Επίλυση u=x^2y^2-5x^2-8xy-5y^2

$$u = x ^ { 2 } y ^ { 2 } - 5 x ^ { 2 } - 8 x y - 5 y ^ { 2 }$$

Λύση ως προς x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{5y^{4}+uy^{2}-9y^{2}-5u}+4y}{y^{2}-5}\text{; }x=\frac{-\sqrt{5y^{4}+uy^{2}-9y^{2}-5u}+4y}{y^{2}-5}\text{, }&y\neq -\sqrt{5}\text{ and }y\neq \sqrt{5}\\x=\frac{-5y^{2}-u}{8y}\text{, }&y=-\sqrt{5}\text{ or }y=\sqrt{5}\end{matrix}\right,$$

Λύση ως προς u

$$u=\left(xy\right)^{2}-5y^{2}-8xy-5x^{2}$$

Λύση ως προς x

$$\left\{\begin{matrix}x=\frac{\sqrt{5y^{4}+uy^{2}-9y^{2}-5u}+4y}{y^{2}-5}\text{; }x=\frac{-\sqrt{5y^{4}+uy^{2}-9y^{2}-5u}+4y}{y^{2}-5}\text{, }&\left(y\neq -\sqrt{5}\text{ and }u=-\frac{y^{2}\left(5y^{2}-9\right)}{y^{2}-5}\text{ and }|y|<\sqrt{5}\right)\text{ or }\left(u\leq -\frac{y^{2}\left(5y^{2}-9\right)}{y^{2}-5}\text{ and }y>-\sqrt{5}\text{ and }|y|<\sqrt{5}\right)\text{ or }\left(u\geq -\frac{y^{2}\left(5y^{2}-9\right)}{y^{2}-5}\text{ and }|y|>\sqrt{5}\right)\text{ or }\left(u=-\frac{y^{2}\left(5y^{2}-9\right)}{y^{2}-5}\text{ and }|y|\neq \sqrt{5}\right)\\x=\frac{-5y^{2}-u}{8y}\text{, }&|y|=\sqrt{5}\end{matrix}\right,$$