cients of a quadratic equation involve letters, the equation is

called a literal quadratic equation.

Such equations are solved in the usual way.

Completing the square, x2 + 6 mx + 9 m2 = 9 m2 - 8.

(1)

..x=- ·3 m± √9 m2 — 8.

t2+gt+h=0.

2. Solve:

Here,

a = 1, b

=g, c=h.

Hence, by Sec. 177,

179. Collected Methods. We have used three methods of solving quadratic equations:

ax2+ bx + c = 0. See Sec. 371, Part I. x=

Solve and test, using whichever of the methods in Sec. 179

45. The product of two consecutive positive integers is 306.

Since the integers are to be positive, the value 18 is not admissible. x = 17, .. x + 1 = 18, and the integers are 17 and 18.

TEST. 17. 18306.

46. There is also a pair of consecutive negative integers whose product is 306. What are they?

47. If the square of a certain number is diminished by the number, the result is 72. Find the number.

48. A certain number plus its reciprocal is -2. What is the number?

49. A certain positive number minus its reciprocal is §. What is the number? What negative number has the same property?

25 IN.

50. The perimeter of the rectangle x shown in the figure is 62 in. Find the

sides.

51. One perpendicular side of a certain right triangle is 31 units longer than the other; the square of their sum exceeds the square of the hypotenuse by 720. Find the sides.

5%. In a right triangle of area 60 sq. ft.; the difference between the perpendicular sides is 7. Find the three sides.

NOTE. Those who have studied geometry may take up some of the problems based upon geometric properties found in Chapter X.

RELATIONS BETWEEN ROOTS AND COEFFICIENTS

180. Relation of Roots to Coefficients. values found for the roots (Sec. 177), we obtain

By adding the

4 ac

α

Applying these results to the equation x2 + px + q = 0, we

In the equation x2 + px+q = 0, the coefficient of x with its sign changed is the sum of the roots, and the absolute term is their product.

Every quadratic equation can be put into the form x2 + px + q = 0 by dividing both members by the coefficient of x2.

181. Symmetric Functions of the Roots. It is apparent that the equations in Section 180 remain unchanged if î, and r1⁄2 are interchanged. On this account the expressions +r and r12 are called symmetric functions of the roots of the quadratic equation. There are other such functions, but these only will be treated here. The two following sections show some of their

uses.