 


28. 7.888+ 




-V3 


20. .9449+ 


23. -.18036 


16. 


2 


21. .6666+ 
EXERCISE 48 


24. .4031+ 


1. 


2\/2-3 


9. 2\/l6-3V3 


IT. 2-V8 


8. 


V5 + 2V6 


10. JV2 + iV6 


18. 3-V2 


3. 


3V3-2\^ 


11. fV3-jV6 


19. 1+}V6 


4. 


V0-\/3 


12. i+V8 


20. 1+v^ 

21. V7-V2 


B. 


Vl4 + 2v^ 


IS. 2-.JV3 


22. 3 + 2\/3 


6. 


3V6 - 2V7 


14. Vm + n + Vwi 


~^ 23. 3V2 -V6 


7. 


2v/6+v/6 


IB. a + SVd'-i + l 


24. 2V3-V6 


8. 


3V6-4v/2 


16. V3-1 

EXERCISE 46 


2B. 2+V8 


1. 


5V-1 


8. 8V-2 


5. aV-1-6 


S. 


6^/33 


4. 


6. -(a 4-3 6) V:^ 



ANSWERS 



XXI 



47. 
48. 
49. 
50. 
51. 
5S. 
58. 



V0-2 




12. 28 V0 

13. -lOViO 

14. (y-a:)V^ 

15. (i(a-l)2V31 

18 v^ZT - 2\/^r6 + 9 V2 - 2 V3 



16. 
17. 
18. 



3+\/2 

-6-.5V6 

24 



35. 
86. 

87. 

38. 

89. 



a2_52^2a6\/Zl 

a^ + ft^ 
24 + 7\/lO 
2 

6 



40. 

41. 
42. 



13 

VTO + SV^li' 
14 

3_VZT 

43. 2V3 + 3>/Z^ 

44. 2^/Z^-3\/Z2' 

45. 4-3V^^ ' 

46. 5v^^ + 4\/2 

(a±ib); (a^±ib^); (x±V-^); ((w;±i6); («±i); (a:*+t*)(x*-t-) 
a;>2; a>V2 

V^T; 1 ; - 1 ; V^ ; vCn ; V^T; - 1 
53. Ti-l 56. 2-2V^n[ 

4-Qi 54. eV^l 57. fVin+4; 

3 55. \/^-ll W^-i 

V^-l, -V^l, 1 



1. 

2. J%j5->/Zr2 - 6\/2 

3. 4V2-jV^^ 

4. (3a2+62)VaaZ 



6-^ 



EXERCISE 47 

8v''676 
-QVQ 

Vn 



^- "IT 
10. -,vv 



XXll DURELL'S ALGEBRA : BOOK TWO 

H g + 5 + 2^05 jj 6(y/a—s/b) 

a—b ' a—b 

13 8v^ + 2+>/6 + 2\/3 l-\C~2-VZr3 

2 "• 2 

«. i^ 19. v':r6+v'r2 «*. 5 
"• y "• ieV^-12 M. 
18. J^ « « " i 

»"• ""•y:y' v5^^ 31. KVWTi+VVTTi) 

S9. v^; 1^; a^ SI. '=* + y^ 

80. -l + lOVri ^ *« + a=J, + j^ 

39. O, 

BXSRCISE 49 

1. ±4 4. ±2>/2 ^ j_26 9. ± V^^ 14. 50 ft., 200 ft. 
«• ±1 ». ±J>/3 a 10. ±4 16. 100 yd., 400 yd. 

8. ±1 e. ±5V2 8. ±6 11. _t6 

EXSRCISB 80 

i. 3, 4 10. 3, } 19. -JijVIf 

8. j, I 11. 3, - J 80. J ± j\/=l 

8. }, - i 12. 1, - J^ 21. I ± } Vi93' 

