Solve sqrt{x+12}-sqrt{x-15}=3

$$\sqrt { x + 12 } - \sqrt { x - 15 } = 3$$

Solve for x

$$x=24$$

Solution Steps

Subtract -\sqrt{x-15} from both sides of the equation.

$$\sqrt{x+12}=3+\sqrt{x-15}$$

Square both sides of the equation.

$$\left(\sqrt{x+12}\right)^{2}=\left(3+\sqrt{x-15}\right)^{2}$$

Calculate \sqrt{x+12} to the power of 2 and get x+12.

$$x+12=\left(3+\sqrt{x-15}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{x-15}\right)^{2}.

$$x+12=9+6\sqrt{x-15}+\left(\sqrt{x-15}\right)^{2}$$

Calculate \sqrt{x-15} to the power of 2 and get x-15.

$$x+12=9+6\sqrt{x-15}+x-15$$

Subtract 15 from 9 to get -6.

$$x+12=-6+6\sqrt{x-15}+x$$

Subtract 6\sqrt{x-15} from both sides.

$$x+12-6\sqrt{x-15}=-6+x$$

Subtract x from both sides.

$$x+12-6\sqrt{x-15}-x=-6$$

Combine x and -x to get 0.

$$12-6\sqrt{x-15}=-6$$

Subtract 12 from both sides.

$$-6\sqrt{x-15}=-6-12$$

Subtract 12 from -6 to get -18.

$$-6\sqrt{x-15}=-18$$

Divide both sides by -6.

$$\sqrt{x-15}=\frac{-18}{-6}$$

Divide -18 by -6 to get 3.

$$\sqrt{x-15}=3$$

Square both sides of the equation.

$$x-15=9$$

Add 15 to both sides of the equation.

$$x-15-\left(-15\right)=9-\left(-15\right)$$

Subtracting -15 from itself leaves 0.

$$x=9-\left(-15\right)$$

Subtract -15 from 9.

$$x=24$$

Substitute 24 for x in the equation \sqrt{x+12}-\sqrt{x-15}=3.

$$\sqrt{24+12}-\sqrt{24-15}=3$$

Simplify. The value x=24 satisfies the equation.

$$3=3$$

Equation \sqrt{x+12}=\sqrt{x-15}+3 has a unique solution.

$$x=24$$