Solve sqrt{x-8}+sqrt{x}=4

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

Solve for x

$$x=9$$

Solution Steps

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

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

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(4-\sqrt{x}\right)^{2}.

$$x-8=16-8\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$x-8=16-8\sqrt{x}+x$$

Add 8\sqrt{x} to both sides.

$$x-8+8\sqrt{x}=16+x$$

Subtract x from both sides.

$$x-8+8\sqrt{x}-x=16$$

Combine x and -x to get 0.

$$-8+8\sqrt{x}=16$$

Add 8 to both sides.

$$8\sqrt{x}=16+8$$

Add 16 and 8 to get 24.

$$8\sqrt{x}=24$$

Divide both sides by 8.

$$\sqrt{x}=\frac{24}{8}$$

Divide 24 by 8 to get 3.

$$\sqrt{x}=3$$

Square both sides of the equation.

$$x=9$$

Substitute 9 for x in the equation \sqrt{x-8}+\sqrt{x}=4.

$$\sqrt{9-8}+\sqrt{9}=4$$

Simplify. The value x=9 satisfies the equation.

$$4=4$$

Equation \sqrt{x-8}=-\sqrt{x}+4 has a unique solution.

$$x=9$$