Solve x=sqrt{2+x+y}

$$x= \sqrt{ 2+x+y }$$

Solve for y

$$y=\left(x-2\right)\left(x+1\right)$$

$$x\geq 0$$

Steps for Solving Radical Equations

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

$$\sqrt{2+x+y}=x$$

Square both sides of the equation.

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

Subtract 2+x from both sides of the equation.

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

Subtracting 2+x from itself leaves 0.

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

Subtract 2+x from x^{2}.

$$y=\left(x-2\right)\left(x+1\right)$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{-\sqrt{4y+9}+1}{2}\text{, }&arg(\frac{-\sqrt{4y+9}+1}{2})<\pi \text{ or }y=-2\\x=\frac{\sqrt{4y+9}+1}{2}\text{, }&arg(\frac{\sqrt{4y+9}+1}{2})<\pi \end{matrix}\right.$$

Solve for y (complex solution)

$$y=\left(x-2\right)\left(x+1\right)$$

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

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{4y+9}+1}{2}\text{, }&y\geq -\frac{9}{4}\\x=\frac{-\sqrt{4y+9}+1}{2}\text{, }&y\geq -\frac{9}{4}\text{ and }y\leq -2\end{matrix}\right.$$