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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
•pwe:
«•>>:<
.•.•-'«• ".