Решите систему уравнений: x^2+y^2=5; xy=-2.

\[(1) + (2):\ \ \ \ \ \ (x + y)^{2} = 1\]

\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x + y = 1\]

\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x + y = - 1\]

\[1)\ \left\{ \begin{matrix} x + y = 1 \\ xy = - 2\ \ \\ \end{matrix}\text{\ \ \ \ } \right.\ \text{\ \ \ \ \ \ \ }\left\{ \begin{matrix} x = 1 - y \\ xy = - 2\ \ \\ \end{matrix} \right.\ \]

\[(1 - y)y = - 2\]

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

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

\[D = ( - 1)^{2} - 4 \cdot 1 \cdot ( - 2) =\]

\[= 1 + 8 = 9\]

\[y_{1} = \frac{- 1 + \sqrt{9}}{2} = \frac{1 + 3}{2} = \frac{4}{2} = 2\]

\[y_{2} = \frac{1 - \sqrt{9}}{2} = \frac{1 - 3}{2} = \frac{- 2}{2} = - 1\]

\[y_{1} = 2 \Longrightarrow \text{\ \ \ \ \ \ \ \ }x_{1} = 1 - 2 = - 1.\]

\[y_{2} = - 1 \Longrightarrow \text{\ \ \ \ \ }x_{2} = 1 - ( - 1) =\]

\[= 1 + 1 = 2.\]

\[( - 1 - y)y = - 2\]

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

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

\[D = 1^{2} - 4 \cdot 1 \cdot ( - 2) = 1 + 8 =\]

\[= 9\]

\[y_{3} = \frac{- 1 + \sqrt{9}}{2} = \frac{- 1 + 3}{2} = \frac{2}{2} = 1\]

\[y_{4} = \frac{- 1 - \sqrt{9}}{2} = \frac{- 1 - 3}{2} = \frac{- 4}{2} =\]

\[= - 2\]

\[y_{3} = 1 \Longrightarrow \text{\ \ \ \ \ \ }x_{3} = - 1 - 1 = - 2.\]

\[y_{4} = - 2 \Longrightarrow \text{\ \ }x_{4} = - 1 - ( - 2) =\]

\[= - 1 + 2 = 1.\]

\[Ответ:\ \ ( - 1;2),\ (2;\ - 1),\ ( - 2;1),\]

\[(1;\ - 2).\]