Let a and b be the roots of the quadratic 2x^2 - 8x + 7 = x^2 - 14x + 1. Compute a^3 + b^3.
+1622 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
