Charles Davies.

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EXTRACTION OF THE SQUARE ROOT. 147

EXAMPLES.

1. Reduce \/75a :i bc to its simplest form.

Ans. 5a -\/3abc.

2. Reduce \/l28b 5 a 6 d 2 to its simplest form.

Ans. 8b 2 a 3 d^2f.

3. Reduce y^32a 9 b 8 c to its simplest form. ^

Ans. 4a*b* -\/2ac.

4. Reduce -\/256a 2 & 4 c 8 to its simplest form.

Ans. 16a& 2 c*.

5. Reduce V 102 4 a 9 ^> 7 c T to its simplest form. ^

Ans. 32a*b 3 c 2 t/Wc.

6. Reduce -\/ 7 29 a 1 b 5 c e d to its simplest form.

Ans. 27 a 3 b 2 c 3 -y/abJ.

7. Reduce -\Z675a?b 5 c 2 d to its simplest form.

Ans. 15a 3 b 2 c\/3abd.

8. Reduce ^/T^AbcPcH* to its simplest form.

Ans. 17 ac^d 2 -\ZWa.

9. Reduce ^/ 1008a 9 d : m 8 ' to its simplest form.

Ans. \2aH 3 m* <y/7al.
10. Reduce ^/2\5&a w b 8 cP to its simplest form.

Ans. 14a 5 b*c 3 vTT

11. Reduce y r 405a^ 6 d 8 to its simplest form.

Ans. 9a 3 b 3 d^^5a.

148 FIRST LESSONS IN ALGEBRA.

106. Since like signs in both the factors give a plus
sign in the product, the square of — a, as well as that of
+ a, will be a 2 ; hence the root of a 2 is either -\-a or — a.
Also, the square root of 2ba 2 b % is either -\-5ab 2 or — bob 2 .
Whence we may conclude, that if a monomial is positive,
its square root may be affected either with the sign + or
— ; thus, V9« 4 =±3« 2 ; for, -\-3a 2 or —3a 2 , squared,
gives 9a 4 . The double sign ± with which the root is
affected is read plus or minus.

If*-the proposed monomial were negative, it would be im-
possible to extract its root, since it has just been shown that
the square of every quantity, whether positive or negative,
is essentially positive. Therefore,

^/ZTq, V— 4a 2 , ^/'^8a T b,

are algebraic symbols which indicate operations that cannot
be performed. They are called imaginary quantities, or
rather imaginary expressions, and are frequently met with
in the resolution of equations of the second degree. These
symbols can, however, by extending the rules, be simplified
in the same manner as those irrational expressions which
indicate operations that cannot be performed. Thus, y— 9
may be reduced by (Art. 104). Thus,

V^9 = t/9x V^T=z 3 V^T,

and V— • 4a 2 = -y/Acfix -\J —\=2a-^~—i~\ also,

V — 8a 2 6= \/4a 2 X— 2b— 2a ^~2b sz 2a y/Wx ^^T.

Quest. — 106. What sign is placed before the square root of a mono-
mial] Why may you place the sign plus or minus'? What is an ima-
ginary quantity 1 Why is it called imaginary 1

RADICALS OF THE SECOND DEGREE 149

Of the Calculus of Radicals of the Second Degree.

107. A radical quantity is the indicated root of an
imperfect power.

The extraction of the square root gives rise to such ex-
pressions as -yja, 3 ^/b, 7 -\/2, which are called irra-
tional quantities, or radicals of the second degree. We will
now establish rules for performing the four fundamental
operations on these expressions.

108. Two radicals of the second degree are similar,
when the quantities under the radical sign are the same in
both. Thus, 3 y/b and 5c \/b are similar radicals ; and
so also are 9 -\/2 and 7 t/2.

together by the following

RULE.

the sum annex the common radical.

II. If the radicals are not similar, connect them together
with their proper signs.

Thus, 3a^/b'+5c^b = (3a+5c)^/b.

Quest. — 107. What is a radical quantity 1 What are such quantities
called 1 — 108. When are radicals of the second degree similar 1 —
109. How do you add similar radicals of the second degree 1 How do

13*

150 FIRST LESSONS IN ALGEBRA.

In like manner.

