# How do you solve sqrt(x+5 ) + sqrt(x-5) = 10?

Sep 8, 2015

Rearrange and square a couple of times to find:

$x = \frac{101}{4}$

#### Explanation:

Subtract $\sqrt{x - 5}$ from both sides to get:

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

Square both sides to get:

$x + 5 = 100 - 20 \sqrt{x - 5} + \left(x - 5\right)$

Add $20 \sqrt{x - 5}$ to both sides to get:

$20 \sqrt{x - 5} + x + 5 = 100 + x - 5$

Subtract $x + 5$ from both sides to get:

$20 \sqrt{x - 5} = 100 - 10 = 90$

Divide both sides by $20$ to get:

$\sqrt{x - 5} = \frac{90}{20} = \frac{9}{2}$

Square both sides to get:

$x - 5 = {\left(\frac{9}{2}\right)}^{2} = \frac{81}{4}$

Add $5$ to both sides to get:

$x = \frac{81}{4} + 5 = \frac{81 + 20}{4} = \frac{101}{4}$

Since we have squared the equation a couple of times, we should check that the solution is correct and not spurious:

Let $x = \frac{101}{4}$

Then:

$\sqrt{x + 5} + \sqrt{x - 5}$

$= \sqrt{\frac{101}{4} + 5} + \sqrt{\frac{101}{4} - 5}$

$= \sqrt{\frac{121}{4}} + \sqrt{\frac{81}{4}}$

$= \frac{11}{2} + \frac{9}{2} = \frac{20}{2} = 10$