Solve -xleft(x+2right)=sqrt{y}

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

Solve for y

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

$$x\geq -2\text{ and }x\leq 0$$

Steps for Solving Radical Equations

Use the distributive property to multiply -x by x+2.

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

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

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

Multiply x and x to get x^{2}.

$$\sqrt{y}=-x^{2}+2\left(-1\right)x$$

Multiply 2 and -1 to get -2.

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

Square both sides of the equation.

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

Solve for x (complex solution)

$$x=\sqrt{-\sqrt{y}+1}-1$$

$$x=-\sqrt{-\sqrt{y}+1}-1$$

Solve for y (complex solution)

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

$$arg(x\left(x+2\right))\geq \pi \text{ or }x=-2\text{ or }x=0$$

Solve for x

$$x=\sqrt{-\sqrt{y}+1}-1$$

$$x=-\sqrt{-\sqrt{y}+1}-1\text{, }y\geq 0\text{ and }y\leq 1$$