Question 18028
{{{ sqrt( x+10 ) + sqrt( x-6 ) = 8 }}}

Let w = sqrt( x+10 )
Let z = sqrt( x-6 )

Then w+z = 8
multiply both sides by ( w-z )

( w+z )( w-z ) = 8 ( w-z )

multiplying left side:
w^2 - z^2 = 8( w-z )
w^2 = x+10
z^2 = x-6

substituting:
x+10 - (x-6) = 8( w-z )
16 = 8( w-z )

dividing both sides by 8:
w-z = 2
recall, w+z = 8

w-z = 2
w+z = 8

adding:
2w = 10
w = 5
z = 3

substituting:
sqrt(x-6) = 3

square both sides:
x-6 = 9
x = 15

also:
sqrt(x+10) = 5

square both sides:
x+10 = 25
x = 15

This confirms the result. 
x = 15 plugged into the original equation gives the solution:

{{{ sqrt( 15+10 ) + sqrt( 15-6 ) = 8 }}}

Note the the positive square roots must be taken.