Nearly a quarter century year old problem
Algebra
Level
4
The product of two of the four roots of is 24. Find .
The answer is 41.
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1 solution
Given the initial equation, we could use Vieta to create a system of equations relating the four roots and the four coefficients.
However, the calculations are less involved if we factor the quartic into the product of 2 quadratics first.
If we identify the four roots as a, b, c, and d, we can choose any pair to have a product, 24.
So let c * d = 24. Then, from Vieta, a * b = -1992/24 = -83.
Let c + d = p and a + b = q.
Then, the equation can be written as:
(x^2 + p x + 24) (x^2 + q x - 83) = 0
Expanding this again yields:
x^4 + (p+q) x^3 + (pq-59) x^2 + (24q-83p) x -1992 = 0
p + q = -20, so p = -(q + 20) and:
(24 q + 83 (q + 20)) = 590, so 107 q = -1070
Therefore, q = -10 and p = -10
k = p q - 59 = 100 - 59 = 41