Question

If $\alpha$ and $\beta$ are the roots of $4x^2+3x+7=0$ the value of $\frac{1}{{\alpha }^3}+\frac{1}{{\beta }^3}$ is

• A. $\frac{225}{343}$
• B. $\frac{223}{345}$
• C. $\frac{223}{343}$
• D. $\frac{225}{345}$

Answer 1

The correct option is A $\frac{225}{343}$

$\alpha +\beta =-3/4$ and $\alpha \beta =7/4$

$\frac{1}{{\alpha }^3}+\frac{1}{{\beta }^3}=\frac{1}{{\alpha }^3{\beta }^3}({\alpha }^3+{\beta }^3)$

$=\frac{{(\alpha +\beta )}^3-3ab(\alpha +\beta )}{{\left(\alpha \beta \right)}^3}$

=$\frac{-\frac{27}{64}-3\times \frac{7}{4}\times \frac{3}{4}}{{\left(\frac{7}{4}\right)}^3}$

$=\frac{225}{343}$