Solve sqrt{x-5}-sqrt{x+10}=-3

$$\sqrt { x - 5 } - \sqrt { x + 10 } = - 3$$

Solve for x

$$x=6$$

Solution Steps

Subtract -\sqrt{x+10} from both sides of the equation.

$$\sqrt{x-5}=-3+\sqrt{x+10}$$

Square both sides of the equation.

$$\left(\sqrt{x-5}\right)^{2}=\left(-3+\sqrt{x+10}\right)^{2}$$

Calculate \sqrt{x-5} to the power of 2 and get x-5.

$$x-5=\left(-3+\sqrt{x+10}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(-3+\sqrt{x+10}\right)^{2}.

$$x-5=9-6\sqrt{x+10}+\left(\sqrt{x+10}\right)^{2}$$

Calculate \sqrt{x+10} to the power of 2 and get x+10.

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

Add 9 and 10 to get 19.

$$x-5=19-6\sqrt{x+10}+x$$

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

$$x-5+6\sqrt{x+10}=19+x$$

Subtract x from both sides.

$$x-5+6\sqrt{x+10}-x=19$$

Combine x and -x to get 0.

$$-5+6\sqrt{x+10}=19$$

Add 5 to both sides.

$$6\sqrt{x+10}=19+5$$

Add 19 and 5 to get 24.

$$6\sqrt{x+10}=24$$

Divide both sides by 6.

$$\sqrt{x+10}=\frac{24}{6}$$

Divide 24 by 6 to get 4.

$$\sqrt{x+10}=4$$

Square both sides of the equation.

$$x+10=16$$

Subtract 10 from both sides of the equation.

$$x+10-10=16-10$$

Subtracting 10 from itself leaves 0.

$$x=16-10$$

Subtract 10 from 16.

$$x=6$$

Substitute 6 for x in the equation \sqrt{x-5}-\sqrt{x+10}=-3.

$$\sqrt{6-5}-\sqrt{6+10}=-3$$

Simplify. The value x=6 satisfies the equation.

$$-3=-3$$

Equation \sqrt{x-5}=\sqrt{x+10}-3 has a unique solution.

$$x=6$$