One solution of the equation
If a, b, c, and d are different integers, find the value of a+b+c+d
Since x=7 is a solution of the given equation, it follows that:
P*Q*R*S = 25, where (P, Q, R, S) = (7-a, 7-b, 7-c, 7-d)(say) ...(#)
Since a, b, c, d are distinct integers, so are P, Q, R, S, and accordingly, we must have:
(P, Q, R, S) (- (5, -5, 1, -1). In other words, (P, Q, R, S) corresponds to a perm of (5, -5, 1, -1) in consonance with (#). No other factorization of 25 will yield distinct integer values for the quadruplet (P, Q, R, S).
Thus, P+Q+R+S = 5-5+1-1 = 0, giving:
28-(a+b+c+d) = 0, so that:
a+b+c+d = 28
Consequently, the required value of a+b+c+d is 28.
Edited on July 10, 2008, 6:42 am