Question 1119231
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In your original equation 

x^2 - y^2 = 2xy

divide both sides by y^2.  You will get


{{{x^2/y^2 - 1}}} = {{{2*(x/y)}}},   or


{{{(x/y)^2 - 2*(x/y) - 1}}} = 0.


In the last equation, introduce new variable t = {{{x/y}}}.  Then the equation takes the form

t^2 - 2t - 1 = 0.


Apply the quadratic formula:

{{{t[1,2]}}} = {{{(-(-2) +- sqrt((-2)^2 -4*1*(-1)))/2}}} = {{{(2 +- sqrt(8))/2}}} = {{{1 +- sqrt(2)}}}.


Since t = x/y  and  x > 0, y > 0,  only positive root  t = {{{1 + sqrt(2)}}}   works and gives the 


<U>Answer</U>.  {{{x/y}}} = {{{1 + sqrt(2)}}}.
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Solved.