Question 372500
You seem to have the right idea, will go thru it step by step:
{{{sqrt(10 - 2x) - sqrt(5x + 16)}}} = 3
:
{{{sqrt(10 - 2x)}}} =  {{{sqrt(5x + 16)}}} + 3; I think you have the wrong sign here
:
Square both sides, FOIL the right side
10 - 2x = (5x+16) + {{{6sqrt(5x + 16)}}} + 9
:
10 - 2x = 5x + 25 + {{{6sqrt(5x + 16)}}}
:
10 - 25 - 2x - 5x = {{{6sqrt(5x + 16)}}}
:
-7x - 15 = {{{6sqrt(5x + 16)}}}
:
square again, FOIL the left side
49x^2 + 105x + 105x + 225 = 36(5x + 16)
:
 49x^2 + 210x + 225 = 180x + 576  
:
Combine on the right
49x^2 + 210x - 180x + 225 - 576 = 0
;
A quadratic equation
49x^2 + 30x - 351 = 0 
:
Use the quadratic formula to find x. a=49; b=30; c=-351
{{{x = (-30 +- sqrt(30^2-4*49*-351 ))/(2*49) }}}
:
{{{x = (-30 +- sqrt(900-(-68796) ))/98 }}}
:
{{{x = (-30 +- sqrt(69696 ))/98 }}}
two solutions
{{{x = (-30 + 264)/98 }}}
x = {{{234/98}}}
x = +2.387755
and
{{{x = (-30 - 264)/98 }}}
x = {{{(-294)/98}}}
x = -3
:
I checked both these solutions in the original problem. You can confirm this yourself, however.
:
I'm not surprised that you had difficulty, a dozen ways to go wrong here.