Решите систему уравнений: 4x^2-y^2=32; xy=6.

Ответ:

\[\left\{ \begin{matrix} 4x^{2} - y^{2} = 32 \\ xy = 6\ \ \ \ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\text{\ \ \ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} 4x^{2} - \frac{36}{x^{2}} - 32 = 0 \\ y = \frac{6}{x}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix}\text{\ \ \ \ \ \ \ } \right.\ \]

\[\left\{ \begin{matrix} 4x^{4} - 32x^{2} - 36 = 0\ \ |\ :4 \\ y = \frac{6}{x}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ } \\ \end{matrix} \right.\ \]

\[x^{4} - 8x^{2} - 9 = 0\]

\[x_{1}^{2} + x_{2}^{2} = 8,\ \ x_{1}^{2} \cdot x_{2}^{2} = - 9\]

\[x_{1}^{2} = - 1\ (нет\ корней).\]

\[x² = 9 \rightarrow \ \ x = \pm 3.\]

\[\left\{ \begin{matrix} x = 3 \\ y = 2 \\ \end{matrix}\text{\ \ \ \ \ \ } \right.\ \text{\ \ \ \ }\left\{ \begin{matrix} x = - 3 \\ y = - 2 \\ \end{matrix} \right.\ \]

\[Ответ:(3;2);\ \ \ ( - 3;\ - 2)\text{.\ }\]