Page:Elementary algebra (1896).djvu/255

237 Rh 6. {x - a}{x + a} + {x + a}{x - a}= 5. 7. {3(x2-1)}{x^2 - 1} + {4(x2-4)}{x2 + 3} - {3(9x^-1)}{(x2-1)(x2 + 3)} = 7. 8. (2x - c) (x + d )+ (2x + c) (x - d) = 2 cd(2 cd - 1).

AFFECTED QUADRATIC EQUATIONS.

285. The equation x^2 = 36 is an instance of the simplest form of quadratic equations. The equation (x — 3)^2 = 25 may be solved in a similar way ; for taking the square root of both sides we have two simple equations, x-3 = ±5.

Taking the upper sign, x— 3 = + 5, whence x = 8 ; taking the lower sign, x — 3 = — 5, whence x = — 2. the solution is x = 8, or — 2.

Now the given equation (x — 3)^2 = 25 may be written x^ — 6x+ (3)^2 = 25, or x^2 — 6x = 16.

Hence, by retracing our steps, we learn that the equation x^2 - 6x = 16 can be solved by first adding (3)^2 to each side, and then extracting the square root ; and we add 9 to each side because this quantity added to the left side makes it a perfect square.

Now whatever the quantity a may be, x^2 + 2 ax + a^2 = (x + a)^2, and x^2 — 2ax + a^2 = (x — a)^2; so that, if a trinomial is a perfect square, and its highest power, x^2, has unity for a coefficient, the term without x must be equal to the square of half the coefficient of x.

Ex. 1. Solve 7x = x^2 - 8.

Transpose so as to have the terms involving x on one side, and the square term positive. Thus x^2 - 7x = 8.