Solve sqrt{x+9}+sqrt{x+2}=7

$$\sqrt { x + 9 } + \sqrt { x + 2 } = 7$$

Solve for x

$$x=7$$

Solution Steps

Subtract \sqrt{x+2} from both sides of the equation.

$$\sqrt{x+9}=7-\sqrt{x+2}$$

Square both sides of the equation.

$$\left(\sqrt{x+9}\right)^{2}=\left(7-\sqrt{x+2}\right)^{2}$$

Calculate \sqrt{x+9} to the power of 2 and get x+9.

$$x+9=\left(7-\sqrt{x+2}\right)^{2}$$

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

$$x+9=49-14\sqrt{x+2}+\left(\sqrt{x+2}\right)^{2}$$

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

$$x+9=49-14\sqrt{x+2}+x+2$$

Add 49 and 2 to get 51.

$$x+9=51-14\sqrt{x+2}+x$$

Add 14\sqrt{x+2} to both sides.

$$x+9+14\sqrt{x+2}=51+x$$

Subtract x from both sides.

$$x+9+14\sqrt{x+2}-x=51$$

Combine x and -x to get 0.

$$9+14\sqrt{x+2}=51$$

Subtract 9 from both sides.

$$14\sqrt{x+2}=51-9$$

Subtract 9 from 51 to get 42.

$$14\sqrt{x+2}=42$$

Divide both sides by 14.

$$\sqrt{x+2}=\frac{42}{14}$$

Divide 42 by 14 to get 3.

$$\sqrt{x+2}=3$$

Square both sides of the equation.

$$x+2=9$$

Subtract 2 from both sides of the equation.

$$x+2-2=9-2$$

Subtracting 2 from itself leaves 0.

$$x=9-2$$

Subtract 2 from 9.

$$x=7$$

Substitute 7 for x in the equation \sqrt{x+9}+\sqrt{x+2}=7.

$$\sqrt{7+9}+\sqrt{7+2}=7$$

Simplify. The value x=7 satisfies the equation.

$$7=7$$

Equation \sqrt{x+9}=-\sqrt{x+2}+7 has a unique solution.

$$x=7$$