Solve sqrt{5+x}-sqrt{x}=1

$$\sqrt { 5 + x } - \sqrt { x } = 1$$

Solve for x

$$x=4$$

Solution Steps

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

$$\sqrt{5+x}=1+\sqrt{x}$$

Square both sides of the equation.

$$\left(\sqrt{5+x}\right)^{2}=\left(1+\sqrt{x}\right)^{2}$$

Calculate \sqrt{5+x} to the power of 2 and get 5+x.

$$5+x=\left(1+\sqrt{x}\right)^{2}$$

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

$$5+x=1+2\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$5+x=1+2\sqrt{x}+x$$

Subtract 2\sqrt{x} from both sides.

$$5+x-2\sqrt{x}=1+x$$

Subtract x from both sides.

$$5+x-2\sqrt{x}-x=1$$

Combine x and -x to get 0.

$$5-2\sqrt{x}=1$$

Subtract 5 from both sides.

$$-2\sqrt{x}=1-5$$

Subtract 5 from 1 to get -4.

$$-2\sqrt{x}=-4$$

Divide both sides by -2.

$$\sqrt{x}=\frac{-4}{-2}$$

Divide -4 by -2 to get 2.

$$\sqrt{x}=2$$

Square both sides of the equation.

$$x=4$$

Substitute 4 for x in the equation \sqrt{5+x}-\sqrt{x}=1.

$$\sqrt{5+4}-\sqrt{4}=1$$

Simplify. The value x=4 satisfies the equation.

$$1=1$$

Equation \sqrt{x+5}=\sqrt{x}+1 has a unique solution.

$$x=4$$