Решите уравнение, используя введение новой переменной: (x^2+8)/x-(12x)/(x^2+8)=4.

Ответ:

\[\frac{x^{2} + 8}{x} - \frac{12x}{x^{2} + 8} = 4\]

\[Пусть\ \ \frac{x^{2} + 8}{x} = y:\]

\[y - \frac{12}{y} = 4\ \ \ \ \ \ | \cdot y\]

\[y^{2} - 4y - 12 = 0\]

\[D = 4 + 12 = 16\]

\[y_{1} = 2 + 4 = 6;\ \ \]

\[\ y_{2} = 2 - 4 = - 2.\]

\[Подставим:\]

\[1)\ \frac{x^{2} + 8}{x} = 6^{\backslash x}\]

\[x^{2} - 6x + 8 = 0\]

\[D = 9 - 8 = 1\]

\[x_{1} = 3 + 1 = 4;\ \ \ x_{2} = 3 - 1 = 2.\]

\[2)\ \frac{x^{2} + 8}{x} = - 2^{\backslash x}\]

\[x^{2} + 2x + 8 = 0\]

\[D = 1 - 8 = - 7 < 0\]

\[нет\ корней.\]

\[Ответ:x = 2;\ \ x = 4.\]