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

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Answer 1 · 2015-09-08T08:51:48

Rearrange and square a couple of times to find:

$x = 101/4$

Subtract $sqrt(x-5)$ from both sides to get:

$sqrt(x+5) = 10 - sqrt(x-5)$

Square both sides to get:

$x+5 = 100 - 20sqrt(x-5) + (x-5)$

Add $20sqrt(x-5)$ to both sides to get:

$20sqrt(x-5) + x + 5 = 100 + x - 5$

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

$20sqrt(x-5) = 100 - 10 = 90$

Divide both sides by $20$ to get:

$sqrt(x-5) = 90/20 = 9/2$

Square both sides to get:

$x - 5 = (9/2)^2 = 81/4$

Add $5$ to both sides to get:

$x = 81/4 + 5 = (81+20)/4 = 101/4$

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

Let $x = 101/4$

Then:

$sqrt(x+5) + sqrt(x-5)$

$=sqrt(101/4+5)+sqrt(101/4-5)$

$=sqrt(121/4) + sqrt(81/4)$

$=11/2 + 9/2 = 20/2 = 10$

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