Two radicals, which do not appear to be similar at first
sight, may become so by simplification (Art. 104).
For example,

V48tf6 2 +6 ^/75a=4b <x/3a+5b ^/3a=:9b i/3a;

and 2 t/W+3</5=6 y/t + 3 V^"= 9 V 5 *-

traction can only be indicated. Thus, in order to add
3 -\/b to 5 -\A*> we write

EXAMPLES.

1. What is the sum of V27a 2 and ^/4%tfl

Ans. 7a-\/3.

2. What is the sum of V 50 ^ 2 and V 72 « 4 ^ 2 *

Ans. M&b-y/Y.

3. What is the sum of \l and \l ■ ■ ?

Ans. \a\,

4. What is the sum of Vl25 and V500a 2 ?

Ans. (5 + 10a) V57

RADICALS OF THE SECOND DEGREE. 151

,™ . , . /To - " . /Too .

5. What is the sum of \ / -r—r an d \ /-tzttt •

V 147 V 294

Ans ' 21^

6. What is the sum of y r 98a^ and <y/36x 2 —36a 2 ?

7. What is the sum of V98«% and V 288 ^ 5 •

il;w. (7«+12« 2 ^ 2 )' V / 2^

8 Required the sum of -y/ 7 ^ an( ^ V 128 -

^.^. 14 ^p^.

9. Required the sum of -y/%7 and -\/T47.

.Arcs. 10 <v/~3"-

/2~ /~27~

10. Required the sum of a/—- and

3 V 50

Ans. l ±/T.

1 1 . Required the sum of 2 y/lFb and 3 -\/64for 4 .

.Arcs. (2a+24o: 2 )V r ^

12. Required the sum of <^243 and 10^3637

>lw.s. ii9*/in

13. What is the sum of <\/320a 2 b 2 and V^Ta^^F ?

Tins. (8a5+7a*& 3 )<v/5.

14. What is the sum of ^Ibcfib 1 and ^300a% 5 ?

Ans. (5a 3 £ 3 + 10a 3 Z> 2 ) y^S.

152 FIRST LESSONS IN ALGEBRA.

Subtraction.

110. To subtract one radical expression from another,
we have the following

RULE.

I. If the radicals are similar, subtract their coefficients,
and to the difference annex the common radical.

II. If the radicals are not similar, their difference can only
be indicated by the minus sign.

EXAMPLES.

1. What is the difference between 3a ^b and a\/T ?

Here 3ai/~b — a 'y/~b—2a^/~b7 Ans.

2. From 27a V§7F subtract 6a V 27b 2 .

First, 27a ^27b 2 = 27 ab V~37 and 6a^/27F 2 = lSab^T]
and 27ab^/T—lSab^3 = 9ab^3 Ans.

3. What is the difference of ^/YS and ^~AS 1

Ans. if 3 .

4. What is the difference of -y/~2AaW and ^HWl

Ans. (2ab — 3b 2 )xT6~

Quest. — 110. How do you subtract similar radicals 1 How do you
subtract radicals which are not similar 1

RADICALS OF THE SECOND DEGREE. 153

/ /3 /5

Jr: 5. Required the difference of \ / — and \ /—-.

V 5 V 27

4 ,

Ans. — -yl5.
45

6. What is the difference of <\/T28aW and <^32<F1

Ans. (Sab— 4a 4 ) y^

7. What is the difference of -y/48a 3 6 3 and -\/9ab ?

Ans. Aab -\/3ab — 3 -\J ab.

8. What is the difference of i/242a 5 b 5 and t/2oW1

Ans. (\\a 2 b 2 -ab)^2ab.

a 9. What is the difference of \l —

and \ I — 1

' 9

Ans. -7T\/~3'
o

10. What is the difference of <\/320a 2 and <\/80a 2 ?

Ans. 4a -y/ 5 .

1 1 . What is the difference between

V?20a 3 F and ^/24bab^d 2 ~ ?

