Solve sqrt{x+13}-sqrt{x}+1=2

$$\sqrt { x + 13 } - \sqrt { x } + 1 = 2$$

Solve for x

$$x=36$$

Solution Steps

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

$$\sqrt{x+13}=2-\left(-\sqrt{x}+1\right)$$

To find the opposite of -\sqrt{x}+1, find the opposite of each term.

$$\sqrt{x+13}=2-\left(-\sqrt{x}\right)-1$$

The opposite of -\sqrt{x} is \sqrt{x}.

$$\sqrt{x+13}=2+\sqrt{x}-1$$

Subtract 1 from 2 to get 1.

$$\sqrt{x+13}=1+\sqrt{x}$$

Square both sides of the equation.

$$\left(\sqrt{x+13}\right)^{2}=\left(1+\sqrt{x}\right)^{2}$$

Calculate \sqrt{x+13} to the power of 2 and get x+13.

$$x+13=\left(1+\sqrt{x}\right)^{2}$$

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

$$x+13=1+2\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$x+13=1+2\sqrt{x}+x$$

Subtract 2\sqrt{x} from both sides.

$$x+13-2\sqrt{x}=1+x$$

Subtract x from both sides.

$$x+13-2\sqrt{x}-x=1$$

Combine x and -x to get 0.

$$13-2\sqrt{x}=1$$

Subtract 13 from both sides.

$$-2\sqrt{x}=1-13$$

Subtract 13 from 1 to get -12.

$$-2\sqrt{x}=-12$$

Divide both sides by -2.

$$\sqrt{x}=\frac{-12}{-2}$$

Divide -12 by -2 to get 6.

$$\sqrt{x}=6$$

Square both sides of the equation.

$$x=36$$

Substitute 36 for x in the equation \sqrt{x+13}-\sqrt{x}+1=2.

$$\sqrt{36+13}-\sqrt{36}+1=2$$

Simplify. The value x=36 satisfies the equation.

$$2=2$$

Equation \sqrt{x+13}=\sqrt{x}+1 has a unique solution.

$$x=36$$