Question 1118300
Ans:  {{{ highlight( 34 ) }}}

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Its just a small amount of algebra and rationalization of fractions.

{{{ x + 1/x = 6 }}}
{{{  x^2-6x  = -1 }}}
{{{  x^2-6x+9 = -1 + 9 }}}
{{{  (x-3)^2 = 8 }}}
{{{  (x-3) = +-sqrt(8) }}}  (*)
<b> {{{ blue( x = 3+sqrt(8) ) }}}</b>  <br>

Considering {{{ x=3+sqrt(8) }}}:
Now plug in this value of x into  {{{ x^2 + 1/x^2 }}}:
 {{{ (3 + sqrt(8))^2 = 9+6*(sqrt(8))+8 = 17+6sqrt(8) }}}

{{{ 17 + 6sqrt(8) + green( (1/(17+6sqrt(8))))   =  (17+6sqrt(8)) + green(17-6sqrt(8)) = 17+17 = highlight(34) }}}

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(*) Also,  {{{ 3-sqrt(8) }}} solves  x + 1/x = 6  so that number must be plugged into x^2 + 1/x^2 to get a 2nd answer.   But that answer will also be {{{ highlight(34) }}} 

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Good solution @MathTherapy!