Ans. (l2ab—7cd)-\/5ab.

12. What is the difference between

V968a 2 6 2 and ^200a 2 b 2 ?

Ans. 12ab\/2.

13. What is the difference between

<y/Tl2aW and <}/28aW 1

Ans. 2a*b 3 A/Z

VMVERSn

J?A, °*

154 FIRST LESSONS IN ALGEBRA.

Multiplication.

111. For the multiplication of radicals, we have the
following

RULE.

I. Multiply the quantities under the radical signs together,
and place the common radical over the product.

II. If the radicals have coefficients, ive multiply them to-
gether, and place the product before the common radical.

Thus, ^f~a X \Z~Fz= i/ab ;

This is the principle of Art. 104, taken in the inverse
order.

EXAMPLES.

1 . What is the product of 3 -\/5ab and 4 ^/20a~l

Ans. \20ayfF.

2. What is the product of 2a ^/bc and 3a-\/~bc ?

Ans. 6a 2 bc.

3. What is the product of 2a ^ai+b 2 and — 3a ^a 2 +b 2 ?

Ans. — Qa 2 (a 2 +b 2 ),

Quest. — 111. How do you multiply quantities which are under radi-
cal signs 1 When the radicals have coefficients, how do you multiply
theml

RADICALS OF THE SECOND DEGREE. 155

4. What is the product of 3 \/~2~ and 2 -y/lTT

Ans. 24.

5. What is the product of f -y/%a 2 b and ^ -y/fc' 2 & ?

Arcs. 3 1 Q#6cy'l5.

6. W T hat is the product of 2x + V^ and 2a;— -y/l> ?

JLns. 4a? 2 — 6.

7. What is the product of

Va + 2^/b and Va—2\/b1

Ans. ■y/W—Ab.
8. What is the product of 3a^/27o7 by <y/lla 1

Ans. 9a 3 <^~W.

Division.

112. To divide one radical by another, we have the
following

RULE.

I. Divide one of the quantities under the radical sign by the
other, and place the common radical over the quotient.

II. If the radicals have coefficients, divide the coefficient of
the dividend by the coefficient of the divisor, and place the

Quest. — 112. How do you divide quantities which are under the
radical sign 1 When the radicals have coefficients, how do you divide
them]

156 FIRST LESSONS IN ALGEBRA.

Thus, — =z=\/— ; for the squares of these two

expressions are equal to the same quantity — ; hence
the expressions themselves must be equal.

EXAMPLES.

1. Divide 5a-\/b by 2b-yJ~c. Arts. »\/ — •

2. Divide \2ac-\JWc by Ac \/2F. Ans. 3ai/3c.

3. Divide 6a<\/95tF by 3^/W. Ans. Aab <y/3.

4. Divide Aa 2 ^/bO¥ by 2a 2 <\/W. Ans. 2b 2 ^/lO.

5. Divide 26a 3 b ■y/SlaW by I3a^/9ab7

Ans. 6a 2 b y/ab.

6. Divide 8Aa 3 b± ^Woc by A2ab y/3a.

Ans. 6a 2 b 3 ^/c.

7. Divide Vi^ 2 D y V%- Ans. \a.

8. Divide Qa 2 b 2 -yf20a 3 by 12 -/So. 4** « 3 6 2 >

9. Divide 6«yToF~ by 3-/5T Ajw. 2ab^~2.

10. Divide 484* y3T V 2b2 Vrs- Ans - 36Qb 2 .

11. Divide 8a 2 ¥c 3 y/7d 3 by 2a V^SST

^l/is. 2a¥c 3 d.

12. Divide 96a 4 c 3 -v/98F by A8abc^/W.

Ans. lAaWc*.

RADICALS OF THE SECOND DEGREE. 157

13. Divide 27a 5 b e ^2lb7 by ^/7a~.

Ans. 27a 6 b 6 -^3.

14. Divide 18a 8 & 6 VS5* by 6ab^7

Ans. 6a s b 5 i/2.

To Extract the Square Root of a Polynomial.

