Решите систему уравнений: x^2+y^2=18; xy=8.

Ответ:

\[\frac{64}{y^{2}} + y^{2} - 18 = 0\ \ \ \ \ | \cdot y^{2} \neq 0\]

\[64 + y^{4} - 18y^{2} = 0\]

\[y^{4} - 18y^{2} + 64 = 0\]

\[Пусть\ y^{2} = t \geq 0:\]

\[t^{2} - 18t + 64 = 0\]

\[D_{1} = 81 - 64 = 17\]

\[t_{1} = 9 + \sqrt{17};\ \ \ t_{2} = 9 - \sqrt{17}.\]

\[1)\ y^{2} = 9 \pm \sqrt{17}\]

\[y = \pm \sqrt{9 \pm \sqrt{17}}\text{\ .}\]

\[2)\ x_{1,2} = \pm \frac{8}{\sqrt{9 \pm \sqrt{17}}}.\]