Solve sqrt{x+y}=x

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

Solve for y

$$y=x\left(x-1\right)$$

$$x\geq 0$$

Steps for Solving Radical Equations

Square both sides of the equation.

$$y+x=x^{2}$$

Subtract x from both sides of the equation.

$$y+x-x=x^{2}-x$$

Subtracting x from itself leaves 0.

$$y=x^{2}-x$$

Subtract x from x^{2}.

$$y=x\left(x-1\right)$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{4y+1}+1}{2}\text{, }&arg(\frac{-\sqrt{4y+1}+1}{2})<\pi \text{ or }y=0\\x=\frac{\sqrt{4y+1}+1}{2}\text{, }&arg(\frac{\sqrt{4y+1}+1}{2})<\pi \end{matrix}\right.$$

Solve for y (complex solution)

$$y=x\left(x-1\right)$$

$$arg(x)<\pi \text{ or }x=0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{4y+1}+1}{2}\text{, }&y\geq -\frac{1}{4}\\x=\frac{-\sqrt{4y+1}+1}{2}\text{, }&y\geq -\frac{1}{4}\text{ and }y\leq 0\end{matrix}\right.$$