113. Before explaining the rule for the extraction of
the square root of a polynomial, let us first examine the
squares of several polynomials : we have

(a+b) 2 — a 2 +2ab-\-b 2 ,

(a+b+c) 2 =a 2 +2ab+b 2 +2(a+b)c+c 2 ,

(a+b + c+d) 2 =za 2 +2ab + b 2 +2(a+b)c+c 2

+-2{a+b + c)d+d 2 .

The law by which these squares are formed can be enun-
ciated thus :

The square of any 'polynomial contains the square of the
first term, plus twice the product of the first term by the second,
plus the square of the second ; plus twice the first two terms
multiplied by the third, plus the square of the third ; plus twice
the first three terms multiplied by the fourth, plus the square
of the fourth ; and so on.

Quest. — 113. What is the square of a binomial equal to? What
is the square of a trinomial equal to 1 What is the square of any-
polynomial equal to 1

14

158 FIRST LESSONS IN ALGEBRA.

114. Hence, to extract the square root of a polynomial
we have the following

RULE.

I. Arrange the polynomial with reference to one of its letters
and extract the square root of the first term : this will give the
first term of the root.

II. Divide the second term of the polynomial by double the
first term of the root, and the quotient will be the second term
of the root.

III. Then form the square of the two terms of the root
found, and subtract it from the first polynomial, and then
divide the first term of the remainder by double the first term
of the root, and the quotient will be the third term.

IV. Form the double products of the first and second terms,
by the third, plus the square of the third ; then subtract all
these products from the last remainder, and divide the first
term of the result by double the first term of the root, and the
quotient will be the fourth term. Then proceed in the same
manner to find the other terms.

EXAMPLES.

1 . Extract the square root of the polynomial
49a 2 b 2 — 24a6 3 -f-25a 4 -30a 3 6+16£ 4 .
First arrange it with reference to the letter a.

5a 2 — 3ab-\-4b 2

25a*—30a 3 b+49a 2 b 2 —24ab 3 + l6b±
25a 4 — 30a 3 b+ 9a 2 b 2

10a 2

40a 2 Z> 2 — 24ab 3 +16b± 1st Rem.
40a 2 5 2 — 24«5 3 +166 4

. . . 2d Rem.

t RADICALS OF THE SECOND DEGREE 159

After having arranged trie polynomial with reference to a,
extract the square root of 25a 4 , this gives 5a 2 , which is
placed at the right of the polynomial ; then divide the
second term, — 30a 3 b, by the double of 5a 2 , or 10a 2 ; the
quotient is — 3ab, and is placed at the right of 5a 2 . Hence,
the first two terms of the root are 5a 2 —3ab. Squaring this
binomial, it becomes 25a 4 — 30 a 3 b-{- 9 a 2 b 2 , which, subtracted
from the proposed polynomial, gives a remainder, of which
the first term is 40a 2 & 2 . Dividing this first term by 10a 2 ,
(the double of 5a 2 ), the quotient is + 46 2 ; this is the third
term of the root, and is written on the right of the first two
terms. By forming the double product of 5a 2 — 3ab by 4b 2 ,
and at the same time squaring 4b 2 , we find the polynomial
40a 2 b 2 —24ab 3 -{-16b 4: , which, subtracted from the first re-
mainder, gives 0. Therefore 5a 2 — 3ab + 4b 2 is the required
root.

2. Find the square root of a*-\-4a 3 x-\-§a 2 x 2 -\-4ax 3 -\-x 4c .

A?is. a 2 -{-2ax-{-x 2 .

3. Find the square root of a 4 — 4a 3 x+6a 2 x 2 —4ax 3 -\-x*.

Ans. a 2 — 2ax-\-x 2 .

4. Find the square root of

4# 6 +12;r 5 +5tf 4 — 2x 3 + v 7x 2 — 2x+l.

Ans. 2x 3 +3x 2 —x+\.

5. Find the square root of t

9a 4 -12a 3 6+28a 2 6 2 -16a& 3 +16& 4 .

Ans. 3a 2 —2ab+4b 2 .

