SOLUTION: Solve the following system of equations: a + b +c = 9 ab + bc + ac = 26 abc = 24 answer is [a, b, c]=[2, 3, 4] Thanks.

Question 4397: Solve the following system of equations:
a + b +c = 9
ab + bc + ac = 26
abc = 24
answer is [a, b, c]=[2, 3, 4]
Thanks.
Answer by khwang(438) (Show Source): You can put this solution on YOUR website!
Let a,b,c be the three roots of a cubic polynomial (monic) equation f(x) = 0.
Then f(x) = (x-a)(x-b)(x-c)
By the relationship of roots and coefficients, we have the polyonmial is:
x^3 - 9x^2 + 26x - 24 = 0,
Or x^3 - 2x^2 - 7x^2 + 26x - 24 = 0,
Or x^2(x-2) - (x - 2)(x - 12) = 0
So, (x-2)(x^2 - x + 12) = 0 (Or divide by (x-2) directly)
We have (x-2)(x-3)(x-4) = 0

Hence,2,3 & 4 are roots of f(x) = x^3 - 9x^2 + 26x - 24 = 0.
This means [a,b,c] = [2,3,4] (as a set)
Kenny
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