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Find 'K' (Posted on 2006-05-15) Difficulty: 3 of 5
The product of two of the four roots of the equation x418x3+Kx2+200x1984=0 is 32. Find the value of K.

No Solution Yet Submitted by Ravi Raja    
Rating: 3.0000 (1 votes)

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Solution Puzzle Resolution | Comment 1 of 3

The solution of the puzzle is given below.
Since the product of two of the roots of the given equation is -32; it follows that, the product of the remaining roots is equal to  -1984/ -32 = 62. Accordingly, we can write the given quartic expression as the product of two quadratic expressions, so that:
 x^4 18x^3 + K*x^2 + 200x 1984

= (x^2 + px -32)(x^2 + qx + 62);(say)
Comparing the exponents of x, x^2 and x^3 in the above relationship, we obtain:
(I) p+q = -18; (II) K = 30 + pq; and (III) 62*p 32*q = 200;
Solving  for p and q in  (I) and (III), we obtain, (p,q) = (-4,-14); giving :
K= 30 + (-4)*(-14) = 86.
Note: For a harder variation of quartic methodologies, refer to  "A Quartic Problem"
( Reference:
http://perplexus.info/show.php?pid=4279).

Edited on May 15, 2006, 12:56 pm

Edited on May 15, 2006, 1:00 pm
  Posted by K Sengupta on 2006-05-15 12:54:08

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