SOLUTION: If x > 0 and y> 0 and x^2 - y^2 = 2xy, find the exact value of x/y

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Question 1119231: If x > 0 and y> 0 and x^2 - y^2 = 2xy, find the exact value of x/y
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
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In your original equation 

x^2 - y^2 = 2xy

divide both sides by y^2.  You will get


x%5E2%2Fy%5E2+-+1 = 2%2A%28x%2Fy%29,   or


%28x%2Fy%29%5E2+-+2%2A%28x%2Fy%29+-+1 = 0.


In the last equation, introduce new variable t = x%2Fy.  Then the equation takes the form

t^2 - 2t - 1 = 0.


Apply the quadratic formula:

t%5B1%2C2%5D = %28-%28-2%29+%2B-+sqrt%28%28-2%29%5E2+-4%2A1%2A%28-1%29%29%29%2F2 = %282+%2B-+sqrt%288%29%29%2F2 = 1+%2B-+sqrt%282%29.


Since t = x/y  and  x > 0, y > 0,  only positive root  t = 1+%2B+sqrt%282%29   works and gives the 


Answer.  x%2Fy = 1+%2B+sqrt%282%29.

Solved.