Solve sqrt{x-20}+sqrt{x}=10

$$\sqrt { x - 20 } + \sqrt { x } = 10$$

Solve for x

$$x=36$$

Solution Steps

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

$$\sqrt{x-20}=10-\sqrt{x}$$

Square both sides of the equation.

$$\left(\sqrt{x-20}\right)^{2}=\left(10-\sqrt{x}\right)^{2}$$

Calculate \sqrt{x-20} to the power of 2 and get x-20.

$$x-20=\left(10-\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(10-\sqrt{x}\right)^{2}.

$$x-20=100-20\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$x-20=100-20\sqrt{x}+x$$

Add 20\sqrt{x} to both sides.

$$x-20+20\sqrt{x}=100+x$$

Subtract x from both sides.

$$x-20+20\sqrt{x}-x=100$$

Combine x and -x to get 0.

$$-20+20\sqrt{x}=100$$

Add 20 to both sides.

$$20\sqrt{x}=100+20$$

Add 100 and 20 to get 120.

$$20\sqrt{x}=120$$

Divide both sides by 20.

$$\sqrt{x}=\frac{120}{20}$$

Divide 120 by 20 to get 6.

$$\sqrt{x}=6$$

Square both sides of the equation.

$$x=36$$

Substitute 36 for x in the equation \sqrt{x-20}+\sqrt{x}=10.

$$\sqrt{36-20}+\sqrt{36}=10$$

Simplify. The value x=36 satisfies the equation.

$$10=10$$

Equation \sqrt{x-20}=-\sqrt{x}+10 has a unique solution.

$$x=36$$