Упростите выражение: (10x/(x^2-10x+25)+x/(x-5))(20x/(x+5)-x-5).

Ответ:

\[\left( \frac{10x}{x^{2} - 10x + 25} + \frac{x}{x - 5} \right)\left( \frac{20x}{x + 5} - x - 5 \right) =\]

\[= - x\]

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

\[+ \frac{x^{\backslash x - 5}}{x - 5} = \frac{10x + x^{2} - 5x}{(x - 5)^{2}} = \frac{5x + x^{2}}{(x - 5)^{2}} =\]

\[= \frac{x(x + 5)}{(x - 5)^{2}}\]

\[2)\ \frac{20x}{x + 5} - (x + 5)^{\backslash x + 5} =\]

\[= \frac{20x - x^{2} - 10x - 25}{x + 5} =\]

\[= \frac{- \left( x^{2} - 10x + 25 \right)}{x + 5} = - \frac{(x - 5)^{2}}{x + 5}\ \]

\[3) - \frac{x(x + 5)}{(x - 5)^{2}} \cdot \frac{(x - 5)^{2}}{x + 5} = - x\]