Quest. — 114. Give the rule for extracting the square root of a poly-
nomial 1 What is the first step \ What the second 1 What the third 1
What the fourth 1

160 FIRST LESSONS IN ALGEBRA.

6. What is the square root of

a? 4 — 4ax 3 +4a 2 x 2 — 4x 2 -\-Sax+4.

Ans. x 2 —2ax—2.

7. What is the square root of

9x 2 — l2x-{- 6 xy-{-y 2 — 4y + 4.

Ans. 3x-\-y—2.

8. What is the square root of y 4 — 2y 2 x 2 + 2x 2 — 2y 2 +l
~t~°° ' Ans. y 2 ~x 2 — \.

9. What is the square root of 9a 4 & 4 — 30a 3 b 3 +25a 2 b 2 1

Ans. 3a 2 b 2 —5ab.

10. Find the square root of

25a 4 £ 2 — 40a 3 b 2 c+76a 2 b 2 c 2 —48ab 2 c 3 +36b 2 c*— 30a*bc
+ 24a 3 bc 2 — 3 6a 2 bc 3 + 9a 4 c 2 .

Ans. ba 2 b — 3a 2 c — 4a be -\-6bc 2 .

115. We will conclude this subject with the following
remarks.

1st. A binomial can never be a perfect square, since we
know that the square of the most simple polynomial, viz :
a binomial, contains three distinct parts, which cannot ex-
perience any reduction amongst themselves. Thus, the
expression a 2 -\-b 2 is not a perfect square ; it wants the term
±2ab in order that it should be the square of a±b.

2nd. In order that a trinomial, when arranged, may be a
perfect square, its two extreme terms must be squares, and
the middle term must be the double product of the square
roots of the two others. Therefore, to obtain the square
root of a trinomial when it is a perfect square ; Extract the
roots of the two extreme terms, and give these roots the same
or contrary signs, according as the middle term is positive or

RADICALS OF THE SECOND DEGREE. 161

negative. To verify it, see if the double product of the two
roots gives the middle term of the trinomial. Thus,

9a 6 —48a 4 b 2 -{-64a 2 b 4: is a perfect square,

since *)/9cfl = 3a 3 , and ^64aW=—8ab 2 ,

and also 2 X 3a 3 X — Sab 2 — —48a 4: b 2 = the middle term.

But 4a 2 -\-\4ab-\-9b 2 is not a perfect square : for although
4a 2 and -4-9Z> 2 are the squares of 2a and 3b, yet 2 X2ax 3b
is not equal to I4ab.

3rd. In the series of operations required in a general ex-
ample, when the first term of one of the remainders is not
exactly divisible by twice the first term of the root, we may
conclude that the proposed polynomial is not a perfect
square. This is an evident consequence of the course of
reasoning, by which we have arrived at the general rule for
extracting the square root.

4th. When the polynomial is not a perfect square, it may
be simplified (See Art. 104.)

Take, for example, the expression -\/a 3 b-{-4a 2 b 2 -\-4ab 3 .

The quantity under the radical is not a perfect square ;
but it can be put under the form ab{a 2 -\-4ab-\-4b 2 ). Now,
the factor between the parenthesis is evidently the square
of a +25, whence we may conclude that,

<\/a B b + 4a 2 b 2 +4ab 3 — {a + 2b) <>Jab.
2. Reduce" 2a 2 b — 4ab 2 -\-2b 3 to its simple form.

Ans. {a — b)^2b.

Quest. — 115. Can a binomial ever be a perfect power'? Why not %
When is a trinomial a perfect power 1 When, in extracting the square
root we find that the first term of the remainder is divisible by twice the
root, is the polynomial a perfect power or not ?

14*

162 FIRST LESSONS IN ALGEBRA.

CHAPTER VL

Equations of the Second Degree.

116. An equation of the second degree is an equation
involving the second power of the unknown quantity, or the
product of two unknown quantities. Thus,

x 2 z=za 1 ax 2 -\-bx=zc, and xy=zd 2 ,

are equations of the second degree.

