Solve sqrt{x+16}-sqrt{x}=2

$$\sqrt{ x+16 } - \sqrt{ x } = 2$$

Solve for x

$$x=9$$

Solution Steps

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

$$\sqrt{x+16}=2+\sqrt{x}$$

Square both sides of the equation.

$$\left(\sqrt{x+16}\right)^{2}=\left(2+\sqrt{x}\right)^{2}$$

Calculate \sqrt{x+16} to the power of 2 and get x+16.

$$x+16=\left(2+\sqrt{x}\right)^{2}$$

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

$$x+16=4+4\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$x+16=4+4\sqrt{x}+x$$

Subtract 4\sqrt{x} from both sides.

$$x+16-4\sqrt{x}=4+x$$

Subtract x from both sides.

$$x+16-4\sqrt{x}-x=4$$

Combine x and -x to get 0.

$$16-4\sqrt{x}=4$$

Subtract 16 from both sides.

$$-4\sqrt{x}=4-16$$

Subtract 16 from 4 to get -12.

$$-4\sqrt{x}=-12$$

Divide both sides by -4.

$$\sqrt{x}=\frac{-12}{-4}$$

Divide -12 by -4 to get 3.

$$\sqrt{x}=3$$

Square both sides of the equation.

$$x=9$$

Substitute 9 for x in the equation \sqrt{x+16}-\sqrt{x}=2.

$$\sqrt{9+16}-\sqrt{9}=2$$

Simplify. The value x=9 satisfies the equation.

$$2=2$$

Equation \sqrt{x+16}=\sqrt{x}+2 has a unique solution.

$$x=9$$