Solve sqrt{x}+sqrt{x+7}=7

$$\sqrt { x } + \sqrt { x + 7 } = 7$$

Solve for x

$$x=9$$

Solution Steps

Subtract \sqrt{x+7} from both sides of the equation.

$$\sqrt{x}=7-\sqrt{x+7}$$

Square both sides of the equation.

$$\left(\sqrt{x}\right)^{2}=\left(7-\sqrt{x+7}\right)^{2}$$

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

$$x=\left(7-\sqrt{x+7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7-\sqrt{x+7}\right)^{2}.

$$x=49-14\sqrt{x+7}+\left(\sqrt{x+7}\right)^{2}$$

Calculate \sqrt{x+7} to the power of 2 and get x+7.

$$x=49-14\sqrt{x+7}+x+7$$

Add 49 and 7 to get 56.

$$x=56-14\sqrt{x+7}+x$$

Add 14\sqrt{x+7} to both sides.

$$x+14\sqrt{x+7}=56+x$$

Subtract x from both sides.

$$x+14\sqrt{x+7}-x=56$$

Combine x and -x to get 0.

$$14\sqrt{x+7}=56$$

Divide both sides by 14.

$$\sqrt{x+7}=\frac{56}{14}$$

Divide 56 by 14 to get 4.

$$\sqrt{x+7}=4$$

Square both sides of the equation.

$$x+7=16$$

Subtract 7 from both sides of the equation.

$$x+7-7=16-7$$

Subtracting 7 from itself leaves 0.

$$x=16-7$$

Subtract 7 from 16.

$$x=9$$

Substitute 9 for x in the equation \sqrt{x}+\sqrt{x+7}=7.

$$\sqrt{9}+\sqrt{9+7}=7$$

Simplify. The value x=9 satisfies the equation.

$$7=7$$

Equation \sqrt{x}=-\sqrt{x+7}+7 has a unique solution.

$$x=9$$