Solve sqrt{x+2}+sqrt{x+7}=5

$$\sqrt{ x+2 } + \sqrt{ x+7 } = 5$$

Solve for x

$$x=2$$

Solution Steps

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

$$\sqrt{x+2}=5-\sqrt{x+7}$$

Square both sides of the equation.

$$\left(\sqrt{x+2}\right)^{2}=\left(5-\sqrt{x+7}\right)^{2}$$

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

$$x+2=\left(5-\sqrt{x+7}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-\sqrt{x+7}\right)^{2}.

$$x+2=25-10\sqrt{x+7}+\left(\sqrt{x+7}\right)^{2}$$

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

$$x+2=25-10\sqrt{x+7}+x+7$$

Add 25 and 7 to get 32.

$$x+2=32-10\sqrt{x+7}+x$$

Add 10\sqrt{x+7} to both sides.

$$x+2+10\sqrt{x+7}=32+x$$

Subtract x from both sides.

$$x+2+10\sqrt{x+7}-x=32$$

Combine x and -x to get 0.

$$2+10\sqrt{x+7}=32$$

Subtract 2 from both sides.

$$10\sqrt{x+7}=32-2$$

Subtract 2 from 32 to get 30.

$$10\sqrt{x+7}=30$$

Divide both sides by 10.

$$\sqrt{x+7}=\frac{30}{10}$$

Divide 30 by 10 to get 3.

$$\sqrt{x+7}=3$$

Square both sides of the equation.

$$x+7=9$$

Subtract 7 from both sides of the equation.

$$x+7-7=9-7$$

Subtracting 7 from itself leaves 0.

$$x=9-7$$

Subtract 7 from 9.

$$x=2$$

Substitute 2 for x in the equation \sqrt{x+2}+\sqrt{x+7}=5.

$$\sqrt{2+2}+\sqrt{2+7}=5$$

Simplify. The value x=2 satisfies the equation.

$$5=5$$

Equation \sqrt{x+2}=-\sqrt{x+7}+5 has a unique solution.

$$x=2$$