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.