Solve x=sqrt{xy}

$$x = \sqrt { x y }$$

Solve for y

$$\left\{\begin{matrix}y=x\text{, }&x\geq 0\\y\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Radical Equations

Swap sides so that all variable terms are on the left hand side.

$$\sqrt{xy}=x$$

Square both sides of the equation.

$$xy=x^{2}$$

Divide both sides by x.

$$\frac{xy}{x}=\frac{x^{2}}{x}$$

Dividing by x undoes the multiplication by x.

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

Divide x^{2} by x.

$$y=x$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=y\text{, }&arg(y)<\pi \text{ or }y=0\end{matrix}\right.$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=x\text{, }&arg(x)<\pi \text{ or }x=0\\y\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=y\text{, }&y\geq 0\end{matrix}\right.$$