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#