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Get The Quadruplets (Posted on 2007-04-06) Difficulty: 2 of 5
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: (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.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
SolutionAll Answers AnalyticallyBrian Smith2007-04-09 14:28:21
Some Thoughtsre: to begin with ....spoilerDej Mar2007-04-06 22:20:52
Solutionto begin with ....spoilerAdy TZIDON2007-04-06 11:54:07
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