Question 102444
{{{X+1/X=5/2}}} 
{{{X^2+1=(5/2)*X}}}Multiply both sides by X.
{{{X^2-(5/2)*X+1=cross((5/2)*X)-cross((5/2)*X)}}}Additive inverse of {{{(5/2)*X}}}
{{{X^2-(5/2)*X+1=0}}}Quadratic equation. 
We can solve this equation using the quadratic formula where the solution for 

{{{aX^2+b*X+c=0}}} is solved by

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

In this case {{{a=1}}},{{{b=-5/2}}},{{{c=1}}}
{{{x = (-(-5/2) +- sqrt( (5)^2/(2)^2-4*1*1))/(2*1) }}}Substitute into the quadratic formula. 
{{{2x = (5/2 +- sqrt( 25/4-4))) }}}Multiply both sides by 2, just for convenience. 
{{{2x = (5/2 +- sqrt( 25/4-16/4))) }}}Subtract fractions in the square root, need same denominator.
{{{2x = (5/2 +- sqrt( 9/4))) }}}Simplify.
{{{2x = (5/2 +-  3/2) }}}
Two solutions are
1.{{{2x = (5/2 +  3/2) }}}
1.{{{2x=8/2}}}
1.{{{x=2}}}
and
2.{{{2x = (5/2 - 3/2) }}}
2.{{{2x = (2/2) }}}
2.{{{x = 1/2) }}}
Check your answers
{{{X + 1/X = 5/2}}}
{{{2 + 1/2 = 5/2}}}
{{{5/2 = 5/2}}}
True, good answer. 
{{{X + 1/X = 5/2}}}
{{{1/2 + 1/(1/2) = 5/2}}}
{{{1/2 + 2 = 5/2}}}
{{{5/2 = 5/2}}}
True, good answer.