If roots of equation ax^2 + bx + c = 0 are alpha and beta, Find Alpha^3 + Beta^3.

We know that for the standard equation ax^2 + bx + c = 0,

Sum of roots = -b/a

Product of roots = c/a

Here, the roots are alpha and beta, so 

α + β  = -b/a

αβ = c/a

Now, we'll find α^3 + β^3 by using the formula a^3 + b^3 = (a + b)(a^2 - ab + b^2). We'll keep substituting the α + β and αβ with the results we found.

α^3 + β^3

=> (α + β)(α^2 - αβ + β^2)

=> -b/a (α^2 - αβ + β^2 + 3αβ - 3αβ)

(This addition and subtraction of 3αβ makes sure that the expression is not altered and we get to simplify terms to make (a + b)^2 form)   

=> -b/a (α^2 + 2αβ + β^2 - 3αβ)

=> -b/a ((α + β)^2 - 3αβ)

=> -b/a ((-b/a)^2 - 3(c/a))

=> -b/a (b^2/a^2 - 3c/a)

=> -b/a (b^2 - 3ac)/a^2

=> (3abc - b^3)/a^3

 (3abc - b^3)/a^3 will be the answer.