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 Get The Quadruplets (Posted on 2007-04-06)
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 No Rating Solution: (Hide) 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.

 Subject Author Date All Answers Analytically Brian Smith 2007-04-09 14:28:21 re: to begin with ....spoiler Dej Mar 2007-04-06 22:20:52 to begin with ....spoiler Ady TZIDON 2007-04-06 11:54:07

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