Simplify : 1/(x^2 - 8x + 15) - 1/(x^2 - 25)

Question

Simplify : \(\frac{1}{x^2-8x+15}-\frac{1}{x^2-25}\)

Answer 1

x² - 8x + 15 = x² - 5x - 3x + 15

= x(x - 5) - 3(x - 5) = (x - 5)(x - 3)

x² - 25 = (x - 5)(x - 5)

∴ \(\frac{1}{x^2-8x+15}-\frac{1}{x^2-25}\) = \(\frac{1}{(x-5)(x-3)}\) - \(\frac{1}{(x-5)(x-5)}\)

= \(\frac{(x-5)-(x-3)}{(x-3)(x-5)(x-5)}\)             (Note the step)

= \(\frac{x-5-x+3}{(x-3)(x-5)(x-5)}\) = \(\frac{-2}{(x-3)(x-5)(x-5)}\)