*. -}, -V IS. 2, 6 22. -2±v^ 

8. .3, .2 14. 4, -f 23. - 1, 1* 

6. -.4, -.05 15. 2,-5 24. 1.1400+, -6.1400+ 

7. ♦, -i 16. 2, 2 26. .9067+, - 2.5734- 

8. 2, -3 IT. J, } 26. 6 

8. 3, -J 18. 2, -Y 87. 8,9,10 



EXERCISE 61 

a&, 2 a 
_6£ 



1. 3a, 4a 3. -8a6, 2a6 - 2d 2d 



«. 6c,-2c «.2i>, , ,^ 3^ 



c ' 3c 



* 4a' 2a 



ANSWERS 


XXIU 


7. a, — 6 n. JL ?_^ 13 -£ , £5 

8. a, — & * 2 6' 6 * a6' c 


15. «=t6 
2 


». -3«,_26 «. -£ 6^ 14. „,-2 
10.p,-« ' »' 7 


IS. Sac±2ab 
IT. 2a, a- 6 

10. b^iP+J^ 
P+b 


'"• -2i^fa^^-^«^ ^^- -|i|^^^-^^ 



EXERCISE 51 

1. 2, 4 8. 0, 2, 4 S. i, - 4 4. 3, } 5. 0, 1, - 2, 3 6. 0, ± 2 

7. 3, -|±iVZl 9. 2, -2, ±2V^ 11. -1, ±1, ±V^ 

as, -S±iV^ 10. ±a, ±2a 12. 2, - 1 

18. 0,1,2,-} IS. 2,4, -3 

14. 2, - 1 ± J Vl6 IT. 3, i, i 

15. -2,-1,3 18. 1,2,6 

19. 2, - 2, 1 ± V^s, - 1 ± V^rs 
«o. 1, -iijv^s; 3, -fijVirs 

21. 9, -.J±J>/i:8 «*. 86, -2 ,g ^^ £_ 

M. 2a, 66 SB .^ _c ^ + ^ 

IS. 3«;-6a ''' ^ "-'^' + '' 

EXERCISE 53 
1. ±2, ±3 8. ±1, ±6 5. 1, 2, -i±jV^-l±V:r3 

«. ± 1, ± V2 4. ± 1, ± i 6. 2, \^, - r db V^^ 

7. 1, 16* ». 1, 64 11. 27, - J 18. 1, ^- 

8. 16,H 10.1,32 1*. 4,A* 14. 1, (-})* 
15. 64, i 16. 14, - 1* IT. 1, - J, - 1, f 
18. 2, -3, - J ± iVTS M- 2, - 3, J, - I 

19. 1, im **• ^' A* 

j^ 1 1 3j:V:r603 w. -A»i^J^ 

• '2' 4 26. ±1, ±,^V310 

11. 7,-4,4, -1 «T. 1, -3, -1±V3 

M. 14, - 1* S8. 1, - J, 1 ± V3 

EXERCISE 54 
1- - 1» 34* 8. 7, J* 8. J Vl5, - jVir* 



XXiv DURELL'S ALGEBRA : BOOK TWO 



T. 9, - V* 


10. H.o* 


18. 1 V3, - f V3* 


». 1*, A 


11. 3, - 6* 


14. 3, J 


9. Va«-6«, 


13. 0, «• 




-.>/aa-ft» 


EXERCISE 65 




1. 1,7 


6. - 2 a, 6 a 


8. ±1, ±1 




. 35 4& 


9. -8, A 

10. -«, ^ 


4. ♦.-* 


7. 2, - 3 6 

EXERCISE 57 


1-a'l + a 


i. h-i 


4. 0,2, -l±V-3 


7. -l,ta±iV85 


S. 6 a, - 2 a 


5. ±V2, ±2 


a 2 a, - 3 6 


3. 6,jyi 


6. 4,1 


3. 1, A 10- s. J 


11. 0,^1,^^;^-^ 


-l±V-3 
2 


13. « ^ 

6 a 


13. 4, -J 


15. -i2» 


IT. -2,J,ldbjViO 


14. 0, 2, ± 1 


16. - 1, - a 


18. 0,5 


w. 0,5,1, -¥ 






30. 4, 138tV 


jg a + 26 a- 2 6 


34. 2a 6, ^'-l^^' 


31. a2, - 4 a«» 


2a- 6' 2a + 6 


b 


33. 1,W 


EXERCISE 58 




1. 9, 13 


10. 35 in. 


19. 9 in. 


3. 6, 7, 8 


11. 3 


SO. 20 


3. 13 


13. 3,6 


31. 10 rd. 


4. 8 


13. 4 hr., 16 hr. 


S3. 15 ft. 


6. 11 yd., 16 yd. 


14. 6 mi. per hr. 
10 mi. per hr. 

15. 6 mi. per hr. 


S3. 8 ft., 6 ft. 
«4. -f 


7. 8 ft. 


16. 82 

17. 31 


2 


8. 50 yd., 100 yd. 


36. ii + jVl+4 6 


9. 10 rd. 


18. 25 


36. -a + Va2 + 6 



ANSWERS 
EXERCISE 59 



XXV 



1. B^yj^; D=yl^ 

2. 83.22+ ft. 

8. 4.10+ ; i?=-Y±\ 4+- 

4. 1.782+ 



5. E^^-±^— + - 



6. 12.8+ sec. 

ff 

8. « = 1 (t? ± Vt;2 + 2 agr - 2 grA) 

9. 121 ft.; less 

10. 191.26+ ft. 

11. 11.34+ sec. 



8. 



l-2,J 
{5, -1 
12,1 

fl,i 
10, -i 



±1, ±4 
± 4, T 14 



f±2, iiVIO 
I ±3, ±iVlO 

J -12, ±Jj^V6 
1 ± 5, T iV6 

f±4, ±fV3 
U3, ±JV3 



f±6, 1- 
U4, ± 



L-V3 
3>/3 



1-3, 



-3 
5 



EXERCISE 60 



2,7 

0,7 



T. 



