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Real Variable Validation (Posted on 2016-06-11) Difficulty: 3 of 5
Each of P, Q and R is a positive real number satisfying this system of equations:

P + Q = 13, and:
Q2 + R2 - Q*R = 25, and:
P2 + R2 + P*R = 144

Find R.

No Solution Yet Submitted by K Sengupta    
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Solution Solution | Comment 1 of 4
I started by competing the square on the second and third equations:
(2Q-R)^2 = 100-3R^2
(2P+R)^2 = 576-3R^2

Then expressed each P and Q in terms of R:
Q = (R +/- sqrt(100-3R^2))/2
P = (R +/- sqrt(576-3R^2))/2

Then substitute into the first equation:
(R +/- sqrt(100-3R^2))/2 + (R +/- sqrt(576-3R^2))/2 = 13

This simplifies to a quartic in R:
27R^4 + 2028R^2 - 57600 = 0

The one positive real root is R = sqrt[2*sqrt(71761)-169]/3

Normally these problems have a tidier final form.  Did I miss something?

  Posted by Brian Smith on 2016-06-11 11:29:27
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