Question

$$2x^{2}+y^{2}-6x+uy+u=0$$

Solve for u (complex solution)

$\left\{\begin{matrix}u=-\frac{2x^{2}-6x+y^{2}}{y+1}\text{, }&y\neq -1\\u\in \mathrm{C}\text{, }&\left(x=\frac{3-\sqrt{7}}{2}\text{ or }x=\frac{\sqrt{7}+3}{2}\right)\text{ and }y=-1\end{matrix}\right.$

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Solve for u

$\left\{\begin{matrix}u=-\frac{2x^{2}-6x+y^{2}}{y+1}\text{, }&y\neq -1\\u\in \mathrm{R}\text{, }&\left(x=\frac{3-\sqrt{7}}{2}\text{ or }x=\frac{\sqrt{7}+3}{2}\right)\text{ and }y=-1\end{matrix}\right.$

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Solve for x (complex solution)

$x=\frac{\sqrt{9-2u-2uy-2y^{2}}+3}{2}$
$x=\frac{-\sqrt{9-2u-2uy-2y^{2}}+3}{2}$

Solve for x

$x=\frac{\sqrt{9-2u-2uy-2y^{2}}+3}{2}$
$x=\frac{-\sqrt{9-2u-2uy-2y^{2}}+3}{2}\text{, }y\geq \frac{-\sqrt{u^{2}-4u+18}-u}{2}\text{ and }y\leq \frac{\sqrt{u^{2}-4u+18}-u}{2}$