[4,-1 
1-1,1 
f-4,- 
10, -1 



f4, -A 
16, -A 



12, -i 



BZSSCISB 61 

•• l±3, ±f 

- f±l, ±iV5 
iTl, ±1V3 



±16 



10. 



±3.±JL 

V91 



±1, ± 



11 
V9l 



EXERCISE 6S 



ii 



11. 



U. 1,2 



l<» 



-\a 



a f±2,±7V2 

• \±5, TeV2 



IS. 



-3,7 
-t,4 



f±4, ±7^=2 
Ul,=F3V=2 

U. 36 



l=F3, 



T3 
±6 



XXVI 



DURELL'S ALGEBRA : BOOK TWO 



4. 



10. 

U. 
IS. 



f4,2 
12,4 

f±2,±J 
1=FJ,=F2 

r3a, a 
\a,Sa 

U.I 

(4,-8 

1-8,4 

[a±h 

la=F6 

/5,2 

12,5 

4,-2 

-2,4 

± i. ± i 



f8, 1, 2^:8^^ 



u. 



14. 



IS. 



16. 



17. 



18. 



19. 



11,1 
[h-i 

r±i. ±2 

l±2, ±1 

f»,4 

14,» 

f4, -20 

120,-4 

f4,8 

\-8,-4 

la + b, -a + 
\a — b, —a — 

P-2g,| 
_«, 2«-j) 





11. 


f±8 
±3 




SI. 


f ± 8 6, ± 2 6 




l±26, ±86 




u. 


[±1, ±JVS 




Its, t}V3 




S4. 






S6. 


[I.I 
it. i 




M. 


[4, - V 

i2.-» 


b 


«. 


f±2, ±8 


b 


[=F4, Tl 




S8. 


8, -1 

ll, -8 




S9. S 


.,» 



BZBKCISE 6S 




161 
161 



7. 



9. 



10. 



|i,i±vc:2 

l-i,-i±v:r2 

|8.2,i±ivr 

f-46,2 
116,3 

(±2, ±V3 
l±l,0 

f4, -I 

U. -♦ 

f±3,±l 
iTl, =F8 



EXSKCISX 64 



1. 



fl,8,i(9iv^)» 
t8,l,i(9TVgO)* 



fl,_6,-4,-l 
■ l-8,i,-J,-2 



ANSWERS 



XXVU 



8 f4,l,J(5±V-ll)* 

• 1 1, 4, }(6T v^ni)» 

, f5, -l,i(9±Vl01)» 



1. 



f3, -4, ±2-v^ 
1-4,8, T 



7. 



2V3 



|±v^,_±l 



10. /3.0, =FVi 

(-2,1,- 
U. / 26, 4, 43 ± 30V2 

1 4, 25, 43 T 



3V2, Tl 

9,-l,4±ViO,-2,-3, =li|^ 
1,_9,_4±VI0,3,2, ^±^ 



v/6 

2±v^ 



30 v^ 



f±2 

l±4 
U. f±4, ±V2 

1 ± 2, T *V _ 
14 f3,2,i(6±>/73) 
' 1 1,1, 



\V2 
J(5T-V73) 



f8,-4 
12,-4 

l-i. -4 



li,l 

10,2 



le, |5,4, 8±2Vn* 

U, 5, 8T2V1T* 

'M5 



-3 



a 

2 

6 

12 



EXERCISE 65 

MS 



f2,2, f, 
I 2, 6, 2, 



f 2, -6,2,6 
ll,-3, -1,-S 
f - 1, 
10,1 



fj(2±V3),0 
l2±V3, ±\/3i 






-2 
6 



EXERCISE 66 



3 f±l, TAV80 
1 ± 2, ± f V80 



XXVIU DUBELL'S ALGEBRA: BOOK TWO 



7. 



(2.1, 

11,2, 

f 1, 1, 6, J 
l«,4,2,2 

1 1,8 
1-8,-1 

f6,2 
18,6 



10. 
11. 
It 
IS. 
14. 



16. 
17. 



i(8±V^^66) 
i(8=FvCr55) 



6 
2 

f 6, -«if«, V*, 10« 
1 4, i«, V*, «• 

/l,-3 
1-3,-7 

(8,2 

IL* 

(l,i 

(6,i 
1-2,-} 

(l,f,i(7±Vl9) 
l3, f J(7T%/I9) 

(4,2 

18.4 ,^ ,. 




»• r f* 

1±3, 



86. 



1 



18. |2,i,6,-! 
l3,}, -10, -f 

19. JM _ "■ 11,-2 

28 8 2 37. /3.-2,SM« 

f27, 1 '»• ±J, ±1, ±« 

11,27 89. 1,2,8 



n. 



