Решите уравнение методом замены переменной: (x-5)/(x+3)+(x+3)/(x-5)=-21/2.

Ответ:

\[\frac{x - 5}{x + 3} + \frac{x + 3}{x - 5} = - 2\frac{1}{2}\ \]

\[Пусть\ \ t = \frac{x - 5}{x + 3};\ x \neq - 3;\ \ \]

\[x \neq 5:\]

\[x + \frac{1}{t} = - \frac{5}{2}\ \ \ \ \ \ \ \ \ \ | \cdot t\]

\[t^{2} + \frac{5}{2}t + 1 = 0\ \ \ \ \ \ \ | \cdot 2\]

\[2t^{2} + 5t + 2 = 0\]

\[D = 25 - 16 = 9\]

\[t_{1} = \frac{- 5 + 3}{4} = - \frac{2}{4} =\]

\[= - \frac{1}{2} = - 0,5;\]

\[t_{2} = \frac{- 5 - 3}{4} = - \frac{8}{4} = - 2.\]