Solve sqrt{x-9}+sqrt{x+6}=5

$$\sqrt { x - 9 } + \sqrt { x + 6 } = 5$$

Solve for x

$$x=10$$

Solution Steps

Subtract \sqrt{x+6} from both sides of the equation.

$$\sqrt{x-9}=5-\sqrt{x+6}$$

Square both sides of the equation.

$$\left(\sqrt{x-9}\right)^{2}=\left(5-\sqrt{x+6}\right)^{2}$$

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

$$x-9=\left(5-\sqrt{x+6}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5-\sqrt{x+6}\right)^{2}.

$$x-9=25-10\sqrt{x+6}+\left(\sqrt{x+6}\right)^{2}$$

Calculate \sqrt{x+6} to the power of 2 and get x+6.

$$x-9=25-10\sqrt{x+6}+x+6$$

Add 25 and 6 to get 31.

$$x-9=31-10\sqrt{x+6}+x$$

Add 10\sqrt{x+6} to both sides.

$$x-9+10\sqrt{x+6}=31+x$$

Subtract x from both sides.

$$x-9+10\sqrt{x+6}-x=31$$

Combine x and -x to get 0.

$$-9+10\sqrt{x+6}=31$$

Add 9 to both sides.

$$10\sqrt{x+6}=31+9$$

Add 31 and 9 to get 40.

$$10\sqrt{x+6}=40$$

Divide both sides by 10.

$$\sqrt{x+6}=\frac{40}{10}$$

Divide 40 by 10 to get 4.

$$\sqrt{x+6}=4$$

Square both sides of the equation.

$$x+6=16$$

Subtract 6 from both sides of the equation.

$$x+6-6=16-6$$

Subtracting 6 from itself leaves 0.

$$x=16-6$$

Subtract 6 from 16.

$$x=10$$

Substitute 10 for x in the equation \sqrt{x-9}+\sqrt{x+6}=5.

$$\sqrt{10-9}+\sqrt{10+6}=5$$

Simplify. The value x=10 satisfies the equation.

$$5=5$$

Equation \sqrt{x-9}=-\sqrt{x+6}+5 has a unique solution.

$$x=10$$