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Sum Root Sum Settlement (Posted on 2015-05-10) Difficulty: 3 of 5
Find all possible positive real solutions to this equation:

p + q + 1/p + 1/q + 4 = 2(√(2p+1) + √(2q+1))

No Solution Yet Submitted by K Sengupta    
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Solution No Subject | Comment 1 of 5
By symmetry all solutions must have p=q.   So the equation simplifies to
2p + 2/p + 4 = 4*sqrt(2p+1)
And then to the polynomial 
16p^4 + 8p^3 + 8p^2 + 4p + 1 = 0
Which has no real roots. 

If I have made no errors the equation has no real solutions. 

Edit:  I did make an error.  The polynomial should be
 p^4 - 4p^3 + 2p^2 + 4p + 1 = 0
as xdog pointed out the roots are p=1±2
but the one with the minus is extraneous that arises from squaring both sides.

The only real solution is p=1+2

Edited on May 12, 2015, 9:01 am
  Posted by Jer on 2015-05-11 10:15:26

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