Solve sqrt{x}+sqrt{x+11}=11

$$\sqrt { x } + \sqrt { x + 11 } = 11$$

Solve for x

$$x=25$$

Solution Steps

Subtract \sqrt{x+11} from both sides of the equation.

$$\sqrt{x}=11-\sqrt{x+11}$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(11-\sqrt{x+11}\right)^{2}$$

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

$$x=\left(11-\sqrt{x+11}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(11-\sqrt{x+11}\right)^{2}.

$$x=121-22\sqrt{x+11}+\left(\sqrt{x+11}\right)^{2}$$

Calculate \sqrt{x+11} to the power of 2 and get x+11.

$$x=121-22\sqrt{x+11}+x+11$$

Add 121 and 11 to get 132.

$$x=132-22\sqrt{x+11}+x$$

Add 22\sqrt{x+11} to both sides.

$$x+22\sqrt{x+11}=132+x$$

Subtract x from both sides.

$$x+22\sqrt{x+11}-x=132$$

Combine x and -x to get 0.

$$22\sqrt{x+11}=132$$

Divide both sides by 22.

$$\sqrt{x+11}=\frac{132}{22}$$

Divide 132 by 22 to get 6.

$$\sqrt{x+11}=6$$

Square both sides of the equation.

$$x+11=36$$

Subtract 11 from both sides of the equation.

$$x+11-11=36-11$$

Subtracting 11 from itself leaves 0.

$$x=36-11$$

Subtract 11 from 36.

$$x=25$$

Substitute 25 for x in the equation \sqrt{x}+\sqrt{x+11}=11.

$$\sqrt{25}+\sqrt{25+11}=11$$

Simplify. The value x=25 satisfies the equation.

$$11=11$$

Equation \sqrt{x}=-\sqrt{x+11}+11 has a unique solution.

$$x=25$$