117. Equations of the second degree are of two kinds,
viz : equations involving two terms, which are called incom-
plete equations ; and equations involving three terms, which
are called complete equations. Thus,

x 2 = a and ax 2 =b 1

are incomplete equations ; and

x 2 -\-2axz=zb i and ax 2 -\-bx=zd i

are complete equations.

Quest. — 116. What is an equation of the second degree 1 — 117. How
many kinds are there 1 What is an incomplete equation 1 What is a
complete equation'?

EQUATIONS OF THE SECOND DEGREE 163

118. When we speak of an equation involving two
terms, and of an equation involving three terms, we under-
stand that the equation has been reduced to its simplest
form.

Thus, if we have the equation

3x 2 +4x 2 — 4 — 6,

although in its present form there are four terms, yet it may-
be reduced to an equation containing but two. For, by-
adding 3x 2 to 4x 2 and transposing —4, we have

7a; 2 = 10.

Also, if we have

3x 2 +5x+7x+5z=9,

we get by reducing

3,r 2 -f I2x=z4,

an equation containing but three terms.
Again, if we take the equation

ax 2 +bx 2 +d=zf

we have

(a+b)x 2 =f-d and x 2 =~ b >

an equation of two terms.

Quest. — 118. When you speak of an equation involving two terms,
do you speak of the equation after it has been reduced, or before 1 When
you speak of an equation of three terms, is it the reduced equation to
which you refer 1 To what forms, then, may every equation of the second
degree be reduced ?

164 FIRST LESSONS IN ALGEBRA.

Also, if we have ax 2 +dx 2 -\-fx+b=zc
we obtain (a-\-d)x 2 -{-fx=c — b,

and consequently

a-\-d a+d 1

an equation of three terms.

Hence we may conclude : That every equation of the
second degree may be reduced to an incomplete equation involv-
ing two terms, or to a complete equation involving three terms

Of Incomplete Equations.

1 . What number is that which being multiplied by itself
the product will be 144.

Let x— the number : then

xx tf = a? 2 r=144.

It is plain that the value of x will be found by extracting
the square root of both members of the equation : that is

-\Zx 2 = -\/l44 : that is, x =12.

if the number of dollars be squared and 6 be added, the sum
will be 42 : How much had he ?

Let x=: the number of dollars.

Then by the conditions

rr 2 +6 = 42:

hence, x 2 =z42—6 = 36 and x=6.

Ans. \$6.

EQUATIONS OF THE SECOND DEGREE. 165

3. A person being asked his age said, if from the square
of my age you take 192, the remainder will be the square
of half my age : what was his age ?

Denote his age by x.

Then, by the conditions of the question

-192:

4«>=7

and by clearing the fractions

4x 2 —768=x 2 ;
hence, 4a? 2 —x 2 =768,

and 3x 2 =768

* 2 =256
x = 16.

Ans. 16.

119. There is no difficulty in the resolution of an equa-
tion of the form ax 2 =b. We deduce from it x 2 =. — ,

a

whence

x =Vv-

When — is a particular number, either entire or frac-
a

tional, we can obtain the square root of it exactly, or by ap-
proximation. If — is algebraic, we apply the rules estab-
lished for algebraic quantities.

Quest. — 119. How do you resolve an incomplete equation 1

166 FIRST LESSONS IN ALGEBRA.

Hence, to find the value of x we have the following

RULE.

I. Find the value ofx 2 .

II. Then extract the square root of both members of the
equation.

4. What is the value of x in the equation

3x 2 -\-S=z5x 2 — 10.

By transposition 3x 2 — 5x 2 =—10—8,

by reducing — 2x 2 =z — 18,

by dividing by 2 and changing the signs

x 2 = 9,

by extracting the square root a?=3.

We should, however, remark that the square root of 9,
is either +3, or —3. For,

+ 3x+3 = 9 and — 3x— 3 = 9.

Hence, when we have the equation

x 2 = 9,

we have #= + 3 and xz=z~3.

1 20. A root of an equation is such a number as being
substituted for the unknown quantity, will satisfy the equa-
tion, that is, render the two members equal to each other.
Thus, in the equation

x 2 = 9

there are two roots, +3 and — 3 ; for either of these
numbers being substituted for x will satisfy the equation.

