Solve sqrt{x+8}-sqrt{x}=2

$$\sqrt { x + 8 } - \sqrt { x } = 2$$

Solve for x

$$x=1$$

Solution Steps

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

$$\sqrt{x+8}=2+\sqrt{x}$$

Square both sides of the equation.

$$\left(\sqrt{x+8}\right)^{2}=\left(2+\sqrt{x}\right)^{2}$$

Calculate \sqrt{x+8} to the power of 2 and get x+8.

$$x+8=\left(2+\sqrt{x}\right)^{2}$$

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

$$x+8=4+4\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$x+8=4+4\sqrt{x}+x$$

Subtract 4\sqrt{x} from both sides.

$$x+8-4\sqrt{x}=4+x$$

Subtract x from both sides.

$$x+8-4\sqrt{x}-x=4$$

Combine x and -x to get 0.

$$8-4\sqrt{x}=4$$

Subtract 8 from both sides.

$$-4\sqrt{x}=4-8$$

Subtract 8 from 4 to get -4.

$$-4\sqrt{x}=-4$$

Divide both sides by -4.

$$\sqrt{x}=\frac{-4}{-4}$$

Divide -4 by -4 to get 1.

$$\sqrt{x}=1$$

Square both sides of the equation.

$$x=1$$

Substitute 1 for x in the equation \sqrt{x+8}-\sqrt{x}=2.

$$\sqrt{1+8}-\sqrt{1}=2$$

Simplify. The value x=1 satisfies the equation.

$$2=2$$

Equation \sqrt{x+8}=\sqrt{x}+2 has a unique solution.

$$x=1$$