ANSWERS XXIX 

EXERCISE 67 

1. 8, 3 11. ©60 cu. in. f 6 mi. per hr. 

S. 7,4 la. f I * [2 mi. per hr. 

8. 6, 11 18. 24, 16 «0. 12, 9 

4. 9, 8 14. 10 da. 81. 26 yd., 12 yd. 
•• ®» 2 15 1 ^^ ^^' P®'' ^^- M- 220 ft. 
••«'l • t45ml.perhr. „. g^ rd.. 60 rd. 

7. 16, 12 16. 9 in., 12 in. ^^ - 

5. 30 rd., 26 rd. 17. 36 

9. 26 yd., 9 yd. J 6 mi. per hr. 

10. 48 rd., 20 rd. ^*' \ 1 mi. per hr. **• 28 da., 21 da. 

EXERCISE 69 

In this exercise the results are given as points of intersection. 
1. (0, 0), (9, 9) 4. (2, 2), (8, - 4) 7. (- 4, - 3), (4, 3) 

8. (0, 0), (1, 8) 8. (0, 6), (3, - 4) a (4, 1), (1, 4) 

3. (0, 0), (9, - 9) 6. (- 4, - 3), (3, 4) 9. (4, - 1), (- 1, 4) 

10. (- 3, 1), (1, - 3) 16. (0, 0^, (4, 2) 

11. (1, - 1), (- 1, 1) 16. (2V3, ±2), (- 2V3, ± 2) 
18. (±4, :f3), (T3, ±4) 17. (0, ±V6) 

IS. (3,4), (-4, -3) 18. (i, -i) 

14. (2,1), (-2,1) 19. (3, -3), (6,0) 

80. (-2, -6), (-6,-1) 

81. (1) Results are imaginary. 

(2) (3,0), (-3,0) _ 

(3) (±AV66, ±AV91) 

88. (1, 2), (2.1+, - 1.6+), (- .7+, 2.4+), (- 2.3+, - 2.7+) 

EXERCISE 70 
1. ± 2 6. 0, ± 1.4+ 

8. 4, - 1 7. - 1, 1.61+, - .61+ 

3. i, - f 8. 0, 3, - 2 

4. 3 9. 2, ± 1.4+ 
8. 3.73+, .26+ 10. db 1, ± 4 



XXX DURELL'S ALGEBRA : BOOK TWO 

BXERCISB 7S 

1. Real; uneq. 6. Real; oneq. 11. Equal 

S. Beal ; uneq. 7. Real ; eq. IS. Imag. 

S. Real ; eq. 8. Equal. IS. (1) Real ; uneq. 

4. Imag. 9. Real; uneq. (2) Equal. 

5. Real ; uneq. 10. Real ; uneq. (3) Imag. 

15. Imag. 18. ±10 il. 3, >- 4 

16. i !•. ± ♦ aa. - 3, - J 

17. Y 10.-2 23. - 1, V 

15. Greater than i ; less than i 

16. > V, < V ; none, all (except w = — 2) 
19. (1)9 (2)10 (3)- 16 

EXERCISE 78 

1. Sum=-3 •.!»-* 8- -A. -A 

Product =6 ^ a-1 a« 

1.1,7 6.1,1 ••—'7 

3. 6,-10 10 — - ^""^^ 

4. 3, -i 7. 4,i a' 2a 
11. x2 — 6a; + 6=0 18. x2-4x + 6 = 

11. x2-a;-6 = 17. x?- 3x2 + a;- 3 = 



M. -J 



IS. x2 + 6x+5 = 

14. xa-6.04x+.2 = ^.^_^ 

15. xa + Jx + } = *•• J—^ 

16. xa-ix-l = 3Q 3x2-10x-8 = 

17. x2+.12x-. 016 = 



31. (x-4)(3x + 2) 
l±v^) 
1±V6\ 



18. x2-(o6-a)x-a26 = ^ 

19. ax2-(a2-62)x-a = ^- (;« + l±^) 

10. x2-2x--l = 
81. x2 + 6x + 6 = 

11. 2x2-4x+l = 
IS. 2x2-2x+l=0 
14. 2xa + 4x + 3=0 

15. 4x2 + a2 + 4(J26 = 4ax 87. -3(x-J±jV-^ 

88. f;i; V;f;¥; W;!; -»; -it 




ANSWERS XXXI 









q q g2 

43. p=— 2; rootB = 17, 

£XERCIS£ 74 

1. 16:6 6. Scfi'.l 11. 3:2; 6:3 

2. 7:2 7. 9:1 U. o:l; 6:1 

3. 7:3 8. 14:11 18. 3:7; g:p 

4. -1:1 9. 4:3 14. 19:40 

5. a; : 200,000 10. -11:7 

15. First two, commensurable ; last, incommensurable 

16. 89,660 (g ^ fe)8 86. 8, - 1 

17. 2362.6 lb. ' a-\-b 87. 6 a, - 4 a 
Ij c(b-a) ^ (x + 1)» 88. 8 

* 6(6 + c) * x-1 89. 7 



(a-ft)« 


a + 6 
(« + 1)» 


x-1 
.027 
ft-1 


b(b + 1) 
(a-6)» 