EQUATIONS OF THE SECOND DEGREE. 167

5. Again, if we take the equation
ax 2 =zb i

we shall have

c= + v/ — and x= — \/ —

For, ax\ f\/~ I =b, or ax — = b,

and a x |

a

2 * • i

=zb, or «x — = 6.

a

Hence we may conclude,

1st. That every incomplete equation of the second degree
has two roots.

2nd. That those roots are numerically equal but have con-
trary signs.

6. What are the roots of the equation

3x 2 +6=:4x 2 —l0.

Ans. x — -\-4 and a?=— 4.

7. What are the roots of the equation

1 x 2

X 2_ 8=: — +10.

3 9

Ans. a? = + 9 and x=~— 9.

Quest. — 120. What is the root of an equation] What are the roots
of the equation a;2 = 9 1 Of the equation ax2 = b 1 How many roots has
every incomplete eqvation 1 How do those roots compare with each
other?

168 FIRST LESSONS IN ALGEBRA.

8. What are the roots of the equation

6x 2 — 7 = 3a 2 +5.

Arts. x=+2j o?=— -2.

9. What are the roots of the equation.

8-{-5x 2 =~ + 4x 2 +2S.
5

Ans. x=z-\-5, x=—5.

10. Find a number such that one-third of it multiplied
by one-fourth shall be equal to 108 ?

Ans. 36.

1 1 . What number is that whose sixth part multiplied by
its fifth part and product divided by ten, shall give a quo-
tient equal to 3 ?

Ans. 30.

12. What number is that whose square, plus 18, shall be
equal to half its square plus 30^.

Ans. 5.

13. What numbers are those which are to each other as
1 to 2 and the difference of whose squares is equal to 75.

Let #— the less number.

Then 2x = the greater.

Then by the conditions of the question

Ax 2 — # 2 =75,

hence, 3x 2 = 75 ;

and by dividing by 3, x 2 =25 and x =5,

and 2a?=10.

Ans. 5 and 10.

EQUATIONS OF THE SECOND DEGREE. 169

f*.
JC

14. What two numbers are those which are to each other
as 5 to 6, and the difference of whose squares is 44.

Let x= the greatest number.

5

Then — a?= the least,
o

By the conditions of the question
. 25

•*-

~36* = 44 *

by clearing

fractions,

36* 2 -

-25a: 2 = 1584;

hence,

lla 2 =1584,

and

x 2 =144,

hence,

x =12,

and

4, =10.

Ans. 10 and 12

15. What two numbers are those which are to each
other as 3 to 4, and the difference of whose squares is 28 ?

Ans. 6 and 8.

16. What two numbers are those which are to each other
as 5 to 1 1 , and the sum of whose square is 584 ?

Ans. 10 and 22.

17. A says to B, my son's age is one quarter of yours,
and the difference between the squares of the numbers re-
presenting their ages is 240 : what were their ages 1

, (Eldest 16.
Ans. < ■

t Younger 4.

15

170 FIRST LESSONS IN ALGEBRA.

When there are two unknown quantities.

121. When there are two or more unknown quantities,
eliminate one of them by the rule of Article 77 : there will
thus arise a new equation with but a single unknown quantity,
the value of which may be found by the rule already given.

1 . There is a room of such dimensions, that the differ-
rence of the sides multiplied by the less is equal to 36, and
the product of the sides is equal to 360 : what are the

sides 1

Let x= the less side ;
y=z the greater.

Then, by the 1st condition,

(y— x)x=36 ;

and by the 2nd, " #3/ =360.

From the first equation, we have

ocy— # 2 =36 ;

and by subtraction, x 2 =324.

Hence, #=^324=18;

360 ^
=20.

18

Ans. x = 18,y=z20.

Quest. — 121. How do you resolve the equation when there are two
or more unknown quantities 1

EQUATIONS OF THE SECOND DEGREE. 171

2. A merchant sells two pieces of muslin, which together
measure 12 yards. He received for each piece just so

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