Solve sqrt{x-9}-sqrt{x-16}=1

$$\sqrt{ x-9 } - \sqrt{ x-16 } = 1$$

Solve for x

$$x=25$$

Solution Steps

Subtract -\sqrt{x-16} from both sides of the equation.

$$\sqrt{x-9}=1+\sqrt{x-16}$$

Square both sides of the equation.

$$\left(\sqrt{x-9}\right)^{2}=\left(1+\sqrt{x-16}\right)^{2}$$

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

$$x-9=\left(1+\sqrt{x-16}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{x-16}\right)^{2}.

$$x-9=1+2\sqrt{x-16}+\left(\sqrt{x-16}\right)^{2}$$

Calculate \sqrt{x-16} to the power of 2 and get x-16.

$$x-9=1+2\sqrt{x-16}+x-16$$

Subtract 16 from 1 to get -15.

$$x-9=-15+2\sqrt{x-16}+x$$

Subtract 2\sqrt{x-16} from both sides.

$$x-9-2\sqrt{x-16}=-15+x$$

Subtract x from both sides.

$$x-9-2\sqrt{x-16}-x=-15$$

Combine x and -x to get 0.

$$-9-2\sqrt{x-16}=-15$$

Add 9 to both sides.

$$-2\sqrt{x-16}=-15+9$$

Add -15 and 9 to get -6.

$$-2\sqrt{x-16}=-6$$

Divide both sides by -2.

$$\sqrt{x-16}=\frac{-6}{-2}$$

Divide -6 by -2 to get 3.

$$\sqrt{x-16}=3$$

Square both sides of the equation.

$$x-16=9$$

Add 16 to both sides of the equation.

$$x-16-\left(-16\right)=9-\left(-16\right)$$

Subtracting -16 from itself leaves 0.

$$x=9-\left(-16\right)$$

Subtract -16 from 9.

$$x=25$$

Substitute 25 for x in the equation \sqrt{x-9}-\sqrt{x-16}=1.

$$\sqrt{25-9}-\sqrt{25-16}=1$$

Simplify. The value x=25 satisfies the equation.

$$1=1$$

Equation \sqrt{x-9}=\sqrt{x-16}+1 has a unique solution.

$$x=25$$