Solve x=sqrt{x+x^2}

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

Solve for x

$$x=0$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{x+x^{2}} to the power of 2 and get x+x^{2}.

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

Subtract x from both sides.

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

Subtract x^{2} from both sides.

$$x^{2}-x-x^{2}=0$$

Combine x^{2} and -x^{2} to get 0.

$$-x=0$$

Product of two numbers is equal to 0 if at least one of them is 0. Since -1 is not equal to 0, x must be equal to 0.

$$x=0$$

Substitute 0 for x in the equation x=\sqrt{x+x^{2}}.

$$0=\sqrt{0+0^{2}}$$

Simplify. The value x=0 satisfies the equation.

$$0=0$$

Equation x=\sqrt{x^{2}+x} has a unique solution.

$$x=0$$