20 <^(<^-^) 80- 027 40. 8 

" 6(6 +c) ft-1 41. 54,36 

251. ±12axy -5(6 + 1) ,^^ ,^ 

a«- ± -^2 3j (a-6)' "• 8d32^' 3d=2^ 

24. ±(a+6)»(a-6) « + & 43 _«^ _bd__ 

25. ±ix(x-3) 33. 2 • i^a' b-a 

26. ±Jv^3 34. 1, -V 44. 30 

27. Y 85. ±1 45^ 7jft. 

46. 6 ft. 2^ in. 

48. (1) $9600,^6000 49. a: 6 = 4:5 

(2) $7200, 1^4800, $3600 &:c = 6:6 

50. 1260 cu. ft., 4740 cu. ft. 

51. 45 52. 6400 58. 16 54. 25% 



XXxii DURELL'S ALGEBRA : BOOK TWO 







EXERCISE 76 






1. 0,5,-4 




6. 2±lV6 




19. 4, 1 


1. 0,-7,- 


-V 


7. 9, - 12* 




20. a% ab^ 


8. -P,P- 


g 


8. 3i)2 




81. 

86. 64,000 


4. ±3,-2 




17. 6, 13 




86. ^ 


5. 64 




18. 34, 18 
EXERCISE 76 




P 


1. flo; 




9. 00 




la 00 


- ^..0, 


«i" 


10. 00 

11. 00 




"•* 1 . 


ao. (1) -f-V-4ac 


• A 2 








2a 


'•2 = ''0 


= Q0 


la. 0,00 




(2)0 


4. 4 




18. 

U. 00 




(3)« 


5. 12 


81. (1) — LV-4ac 










2a 


6. 6 




16. 2,-00 




(2) -5 
a 


7. i 




16. 




& 




17. 00 
EXERCISE 77 




(3) -00 


t. 31 




la. 673J 




19. ^(6aj-4y 


3. - 26, - 


81 


18. -166 




2 


4. -¥i - 


13 


14. -77V3 




+ 2ry-'rx) 


5. 64,94} 

6. i,0,3f 




15. ^^30 a 
a 




80. 900 

81. 166 strokes 


7. - 7.2, - 


-37.8 


16 »C3n-l) 




88. $26,360 


8. 88 




2 




83. 679.6 ft.; 


9. 164 




17. |(16-3i>) 




3622.6 ft. 


10. -189 






84. 322 ft. 


11. 148t 




18. w(6-n) 
EXERCISE 78 




26. 600 ft. 


1. a = 4; 3 


= 286 


6. a = 6i; d = - 


■2i 


9. n = 21; d=-2 


2. a = -6J 


; «=209 


6. a=-i;d = 


A 


10. n = 18; (l = -A 


8. a = 6; (1 = 4 


7. a = 3}; d = - 


-1 


11. n = 16; d = | 


4. a = 11; 


(l=-3 


8. n = 21 ; d = : 


I 


18. a=-4; n=6 



ANSWERS xxxill 



18. a = 8 ; n = 6 




16. 


a = 


-i;n 


= 16 17. 8 19. 9 sec. 


14. a = 7; n = 6 




16. 


12 


18. i 


tor 9 






EXERCISE 79 


1. d = -2 


4. d 


= i 




7. ^ 


+62 X 10. 1061 yr. 


2. d = i 


6. - 


•Hi 




2ab ' x^-y'^ 11. 130 


8. d = -A 


6. X 






8. 3i 


ft. 






EXERCISE 80 


1. 6,7 








16. 


12 


a. 4i,8 








16. 


-5,-4^... 


3. -V,-V 

4. 6c-76,4c- 


-66 






17. 


fll, 6, 1,-4,-9 

1 - V, - ¥, V, V» 21 


5. »2 








18. 


2, 5, 8, 11, 14 


6. 400 








19. 


134 


- 7n(n + l) 








80. 


30; 13; 150; 100 


^' 2 








81. 


819; 70,336 


9. 102dterm 








88. 


288 ft. 


10. 8, 7J ... 








88. 


W 


11. 3,-2,-7,- 


-12 






84. 


18° 


12. 1,3,6,7 








86. 


0, 357, 826, 212 ft. 


18. 24 days 








86. 


310, 53, 1336, 398 ft. 


14. 1,4,7... 








87. 


948H ft. ; 14,475 a ; 20 sec. 






EXERCISE 81 


1. 486 
8. 192 




6. 


A 
a = 


m 


-ife 


8. -^ 

8 = 33if 
4. 16 

« = 65 




6. 

