2

Then Vx+4x+4=144 by extracting the root;

Or x+2=12,

And therefore x= 12-2=10.

2. Given x2-6x+8=80; to find r.

First, x2-6x=8C-8=72 by transposition;
Then x-6x+9=4+2=81 by completing the square;
And x-3=81-9 by extracting the root;
Therefore x=9+3=12.

3. Given 2x+8x-20=70; to find x.
First, 2x2+8x=70+20=90 by transposition.
Then x2+4x=45 by dividing by 2,

And x2+4x+4=49 by completing the square;
Whence x+2=497 by extracting the root,
And consequently x=7-2=5.

4. Given Sx-3x+6=5; to find x.

Here x2-x+2=17 by dividing by 3,
And x-x=13-2 by transposition;

Also x3-x+1=13—2+1=3 by completing the square,
And x-==by evolution;

Therefore x=+=}•

x2

5. Given +20=42} ; to find x.

I

2

3

7. Given ax3-bx+c=d; to find x.

Here, ax-bx=d-c by transposition,

b dby division;

And x2.

x=

a

a

唔

a

4a2

8. Given x*+2ax3-b; to find x.

Here, x*+2ax3 +a3=b+a3 by completing the square,

And x2+ab+ by evolution;

Whence x2=b+a—a,

And consequently, x=√ : √b+a—a•

9. Given ax®- -bx2-c=—d ; to find x.

First, ax-bx-c-d by transposition,

EXAMPLES FOR PRACTICE.

1. Given x-8x+10=19; to find x. 2. Given x2-x-40=170; to find x. 3. Given 3x+2x-9=76; to find x. 4. Given

Ans. x=9.

Ans. x=15.
Ans. x=5.

x

+7=20; to find x.

2

3

10. Given 3x”—2x3—— = ; to find x.

Ans. x=

27

QUESTIONS PRODUCING QUADRATIC
EQUATIONS.

1. To find two numbers, whose difference is 8 and pro

duct 240.

Let equal the less number,

Then will x+8 equal the greater,

And xx x+8=x+8x=240 by the question;

Whence x+8x+16=240+16=256 by completing the square;

Also x+4=256-16 by evolution;

And therefore x=16-4=12= less number, and 12+8=20 greater.

2. To divide the number 60 into two such parts, that their product may be 864.

Let x= greater part,

Then will 60-x-less,

And xx60x60x-x3-864 by the question,

That is, x2-60x——864 ;

Whence 2-60x+-900-864+900-36 by completing the

square;

Also x-3036-6 by extracting the root;
And therefore6+30=36=greater part,

And 60-x-6-36=24-less.

3. Given the sum of two numbers =10 (a), and the sum of their squares =58 (b); to find those numbers.

Let x greater of those numbers,

Then will a-x=less ;
Rr

Vol. I.

2

And x2+a-x=2x2+a3—2ax=b by the question,

Hence these two theorems, being put into numbers, give 7 and 3 for the numbers required.

4. Sold a piece of cloth for 241. and gained as much per cent. as the cloth cost me; what was the price of the cloth? Let x= pounds the cloth cost,

Then 24-x= the whole gain;

But 100 x x: 24-x by the question,

Or x2=100x 24-x=2400-100x ;

Whence

That is, x+100x=2400;

the square,

+100x+2500=2400+2500=4900 by completing

And x+50-4500-70 by extracting the roots, Consequently, x=70-50-20= price of the cloth.

5. A person bought a number of oxen for 801. and if he had bought 4 more for the same money, he would have paid 11. less for each; how many did he buy?