XXV. THEORY OF QUADRATIC EQUA

TIONS.

279. Denoting the roots of the equation 2+px = q by r1 and 2, we have (Art. 267),

That is, if a quadratic equation be reduced to the form x2+px=q, the algebraic sum of the roots is equal to the coefficient of x with its sign changed, and the product of the roots is equal to the second member, with its sign changed.

Example. Required the sum and product of the roots of the equation 2x2 7x-15 0.

The equation may be written in the form

7

Whence, the sum of the roots is 5, and their product is

2'

15

2

280. The principles of Art. 279 may be used to form a quadratic equation which shall have any required roots.

For, denoting the roots of the equation x2 + px - q = 0 by 1 and 2, we have, by the preceding article,

p = - (11+12), and

or,

We may therefore write the equation in the form

x2 - (r1 + r2)x + r1r2 = 0,

x2 - r1x - 2x + 1112 = 0.

That is (Art. 105), (x - 1)(x - 2) = 0.

Hence, to form an equation which shall have any required roots,

Subtract each of the roots from x, and place the product of the resulting expressions equal to zero.

Example. Form the equation whose roots are 4 and

7

4

By the rule,

(x-4) (x+1)=0.

4

Multiplying by 4, (x-4) (4x + 7) = 0.

Or,

4x2-9x-28 = 0, Ans.

EXAMPLES.

281. Find by inspection the sum and product of the roots

282. By Art. 280, the equation x2+px-q=0 may be written in the form (x-ri) (x - 2) = 0, where r1 and r2 are its roots.

It will be observed that the roots may be obtained by placing the factors of the first member separately equal to zero, and solving the simple equations thus formed.

This principle is often used in solving equations:

1. Solve the equation (2x-3) (3x + 5) = 0.

Placing the factors separately equal to zero,

Factoring the first member, x(x - 8) (x+3) = 0.

-1±√-3

Solving (1) by the rules for quadratics, x=

2

9. (x-2)(x2+9x+20)=0. 14. 24 - 2x2 - 12 x = 0.

8. (x2 - 4) (x2 - 9) = 0.

13. (x2-a2) (x2-ax-b) = 0.

15. x(2x+5) (3x - 7) (4x + 1) = 0.

16. (x2 - 5x + 6) (x2 + 7x + 12) (x2 - 3x - 4) = 0.

17.x + 1 = 0.

18.x-1=0.

19. x - x2 - 9x + 9 = 0.

20. 2x + 3x2 - 2 x − 3 = 0.

Note. The above examples are illustrations of the important prin ciple that, the degree of an equation indicates the number of its roots; thus, an equation of the third degree has three roots; of the fourth degree, four roots; etc.

It should be observed that the roots are not necessarily unequal; thus, the equation x2-2x+1=0 may be written (x-1)(x - 1) = 0 and therefore the two roots are 1 and 1.

FACTORING.

283. A Quadratic Expression is a trinomial expression of the form ax2+bx+c. The principles of the preceding articles serve to resolve any such expression into two simple factors. The expression ax2 + bx + c may be written

where r1 and r2 are the roots of the equation,

which, it will be observed, may be formed by placing the given expression equal to zero. Hence,

ax2 + bx + c = a (x - 1) (x - 2).