7. 


32 

1 

616 




9. -68 

10. w 

11. -^^ 


^' «M 








18. 


386,268,750 


18. ^(S+VS) 








19. 


$1657.69, $3773.37 


14. 21V2+28 








80. 


12440.74 


16. 2»-l 








81. 


9,226,406,250+ bu. 


16. 0, 0, .25, .76, 


1.25 






28. 


$10,737,418.24, cost of en- 


17. 2,097,150 








tire shoeing 



XXxiv DURELL'S ALGEBRA : BOOK TWO 

EXERCISE 82 
1. 2, 728 5.-4 9. 6 IS. 6 

t. 6, - 426 g I IQ 5 14. 5 

3. 46, WJ^ 



66 + lOv^ 



7. -} 11. 6 15. 5 



*• *' 9 a -J 11. 6 16. 6 

EXERCISE 83 

1. r = i 6. r = ~i ^ ^ la. .026 

a. r = J 6. ±J • ^}^l 13. 7 

3. r = -2 7. ±6 10. .49 u. r = — 

4. r=-4 8. ±42o2a;j^ 11. .26 o^ 



16. r 



-(I)' 



EXERCISE 84 

1. 3 3. -^ 6. W f' H 

2. t 4. Y •• I «• 6(2 +V2) 

9. J(3V2 + 4) 10. J(3V^ + 2V3) 11. a 

13. ^ 16. 3ilJ 19. Ixh M- 2 

14. Hf ". HSU 20. 1^^ 23. 180 ft. 
16. 6^ 18. 3t¥^ 21. mi «• 240 ft. 

EXERCISE 86 

1. h±i'" Ml- 3.«»12 j^ |2,4,8,12 

2. A»t- !«• 1,3,9,27 ' !¥»¥»*»» 

3. 96, ± 48 ... .. 1 5, 8, 11 W- - 24i, f 

10. 6,16,46 115,8,1 13 (2,4,6,9 

j 7, 14, 28 !*• ^ 15' 3^» ^^' 120 12, J, - }, 9 

"■ [63, -21,7 16. 2V2 + 3 

21. 62J, 0, 2280, 16,660 sq. mi. 24. 7.89 in. 
23. .263+ 26. Between years 1090 and 2000 

EXERCISE 86 

2. ^+i+(n + l)xna ^ C!^J^xn-ia^+ ^^ + ^^^'^ " ^^ o^-^a' + .♦- 

[2 [£ 



ANSWERS XXXV 

£X£RCISS 87 

2. 82 a« - 80 a*x + 80 a»a;2- 40 a2a;8 + 10 aa^-a^ 

8. 1 + 30:4--^ + --+^^ + — + gj 

4. 81x2-216xV+216xy*-96xV + 102/® 
6. a:*-10x* + 40aj»- 80x^ + 80x^-32x6 

6. x"V + 7x"i + 21 x"i + 36 x"^ + 35 + 21 x* + 7 x^ + x^ 

7. Ax6r*-A«V^ + J«*y"*-|xV^ + 4«^-«V 

8. x-io - } x-V + If. x-«y8 - J 5 x-*y9 + ^^ x-SyW « ^Jy yU 

9. 243 a*x"V « 405 a^x-e + 270 a*x~* - 90 ox'S 4 16 a^x"* - 1 

10. 16 x2 + 82 x^y* + 24 x^ + 8 xV + xM 

11. 32 x5y"4 - 40 x^y-i + 20 x^y^ - 5 x^y^ + f x-iy* - ^ x"*y6 

12. 64 a*x-2 + 676 a^'^x"* + 2160 a"^x"* + 4320 a^x* + 4860 a"^x* 
+ 2916 a~^\^^ H- 729 a-^x* 

