حل p^2+p(x+y-2y/x)+xy+y^2/x^2-y-y^2/x=0

$$p ^ { 2 } + p ( x + y - \frac { 2 y } { x } ) + x y + \frac { y ^ { 2 } } { x ^ { 2 } } - y - \frac { y ^ { 2 } } { x } = 0$$

برای x حل کنید (complex solution)

$$\left\{\begin{matrix}x=\frac{y}{y+p}\text{, }&y\neq 0\text{ and }p\neq -y\\x\neq 0\text{, }&y=0\text{ and }p=0\\x=\frac{\sqrt{4y+p^{2}}-p}{2}\text{, }&\left(|\frac{arg(p^{2})}{2}-arg(p)|\geq \pi \text{ and }p\neq 0\right)\text{ or }y\neq 0\\x=\frac{-\sqrt{4y+p^{2}}-p}{2}\text{, }&\left(arg(p)<\pi \text{ and }p\neq 0\right)\text{ or }y\neq 0\end{matrix}\right.$$

برای p حل کنید

$$p=-x+\frac{y}{x}$$

$$p=-y+\frac{y}{x}\text{, }x\neq 0$$

برای x حل کنید

$$\left\{\begin{matrix}x=\frac{y}{y+p}\text{, }&y\neq 0\text{ and }p\neq -y\\x\neq 0\text{, }&y=0\text{ and }p=0\\x=\frac{\sqrt{4y+p^{2}}-p}{2}\text{, }&\left(y\geq -\frac{p^{2}}{4}\text{ and }p<0\text{ and }p\leq -2\sqrt{-y}\text{ and }y\leq 0\right)\text{ or }y>0\text{ or }\left(|p|\geq 2\sqrt{-y}\text{ and }y\geq -\frac{p^{2}}{4}\text{ and }y<0\right)\\x=\frac{-\sqrt{4y+p^{2}}-p}{2}\text{, }&\left(y\geq -\frac{p^{2}}{4}\text{ and }p>0\text{ and }p\geq 2\sqrt{-y}\text{ and }y\leq 0\right)\text{ or }y>0\text{ or }\left(|p|\geq 2\sqrt{-y}\text{ and }y\geq -\frac{p^{2}}{4}\text{ and }y<0\right)\end{matrix}\right.$$