Solve sqrt{x+1}-x-1=0

$$\sqrt { x + 1 } - x - 1 = 0$$

Solve for x

$$x=-1$$

$$x=0$$

Solution Steps

Subtract -x-1 from both sides of the equation.

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

To find the opposite of -x-1, find the opposite of each term.

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

The opposite of -x is x.

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

The opposite of -1 is 1.

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

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

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

Subtract x^{2} from both sides.

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

Subtract 2x from both sides.

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

Combine x and -2x to get -x.

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

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

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

Factor out x.

$$x\left(-1-x\right)=0$$

To find equation solutions, solve x=0 and -1-x=0.

$$x=0 x=-1$$

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

$$\sqrt{0+1}-0-1=0$$

Simplify. The value x=0 satisfies the equation.

$$0=0$$

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

$$\sqrt{-1+1}-\left(-1\right)-1=0$$

Simplify. The value x=-1 satisfies the equation.

$$0=0$$

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

$$x=0 x=-1$$