Question

If alpha and beta are the roots of the equation ax2+bx+c=0 then find the value of alfa^3 + beta^3

Answer 1

we know (a+b)³=a³+3a²b+3ab²+b³

which can be written as (a+b)³=a³+b³+3ab(a+b).

=》a³+b³ = (a+b)³- 3ab(a+b).

similarly

α³+β³= (α+β)³ - 3αβ(α+β).

here The first term is,  ax^2  its coefficient is  a .

The middle term is,  bx  its coefficient is  b.

The last term, "the constant", is  c .

we know in a quadratic equation α+β = -b/a.

and αβ = c/a.

so, substitute these values in (α+β)³ - 3αβ(α+β).

=》(-b/a)³ - 3(c/a)(-b/a).

=》-b³/a³+3bc/a^2.

by taking the LCM OF BOTH..

WE GET.

(-b³+3abc)/a³.

therefore α³+β³ = (-b³+3abc)/a³.