Question

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$ then find the values of
${\alpha }^3+{\beta }^3$

Answer 1

Given that,

$a{x}^2+bx+c=0$

roots are $\alpha$ and $\beta$

then,

sum of roots$=-\frac{b}{a}$

$\alpha +\beta =-\frac{b}{a}$

Product of roots$=\frac{c}{a}$

$\alpha \beta =\frac{c}{a}$

${\alpha }^3+{\beta }^3=?$

Solve ${\alpha }^3+{\beta }^3=(\alpha +\beta {)}^3-3\alpha \beta (\alpha +\beta )$

$={(-\frac{b}{a})}^3-3\times \frac{c}{a}(-\frac{b}{c})$

$=\frac{-b^3}{a^3}+\frac{3bc}{a^2}$

${\alpha }^3+{\beta }^3=\frac{-b^3+3abc}{a^3}$

Hence, this is the answer.