Solve sqrt{x^2+2x+1}-sqrt{4x+1}=0

$$\sqrt { x ^ { 2 } + 2 x + 1 } - \sqrt { 4 x + 1 } = 0$$

Solve for x

$$x=2$$

$$x=0$$

Solution Steps

Subtract -\sqrt{4x+1} from both sides of the equation.

$$\sqrt{x^{2}+2x+1}=\sqrt{4x+1}$$

Square both sides of the equation.

$$\left(\sqrt{x^{2}+2x+1}\right)^{2}=\left(\sqrt{4x+1}\right)^{2}$$

Calculate \sqrt{x^{2}+2x+1} to the power of 2 and get x^{2}+2x+1.

$$x^{2}+2x+1=\left(\sqrt{4x+1}\right)^{2}$$

Calculate \sqrt{4x+1} to the power of 2 and get 4x+1.

$$x^{2}+2x+1=4x+1$$

Subtract 4x from both sides.

$$x^{2}+2x+1-4x=1$$

Combine 2x and -4x to get -2x.

$$x^{2}-2x+1=1$$

Subtract 1 from both sides.

$$x^{2}-2x+1-1=0$$

Subtract 1 from 1 to get 0.

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

Factor out x.

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

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

$$x=0 x=2$$

Substitute 0 for x in the equation \sqrt{x^{2}+2x+1}-\sqrt{4x+1}=0.

$$\sqrt{0^{2}+2\times 0+1}-\sqrt{4\times 0+1}=0$$

Simplify. The value x=0 satisfies the equation.

$$0=0$$

Substitute 2 for x in the equation \sqrt{x^{2}+2x+1}-\sqrt{4x+1}=0.

$$\sqrt{2^{2}+2\times 2+1}-\sqrt{4\times 2+1}=0$$

Simplify. The value x=2 satisfies the equation.

$$0=0$$

List all solutions of \sqrt{x^{2}+2x+1}=\sqrt{4x+1}.

$$x=0 x=2$$