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Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = x^2 - 14x + 1.  Compute a^3 + b^3.

 Sep 20, 2022
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First we combine like terms: (I will not be using quadratic formula or a calculator)

x^2 + 6x + 6 = 0 

First we can use sum of cubes a^3 + b^3 = (a + b)(a^2 - ab + b^2). Then we can see by using Vieté's Formula, it will be easy to solve.

The sum of the roots = -6 = a + b

The product of the roots = 6 = ab

a^2 + b^2 = (a + b)^2 - 2ab

a^3 + b^3 = (a + b)[(a + b)^2 - 2ab - ab]

Plugging in the values, we have:

(-6)[(-6)^2 - 2(6) - (6)]

-6(36 - 18)

-6(18)

a^3 + b^3 = -108

 Sep 20, 2022

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