Solve sqrt{x+sqrt{2x}}=sqrt{2x}

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

Solve for x

$$x=2$$

$$x=0$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Subtract x from both sides of the equation.

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

Combine 2x and -x to get x.

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

Square both sides of the equation.

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

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

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

Subtract x^{2} from both sides.

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

Factor out x.

$$x\left(2-x\right)=0$$

To find equation solutions, solve x=0 and 2-x=0.

$$x=0 x=2$$

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

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

Simplify. The value x=0 satisfies the equation.

$$0=0$$

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

$$\sqrt{2+\sqrt{2\times 2}}=\sqrt{2\times 2}$$

Simplify. The value x=2 satisfies the equation.

$$2=2$$

List all solutions of \sqrt{x+\sqrt{2x}}=\sqrt{2x}.

$$x=0 x=2$$