13. 81 a-862 - 108 o-^ft-a _j. 54 ^-15-8 _ 12 5-10 + aft-i* 

14. x»-3x6 + 9x*-13x8+18x2-12x4-8 

15. 8-86x + 66x2-68x8 + 38x4-9x64-a^ 

16. 16x8 H- 32x'' - 72x6 -136xs+145x* + 204x8- 162x2- 108X + 81 

17. a8 + 8 a^x + 20 cfix^ + 8 a^x* - 26 a*x* - 8 a^x^ + 20 a^a^ - 8 ax^ + x^ 

18. - 14,784 a«xio 21. -198a*s'6T .1 

- 82 23. ^-^ 

19. 1716 x^y* 6920 a"^6* y^ 

20. 3008xi8ye ^ _o_ ^^ -61,236aHx6 

1001 xV ^^'y 

26. >^(n~l)-(n-r + l)^.,^, 

27. (n + 2)rn + 1) . .. (n - r + 4) ^...Sqr-i 

|r-l 

28. -1320x6 32. 24,310 xy" 36. 648.76873636 

29. 1865a*xi8 33. 3.188+ 37. 16,016 

30. mt^x^ 34. 8.916+ _JU 

" 38. -1,298,600 a ^ 

31. -112,640 35. 8.167+ 



XXXVl DURELL'S ALGEBRA : BOOK TWO 

EXERCISE 88 

1. «<! 7. j.^ ab 18. x>8i, y<8} 

»-^>i 9!x>6and<7 «• Either 17 or 18 

4. x>2 10. x<2and>lj 1*- 1^ 

6. x>e 11. a:>261and<801 M. 18 

6. x<i la. ar>59}aiid<66i 18. 88; 8 

EXERCISE 89 



^ * 10. eocVT; « = 2ir-\C 

8. 5 "^ 



8. 6 

3. 8 

4. ±2 



» * 






18. 256 ft. per sec. 




6. 86 






13. Jsec. 




7. 8 






14. .167f 




8. A 






16. 4 




9. « oc «2 ; a = : 


16.1 1« 


16. 9(±V6-2) 










EXERCISE 90 




1. log^ = 2 
logs 27 = 8 
log4 64 = 4 






log4A=-2 
log3i = -2 
logs A=-4 


logio A=-l 
logio .01 =-2 
logio .001 = — 3 


8 log2 32 = 6 


10g2A = 


= -5 log2xiT=-7 


log48 = i 10g8l6 = | 


3. 1 










9. </^=4 






v/1024 = 4 
EXERCISE 91 


v^4096 = 8 


1. 2 


4. 


1 


7. 10. 


-5 13.-4 


8. 4 


6. 





8. 11. 


14. 2 


3. 2 


6. 


-2 


9.-3 12. 


3 19. 1 



16. 4, 3, 6, 2, 1, 6, 1, 2, 



ANSWERS XXXVU 

£XERCISS 92 



1. 


1.6682 


6. 2.2430 


11. 8.8797 - 


10 


16. 9.8914-10 


2. 


1.9294 


7. 1.6172 


12. 3.7619 




17. 8.6309-10 


3. 


0.7782 


8. 0.6767 


13. 7.3366- 


10 


18. 2.3706 


4. 


1.9542 


9. 8.9031- 


10 14. 1.8008 




19. 0.7490 


6. 


2.4771 


10. 0.0086 


15. 0.4774 




20. 3.8911 








EXERCISE 93 








1. 


43 




7. 


4.09 




13. 


2.59 




8. 


770 




8. 


.627 




14. 


30.9 




3. 


236 




9. 


.00803 




15. 


7080 




4. 


3.78 




10. 


.0428 




16. 


77.7 




5. 


8400 




11. 


.00502 




17. 


283.6 




6. 


70.4 




12. 


.000126 




18. 


.4367 










EXERCISE 94 








1. 


105 


11. 


1.427 


21. 


.76183 




31. 


- 9.365 


2. 


34.3 


12. 


2.407 


22. 


.2526 




32. 


.3933 


3. 


.0755 


13. 


.3016 


23. 


4.359 




33. 


.17556 


4. 


7 


14. 


1.324 


24. 


1.4876 




34. 


22.58 


5. 


8 


15. 


.23317 


25. 


1.602 




35. 


- 1.162 


6. 


.04218 


16. 


-4.08 


\ 26. 


.6633 




36. 


3.2714 


7. 


64.7 


17. 


.4287 


27. 


3.936 




37. 


- 2.483 


8. 


.7996 


18. 


12.16 


28. 


.459 




38. 


.873 


9. 


681 


19. 


16.- 


29. 


14.44 




39. 


.35142 


10. 


- .2681 


20. 


197.68 


1 30. 


5.624 




40. 


1.6167 


41. 


$2514.60 




46. 


1.6838 ft. 


51. 


14.2+ yr. ; 


10.24+ yr. 


42. 


$995,200,000,000 


47. 


10.632 ft. 


52. 


7.6717 ft. 




43. 


$5716.30 




48. 


1.4029 


53. 


31.671ft. 




44. 


$5986.70 




49. 


.7333+ 


54. 


1759.2 lb. 




45. 


16.924 ft. 




50. 


216.15 


55. 


467.1ft. 










EXERCISE 95 








1. 
2. 


a + 2x 
3-a 




4. 
5. 


4 x2 - 3 X - 
a^ + ^a- 


-2 

1 


7. 


x-1+1 

X 


3. 


a2 - a - 2 




6. 


2x^-6x- 


3 


8. 


^2 + 2x 3 
^ y 2y2 



XXXVm DURELL'S ALGEBRA : BOOK TWO 











EXSSaSE 96 




1. 


16 


8. 


124 6. 3204 


7. 70.09 


t. 


91 


4. 


362 6. 804.6 


8. .0603 


9. 


.997 






15. 2.704+ 


91. 1.730+ 


10. 


