Solve x^2+2yx-z^2=0

$${ x }^{ 2 } +2yx- { z }^{ 2 } =0$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=\frac{z^{2}}{2x}-\frac{x}{2}\text{, }&x\neq 0\\y\in \mathrm{C}\text{, }&x=0\text{ and }z=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$2yx-z^{2}=-x^{2}$$

Add z^{2} to both sides.

$$2yx=-x^{2}+z^{2}$$

The equation is in standard form.

$$2xy=z^{2}-x^{2}$$

Divide both sides by 2x.

$$\frac{2xy}{2x}=\frac{\left(z-x\right)\left(x+z\right)}{2x}$$

Dividing by 2x undoes the multiplication by 2x.

$$y=\frac{\left(z-x\right)\left(x+z\right)}{2x}$$

Divide \left(z-x\right)\left(z+x\right) by 2x.

$$y=\frac{z^{2}}{2x}-\frac{x}{2}$$

Solve for y

$$\left\{\begin{matrix}y=\frac{z^{2}}{2x}-\frac{x}{2}\text{, }&x\neq 0\\y\in \mathrm{R}\text{, }&x=0\text{ and }z=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract x^{2} from both sides. Anything subtracted from zero gives its negation.

$$2yx-z^{2}=-x^{2}$$

Add z^{2} to both sides.

$$2yx=-x^{2}+z^{2}$$

The equation is in standard form.

$$2xy=z^{2}-x^{2}$$

Divide both sides by 2x.

$$\frac{2xy}{2x}=\frac{\left(z-x\right)\left(x+z\right)}{2x}$$

Dividing by 2x undoes the multiplication by 2x.

$$y=\frac{\left(z-x\right)\left(x+z\right)}{2x}$$

Divide \left(z-x\right)\left(z+x\right) by 2x.

$$y=\frac{z^{2}}{2x}-\frac{x}{2}$$

Solve for x

$$x=\sqrt{y^{2}+z^{2}}-y$$

$$x=-\sqrt{y^{2}+z^{2}}-y$$