Решите уравнение, используя введение новой переменной: (x^2+6)/x-5x/(x^2+6)=4.

Ответ:

\[\frac{x^{2} + 6}{x} - \frac{5x}{x^{2} + 6} = 4\]

\[Пусть\ \frac{x^{2} + 6}{x} = a;\ \ \frac{5x}{x^{2} + 6} = \frac{5}{a}:\]

\[a - \frac{5}{a} = 4\ \ \ \ \ \ \ \ | \cdot a\]

\[a^{2} - 5 - 4a = 0\]

\[a^{2} - 4a - 5 = 0\]

\[a_{1} + a_{2} = 4;\ \ \ \ a_{1} \cdot a_{2} = - 5\]

\[a_{1} = 5;\ \ \ a_{2} = - 1.\]

\[Подставим:\]

\[1)\ \frac{x^{2} + 6}{x} = 5^{\backslash x}\]

\[x^{2} - 5x + 6 = 0\]

\[x_{1} + x_{2} = 5;\ \ \ \ x_{1} \cdot x_{2} = 6\]

\[x_{1} = 3;\ \ x_{2} = 2.\]

\[2)\ \ \frac{x^{2} + 6}{x} = - 1^{\backslash x}\]

\[x^{2} + x + 6 = 0\]

\[D = 1 - 24 = - 23 < 0\]

\[нет\ корней.\]

\[Ответ:x = 2;\ \ x = 3.\]