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A Reciprocal And Square Problem (Posted on 2007-07-10) Difficulty: 2 of 5
Find all real pairs (p, q) satisfying the following system of equations:
p - 1/p - q2 = 0

q/p + pq = 4

See The Solution Submitted by K Sengupta    
Rating: 3.5000 (2 votes)

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Solution Sol | Comment 6 of 7 |
p-1/p=q
q(p+1/p)=4
square the second eq, q*(p+1/p)=16
(p-1/p)*((p-1/p)+4)=16
let p-1/p=x
x+4x-16=0 => (x-2)*(x+2x+8)=0
x+2x+8=0 has no real roots as its discriminant is less than 0
x=2 is the only solution.
Solving p-1/p=2 yields p=(1(+/-)√2)
substitute these in eq(2) to get q=+√2 for p=(1+√2)
and q=-√2 for p=(1-√2)

Edited on July 10, 2007, 11:48 am
  Posted by Praneeth Yalavarthi on 2007-07-10 11:44:22

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