Найдите решения уравнения (x-2x/(x+2))^2=5-(4x^2)/(x+2).

Ответ:

\[\left( x^{\backslash x + 2} - \frac{2x}{x + 2} \right)^{2} = 5 - \frac{4x^{2}}{x + 2}\]

\[\left( \frac{x^{2} + 2x - 2x}{x + 2} \right)^{2} = 5 - 4 \cdot \frac{x²}{x + 2}\]

\[\left( \frac{x^{2}}{x + 2} \right)^{2} = 5 - 4 \cdot \frac{x²}{x + 2}\]

\[y = \frac{x^{2}}{x + 2}:\]

\[y^{2} = 5 - 4y\]

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

\[D = 4 + 5 = 9\]

\[y_{1} = - 2 + 3 = 1;\ \ y_{2} = - 2 - 3 = - 5\]

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

\[x^{2} - x - 2 = 0\]

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

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

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

\[x^{2} + 5x + 10 = 0\]

\[D = 25 - 40 < 0\]

\[нет\ корней.\]

\[Ответ:x = - 1;x = 2.\]