# How do you solve sqrt (x+8) - sqrt (x-4) = 2?

Mar 15, 2016

Rearrange and square a couple of times to find:

$x = 8$

Check the answer since squaring can introduce spurious solutions.

#### Explanation:

Add $\sqrt{x - 4}$ to both sides to get:

$\sqrt{x + 8} = \sqrt{x - 4} + 2$

Square both sides (noting that this may introduce spurious solutions):

$x + 8 = \left(x - 4\right) + 4 \sqrt{x - 4} + 4$

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

Subtract $x$ from both sides and transpose to get:

$4 \sqrt{x - 4} = 8$

Divide both sides by $4$ to get:

$\sqrt{x - 4} = 2$

Square both sides (noting that this may introduce spurious solutions) to get:

$x - 4 = 4$

Add $4$ to both sides to get:

$x = 8$

Check that this works:

$\sqrt{x + 8} - \sqrt{x - 4} = \sqrt{8 + 8} - \sqrt{8 - 4} = \sqrt{16} - \sqrt{4} = 4 - 2 = 2$