Question 1207630
<pre>

{{{sqrt(10 + sqrt(x)) = sqrt(x)}}}

Square both sides:

{{{10 + sqrt(x) = x}}}

Isolate the root:

{{{sqrt(x) = x-10}}}

Square both sides:

{{{x=x^2-20x+100}}}

{{{x^2-21x+100}}}

{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}

{{{x = (-(-21) +- sqrt( (-21)^2-4*1*100 ))/(2*1) }}}

{{{x = (21 +- sqrt(441-400 ))/2 }}}

{{{x = (21 +- sqrt(41))/2 }}}

But when you square both sides of an equation, you have to check your answers,
for some, or even ALL, of your answers may be extraneous.

So we check.  It would be very time consuming to check the answers in exact
form, so get their decimal approximations. 

{{{x = (21 + sqrt(41))/2 }}} is approximately 13.70156212

{{{sqrt(10 + sqrt(x)) = sqrt(x)}}}

{{{sqrt(10 + sqrt(13.70156212)) = sqrt(13.70156212)}}}

{{{sqrt(10 + 3.701562119) = 3.701562119}}}

{{{sqrt(13.701562119) = 3.701562119}}}

{{{3.701562119) = 3.701562119}}}

That checks, so we know that is a solution.

Now let's check the other solution

{{{x = (21 - sqrt(41))/2 }}} is approximately 3.701562119

{{{sqrt(10 + sqrt(x)) = sqrt(x)}}}

{{{sqrt(10 + sqrt(3.701562119)) = sqrt(3.701562119)}}}

{{{sqrt(10 + 1.923944417) = 1.923944417}}}

{{{sqrt(11.923944417) = 1.923944417}}}

{{{3.453106488 = 1.923944417}}}

As we see, this does not check, so there is only 1 solution,

{{{x = (21 + sqrt(41))/2 }}}

Edwin</pre>