Solve sqrt{n+5}-sqrt{n-10}=1

$$\sqrt { n + 5 } - \sqrt { n - 10 } = 1$$

Solve for n

$$n=59$$

Solution Steps

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

$$\sqrt{n+5}=1+\sqrt{n-10}$$

Square both sides of the equation.

$$\left(\sqrt{n+5}\right)^{2}=\left(1+\sqrt{n-10}\right)^{2}$$

Calculate \sqrt{n+5} to the power of 2 and get n+5.

$$n+5=\left(1+\sqrt{n-10}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{n-10}\right)^{2}.

$$n+5=1+2\sqrt{n-10}+\left(\sqrt{n-10}\right)^{2}$$

Calculate \sqrt{n-10} to the power of 2 and get n-10.

$$n+5=1+2\sqrt{n-10}+n-10$$

Subtract 10 from 1 to get -9.

$$n+5=-9+2\sqrt{n-10}+n$$

Subtract 2\sqrt{n-10} from both sides.

$$n+5-2\sqrt{n-10}=-9+n$$

Subtract n from both sides.

$$n+5-2\sqrt{n-10}-n=-9$$

Combine n and -n to get 0.

$$5-2\sqrt{n-10}=-9$$

Subtract 5 from both sides.

$$-2\sqrt{n-10}=-9-5$$

Subtract 5 from -9 to get -14.

$$-2\sqrt{n-10}=-14$$

Divide both sides by -2.

$$\sqrt{n-10}=\frac{-14}{-2}$$

Divide -14 by -2 to get 7.

$$\sqrt{n-10}=7$$

Square both sides of the equation.

$$n-10=49$$

Add 10 to both sides of the equation.

$$n-10-\left(-10\right)=49-\left(-10\right)$$

Subtracting -10 from itself leaves 0.

$$n=49-\left(-10\right)$$

Subtract -10 from 49.

$$n=59$$

Substitute 59 for n in the equation \sqrt{n+5}-\sqrt{n-10}=1.

$$\sqrt{59+5}-\sqrt{59-10}=1$$

Simplify. The value n=59 satisfies the equation.

$$1=1$$

Equation \sqrt{n+5}=\sqrt{n-10}+1 has a unique solution.

$$n=59$$