If (x+k) is the HCF of x^{2}+px+q and x^{2}+qx+p, where p \neq q, then what is the value of k?

Explanation

Since (x+k) is the HCF, x = -k is a root of both equations. So, (-k)^2 + p(-k) + q = 0 and (-k)^2 + q(-k) + p = 0. Subtracting them: -k(p-q) + (q-p) = 0 \implies -k(p-q) = p-q. Since p \neq q, k = -1.

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