The product of two of the four roots of the quartic equation x^4 – 18x^3 + kx^2 + 200x -1984 = 0 is -32. Determine the value of k.

**Solution by Steve Dinh, a.k.a. Vo Duc Dien (dedicated to the lovely Loan Tran)**

An equation having four roots can be expressed as (x – a)(x – b)(x – c)(x – d) = 0, where a, b, c and d are the roots of the equation. In our case, without loss of generality, we may assume ab = -32.

Expanding the equation we have

Now equating the corresponding terms gives

(1) a + b + c + d = 18,

(2) ab (c + d) + cd (a + b) = -200,

(3) abcd = -1984,

(4) k = ab + cd + (a + b)(c + d).

Since ab = -32, from (3), cd = 62. Let y = a + b and z = c + d, equations (1) and (2) become

Solving them we have y = 14 and z = 4. Substituting these values to (4) gives k = 86.