Упростите выражение: a) x^2/(x-5)^2 - 25/(5-x)^2; б) (x^2+25)/(x-5)^3 + 10x/(5-x)^3.

Решение:

1. \(\frac{x^2}{(x-5)^2} - \frac{25}{(5-x)^2} = \frac{x^2}{(x-5)^2} - \frac{25}{(x-5)^2} = \frac{x^2-25}{(x-5)^2} = \frac{(x-5)(x+5)}{(x-5)^2} = \frac{x+5}{x-5}\)
2. \(\frac{x^2+25}{(x-5)^3} + \frac{10x}{(5-x)^3} = \frac{x^2+25}{(x-5)^3} + \frac{10x}{-(x-5)^3} = \frac{x^2+25}{(x-5)^3} - \frac{10x}{(x-5)^3} = \frac{x^2+25-10x}{(x-5)^3} = \frac{(x-5)^2}{(x-5)^3} = \frac{1}{x-5}\)

Ответ: а) (x+5)/(x-5); б) 1/(x-5).