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A Reciprocal And Square Problem (Posted on 20070710) 

Find all real pairs (p, q) satisfying the following system of equations:
p  1/p  q^{2} = 0
q/p + pq = 4

Submitted by K Sengupta

Rating: 3.5000 (2 votes)


Solution:

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The given equations yield;
p – 1/p = q^2
p+ 1/p = 4/q
Thus, 16/(q^2) – q^4 = 4, giving:
u^3 + 4u – 16 = 0, where u = q^2
Or, (u2)(u^2 +2u +8) = 0
Or, u =2, ignoring the complex roots of u which are inadmissible.
or, q = +/ √2)
If q = √2, then p = (1/2)*(q^2 + 4/q) = 1+ √2
Similarly, q =  √2 gives p = 1 – √2
Thus (p, q) = (√2, 1 + √2), (√2, 1 √2) are the only possible solutions.

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