Ex. 13. Given 3x2−2x+12=16-4, to find the values of x.

By transposition, 3x-12x=16-4-12=0;

and by division, x2-4x=0; ... by completing the square, x-4x+4=4, and extracting the root, x-2=2;

..=4, or 0. See (Art. 405). Ex. 14. Given x3-4x+6x=4, to find the values

of x.

(Art. 423), multiplying both sides by x, x1 —4x3 + 6x2-4x=0,

(Art. 422).. (x2 —2x)2 +2(x2 -2x)=0. •*. x2-2x+1=±1, and x=1±√±1; ... the three roots of the proposed equation, are 1, 1+-1, and 1--1. The other value of x, which is equal to 1-1, or 0, belongs to the equation (x2-2x)2+2(x2-2x)=0; hence there are four roots, or four values of x, which will satisfy this last equation.

Adding unity to each side, in order to complete the

square; 81x+18+

1 841 232

+ +16; x2 JC

29

and extracting the root, 9x+1=±22+4).

Let the positive value be taken; then by transpo

28

and 9x-4x=28; by com

pleting the square, &c., we shall have x=2, or

14.

521

But if the negative value be taken, 9x2+4x=-30;

and completing the square, &c., x=

2±√(266)

9

Ex. 16. Given 3x2+2x-9=76, to find the values

values of x.

Ex. 20. Given √(x+5)×√(x+12)=12, to find the values of x.

Ans. x=1, or-28.

Ans. x=4, or

4

Ex. 21. Given 2x2+3x-51(2x2+3x+9)+3=0, to find the values of x.

Ex. 22. Given 9x+2x(16x2 +36x3)=15x2-4,

Ex. 24. Given x1-2x+x=132, to find the

8 -3+/93

or

៖ ៖

Ex. 25. Given x5+x5756, to find the values of x.

Ans. x=243, or (-28).

Ex. 27. Given x+5=√(x+5)+6, to find the values of x. Ex. 28. Given x+16-7(x+1)=10-4√(x +16), to find the values of x.

Ans. x=4, or -1.

the values of x.

Ex. 35. Given

lues of x.

Ans. x=4, or -4.

Ex. 34. Given (4x+5) × √(7x+1)=30, to find

Ex. 36. Given x3+7x344, to find the values of

Ans. x=±8, or ±(—11)a.

Ex. 37. Given 4x+x=39, to find the values of x.

+

78

to find the va

Ans. x=3, or -15.

Ans. x=4, or 1/2.

Ex. 40. Given x2+11+√(x2+11)=42, to find

the values of x.

Ans. x=5, or ±√38.

Ex. 41. Given x2-12x+50=0, to find the values

of x.

of x.

x.

Ans. x=6(-14).

Ex. 42. Given 3x-1=10, to find the values Ans. x=6-4. Ex. 43. Given x-2x3-48, to find the values of Ans. x=2, or 3-6.

Ex. 44. Given x+2x3-7x2-8x=-12, to find the values of x. Ans. 2, or 3, or 1, or —2. Ex. 45. Given x4-10x3+3x2--50x +24=0, to find the values of x. Ans. x=6, or 1; or ±1. Ex. 46. Given x3-8x+19x-12=0, to find the values of x. Ans. x=4, or 2±√−2. Ex. 47. Given

-x

to find the values

,

of x.

Ans. x=4, or 1, or±7.

Ex. 48. Given 4x*+2=4x3+33, to find the va

§ II. SOLUTION OF ADFECTED QUADRATIC EQUATIONS,

INVOLVING TWO UNKNOWN QUANTITIES.

424. When there are two equations containing two unknown quantities, a single equation, involving only one of the unknown quantities, may sometimes be obtained, by the rules laid down for the solution of simple equations; from which equation the values of the unknown quantity may be found, as in the preceding Section. Whence, by substitution, the values of the other may also be determined. In many cases, however, it may be more convenient to solve one or both of the equations first; that is, to find the values

of one of the unknown quantities, in terms of the other and known quantities, as before; when the rules for eliminating unknown quantities, (§ I. Chap. IV), may be more easily applied.

The solution will sometimes be rendered more simple by particular artifices; the proper application of which shall be illustrated in the following examples. to find the values of a and y.

Ex. 1. Given x+2y=7, and x2+3xy-y3=23,

From the 1st equation, x=7-2y; ... x2=49-28y+4y2;

Substituting these values for x and x2 in the 2d equation, then 49-28y+4y2+21y—6y2 —y2=23, or 3y2+7y=49-23=26. (Art. 417), 36y2+84y+49=312+49=361; ... extracting the square root, 6y+7=19, and 6y 19-7=12; y=2, and x= -2y=7—4=3.

2

Ex. 2. Given 4xy=96-x2y3, and x+y=6, to find

the values of x and y.

From the first equation, x2y2+4xy+4=100, and extracting the root, xy+2=±10;

..xy=8, or -12.

Now squaring the second equation,

x2+2xy+y2=36;

but 4xy =32, or -48.

.. by subtraction, x2-2xy+y2=4, or 84; and extracting the root, x-y=±2, or ±√84; but x+y= 6;

.. by addition, 2x=8, or 4, or 6±84; whence, x=4, or 2, or 3±√21; and by subtraction, 2y=4, or 8, or 6/84;

..y=2, or 4, or 321.

Ex. 3. Given x2+x+y=18-y3, and xy-6, to

By transposition, x2+y2++y=18;

find the values of x and y.

and from the second equation, 2xy

=12;