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

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Answer 1 · 2016-03-15T14:38:21

Rearrange and square a couple of times to find:

$x = 8$

Check the answer since squaring can introduce spurious solutions.

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):

$x+8 = (x-4)+4 sqrt(x-4) + 4$

$=x+4 sqrt(x-4)$

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:

$sqrt(x+8)-sqrt(x-4) = sqrt(8+8)-sqrt(8-4) = sqrt(16)-sqrt(4) = 4-2 = 2$

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