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Get The Quadruplets (Posted on 20070406) 

Analytically determine all possible quadruplets (p, q, r, s) of real numbers satisfying the following system of equations:
p+q = 8
pq + r + s = 23
ps + qr = 28
rs = 12

Submitted by K Sengupta

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Solution:

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We observe that,:
(x^2 + px + r)(x^2 + qx + s)
= x^4 + (p+q)x^3 + (pq + r + s)x^2 + (ps + qr)x + rs ......(*)
Accordingly, we consider the polynomial:
L(x)
= x^4 + 8*x^3 + 23*x^2 + 28x + 12
=(x+1)((x+2)^2)(x+3)
Therefore, L(x) can be expressed as the product of two quadratic expressions in four ways and they are:
L(x)
= (x^2 + 4x + 3)(x^2 + 4x + 4)
= (x^2 + 4x + 4) (x^2 + 4x + 3)
= (x^2 + 3x + 2)(x^2 + 5x + 6)
= (x^2 + 5x + 6)(x^2 + 3x + 2).......(i)
Consequently, comparing (i) with (*), we obtain:
(p, q, r, s) = (4,4,3,4); (4,4,4,3); (3,5,2,6); (5,3,6,2) as all possible solutions to the given problem.

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