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

x^4  (a+ b+c+ d) x^3 + [ab + cd + (a +b) (c +d)] x^2 - [ab (c + d) + cd (a + b)]x + abcd = 0.

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

y + z = 18
62 y - 32 z = -200

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