13. The product of two of the four roots of the quartic equation $x^{4}-18 x^{3}+k x^{2}+200 x-1984=0$ is -32 . Determine the value of $k$.
13. Answer: 86
Solution. Let $a, b, c, d$ be the four roots of $x^{4}-18 x^{3}+k x^{2}+200 x-1984=0$ such that $a b=-32$. Then
$$
\left\{\begin{array}{l}
a+b+c+d=18, \\
a b+a c+a d+b c+b d+c d=k, \\
a b c+a b d+a c d+b c d=-200, \\
a b c d=-1984 .
\end{array}\right.
$$

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 that
$$
a+b=4, \quad c+d=14 .
$$

From $a b+a c+a d+b c+b d+c d=k$, we have
$$
\begin{aligned}
k=a b+a c+a d+ & b c+b d+c d=-32+a c+a d+b c+b d+62 \\
& =30+(a+b)(c+d)=86
\end{aligned}
$$