4.217+ 






16. .3968+ 


99. .0536+ 


11. 


1.817+ 






17. .2147+ 


93. (631.3»+ cu. yd.) 


la. 


1.776+ 






18. 1.021+ 


94. (1493«+ cu. ft.) 


18. 


1.642+ 






19. 2.0033+ 


95. (692.8«+ cu. ft.) 


14. 


1.968+ 






SO. 2.901+ 
EXERCISE 97 




1. 


(1) a^±2a 


+ 2 


1 


(4) (a- 


6 + w — n){a — b — n +w) 




(2)a^±2xy- 


ya 


(5) (a5+l)(aj-l)(a; + 3) 




(3) (m-n)(fii 


-n 


+ 5) (6) (z- 


l)(a; + 2)(a--3)(x + 4) 


8. 


J(m-h2n) 






19. P' i 


19. ab 


3. 


-^, _SL 


_ 




90. 24 



p-q 

6 2a 

' aa-62 
^ 2(a + 6) 

a-b 
8. 2*, H 
10. -i,-i 



P + g 



13. f2,8,2±4V=:6 jj agr 
l8,2,2T4>/i:tf 



14. 



15. 



11. 



±8, ±-^ 
V61 



16. 



f5, 11 

1±4, ±4V7 
1,9 
9,1 

9 
26 



[26,- 
19, -i 



P 

99. 2, J 
95. 960 

98. ll-2n, 7-2n 

99. 300 

30. 1 



T6, ± 



13 
V6l 



69,049 

17. |aj'-2a: + 2 = 81. r = ±i,a = },^ 
12x2-2x4-1=0 « 

18. f i. **• -^^'^^^"^ 

83. 8x2_4_8x\/x2i:i 



84. Sixth, - 924 x* ; seventh, 231 x* 35. * = 

36. 8 ,« « 

40. 2x2-4 + 3x"2 

41. 3a + l, a-2 44. J* 
49. (i) 5+12v/iri (2)18 4-V^1 



87. 70 

88. 3, -2* 
39. 66,350 



10. Roots are J, J 
43. -4 



45. 9 x2 + 6 X — 19 = 



13, 
46. Row, 6 mi. ; stream, | mi. 







/^SWERS 


XXXIX 


47. -5} 

49. 0, 1.8, - 2.4, 6.8 
60. ±6V2, 
62. V7 


66. 

67. 
68. 


VS + lla 

iS»5; .16 
n=42 


64. m + n, *"-^~ 
mn 

66. -Y 

66. ir=fi 

4ir 


68. r 

-ft 

a' 


69. 

61. 
62. 
63. 


7,-4,4,- 

±v^, ±1 
3, 9, 15, 21 

1, J, A. - 
or3, },A- 


1 «7 ^_8'^ + irSIJ^ 

68. «' + ^\ 

(a2 - aja)4 

69. il^OO 


„ f9,7, i(25±v'378) 
■ [7, 9, i(25=FV873) 


72. 
73. 


4iV3-AV^V-2 
40 


71. l=J»»t 




74. : 


2x*-3ajiy+2x"V 


, V , IS V^ - 82 83 

'=2^\ 12 

Af,,=2^\ 12 


76. < 
76. 

77. : 





p Vq^ - ipr r 
q P ' P 
3 mi. an hour 


79. r=J 

80. 16 or 14 


81. 
82. 


±2V3 


88 |±4.±i 


84. r = 3, 1 ; series 2, i 


6, 18 


...and2, 2, S 


> ... 


86. 0,1,-2 

86. P'-l 
11,-3 


90. 
91. 


1,27 


94 J^'^ 
96. 10,000 


^- i±3,±! 


92. 


[2,3,4,1 

13,2,1,4 


67 
96. x = i; y = 2 


89 |2»3,3, 2 

• \ 1,-2,0,0 


93. 


±3 


97. n=4; d=-l 


98. (1) 2 + V3 

(2) i\/2+lVlO 




103. 
104. 


J a^i bh 
b-a b--a 


(3) ■y/a-\-b-^ Va 


-6 


180 ft. 


99. p. .7+ 
10,-1.8+ 




106. 
1Q6. 


f2±V-2, -2±v^6 
2 T \/- 2, - 2 T \/6 


100. (1) j±^/l + ^ 


± 


6c — ad 


(?) 6, i 




108. 


V7±V5 109. -1 


(3) 0, 1 




112. 


±2, ii 114. d = 4 


101. «*, 









xl DURELL'S ALGEBRA: BOOK TWO 

115 2nC2n-l)..>Cn + l) ( ^^^ 

117. 64a«-.48a6« ^^^ a*>-y«» 

iig f2,l,l±V^ ' x^'^(x + y) 

1 1, 2, 1 T V- 2 HO. 8.6+ in. ; 81.8+ in, 

m. y^J^M; 23.7+ mi. per hour 



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