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Completing the square,

16

(2x2 - x)2 — 12 (2x2-x)+ =33+ Extracting the square root of each side,

(2x2-x)

This Equation will give

14 ±

53. Find the value of x in the Equation

x38x2+19x=12.

2x2-x-6; whence x= = 2, or -14.

Transposing the 12, and multiplying by x, we have

x48x3+19x2-12x=0.

23

4

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We have now a Biquadratic Equation, which may be reduced to the form. of a Quadratic by the method just exemplified.

x1 — 8x3+19x2 — 12x (x2 —4x
x4

We find the first two terms of the square root of the first member, by the common rule.

Completing the square,

(x2 —4x)2+3(x2 — 4x)=0.

1

16

[blocks in formation]

3x2-12x.

This remainder may be resolved into 3(x2—4x), in which the binomial factor is the same as the part of the square root above found.

Then we shall have the Equation

9

4

=

x2-4x=-3, or 0.

3

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529

16

[blocks in formation]

The Equation 22-4x=-3 will give x=3, or 1.
The Equation x2-4x=0 will give x=4, or 0.

54. Find the value of x in the Equation

x42x3+x=5112.

The first two terms of the square root of the first member are

x2-x;

and the remainder is x2+x; which may be put under the form

-(x2-x).

Hence, we shall have the Equation

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(x2-x)2-(x2-x)=5112..

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--

143

2

x+2x3-7x2-8x=-12.

The first two terms of the square root of the first member are x2+x;

Completing the square, we find

and the remainder is 8x2-8x; which may be expressed by -8(x2+x).

We shall then have the Equation

(x2+x)2-8(x2+x)=-12.

20449

4

(x2+x)2-8(x2+x)+16=4.

Extracting the square root of each side

(x2+x)-4=2; hence

x2+x=6, or 2.

==

The Equation x2+x=6 will give x=2, or -3.
The Equation x2+x=2 will give x=1, or - 2.

-

56. Find the value of x in the Equation

x10x3+35x2—50x+24=0.

Transposing,

x4-10x3+35x2-50x——24.

The first two terms of the square root of the first side of the Equa

tion are

x–5;

and the remainder is 10x2-50x; which may be expressed by

10(2−5x).

Hence, we have the Equation

(2_5)+10(2−5)=−24.

Completing the square,

(–5)+10(–5)+-25=1.

Extracting the square root,

–52+5=±1.

From the Equation x2-5x+5=+1, x will be found =4, or 1. From the Equation x2-5x+5=-1, x will be found 3, or 2.

57. Find the value of x in the Equation

x4-12x3+44x2-48x=9009.

The first two terms of the square root of the first number are

x2-6x;

and the remainder is 8x2-48x; which may be put under the form

8(x2-6x). Hence, we shall have the Equation (x2-6x)2+8(x2-6x)=9009.

Completing the square, we have

(x2-6x)2+8(x2-6x)+16=9025.

Extracting the square root of each side

58. Find the value of x in the Equation

(x2-6x)+495.

From the Equation x2-6x+4=+95, we shall find x=13, or —7.

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x4-8ax3+8a2x2+32a3x=s.

The first two terms of the square root of the first member of the Equation are x2-4ax;

and the remainder is 8a2x2+32a3x; which may be expressed by

-8a2(x2-4ax).

6

Hence, we shall have

(x2-4ax)2-8a2(x2 —4ax)=s.

Completing the square, we find

(x2-4ax)2 — 8a2(x2 — 4ax)+16a2=s+16a1. Extracting the square root of both sides,

(x2-4ax)-4a2±√s+16a.

Transposing -4a2, we have

x2-4ax=4a2±√/s+16a2.

Completing the square,

x2-4ax +4a2=8a2±√/3+16a1. Extracting the square root,

x-2a= 8a2±√/s+16a1. Transposing-2a, we find

x=2a±√/8a2±√/s+16a2.

PROBLEMS

In Pure Equations and Affected Quadratics Containing but One Unknown Quantity.

X

x2

1. Find two numbers such that their product shall be 750, and the quotient of the greater divided by the less, 31.

Let x represent the greater number; then

750

represents the less; and

represents the quotient of the greater divided by the less.

750

We shall then have the Equation

x2

=3},

750

from which we shall find x=50, the greater; ther =15, is the less.

750

50

2. Find a number such that if and of it be multiplied together, and the product divided by 3, the quotient will be 298.

Let a represent the number; then

x2

56

x2

represents the product of and of the number; and

represents this product divided by 3.

168

We shall then have the Equation

x2

168

from which we shall find x=224.

9

4x2

=2983,

3. A mercer bought a piece of silk for £16 4s.; and the number of shillings that he paid per vard, was to the number of yards, as 4 to 9. How many yards did he buy? and what was the price per yard?

Let a represent the number of yards he bought; then

4x

represents the number of shillings per yard; and

represents the number of shillings the whole cost; also

9

£16 4s. is equal to 324 shillings. Hence, we shall have the Equation

4x2

9

12 shillings was the price per yard.

=

=324; whence x=27 yards.

Then of 27

4. Find two numbers which shall be to each other as 2 to 3, and the sum of whose squares shall be 208.

Let x represent the greater number; then

2x

3

represents the less number; and the Equation will be

4x2

9

x2+

=208;

which will give x=12, the greater number; hence of 12=8, is the less number.

5. A person bought a quantity of cloth for $120; and if he had bought 6 yards more for the same sum, the price per yard would have been $1 less. What was the number of yards? and the price per yard?

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