The product of two of the four roots of the quadratic equation $x^{4} - 18^{3} + k x^{2} + 200 x - 1984 = 0$ is $- 32.$ Determine the value of $k .$
(correct answer + 3, wrong answer 0)

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Solution

Let $a, b, c, d$ be the four roots of $x^{4} - 18^{3} + k x^{2} + 200 x - 1984 = 0$ such that $a b = - 32.$
Then, $a + b + c + d = 18$
$a b + b c + c d + d a + a c + b d = k$
$a b c + a b d + a c d + b c d = - 200$
$a b c d = - 1984$
Since $a b = - 32$ and $a b c d = - 1984,$
we have, $c d = 62$
Then, from $a b c + a b d + a c d + b c d = - 200,$ we have
$- 200 = - 32 c - 32 d + 62 a + 62 b = - 32 (c + d) + 62 (a + b)$
Solving this equation together with the equation $a + b + c + d = 18$ gives
$a + b = 4,$ $c + d = 14$

From $a b + b c + c d + d a + a c + b d = k$ , we have
$k = a b + b c + c d + d a + a c + b d = - 32 + a c + a d + b c + b d + 62$
$\Rightarrow k = 30 + (a + b) (c + d) = 30 + 4 \times 14 = 86$
$\Rightarrow k = 86$

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