Question

If $\alpha$ and $\beta$ the zeroes of quadratic (equations) form $3x^2-4x+5$ then find the value of $\frac{1}{{\alpha }^3}+\frac{1}{{\beta }^3}$

Answer 1

Sum of the roots$=\alpha +\beta =\frac{4}{3}$

Product of the roots$=\alpha \beta =\frac{5}{3}$

$\frac{1}{{\alpha }^3}+\frac{1}{{\beta }^3}$

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

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

$=\frac{\left(\frac{4}{3}\right)-3\times \frac{5}{3}\left(\frac{4}{3}\right)}{{\left(\frac{5}{3}\right)}^3}$

$=\frac{\frac{4}{3}-\frac{20}{3}}{\frac{125}{27}}$

$=\frac{\frac{4-20}{3}}{\frac{125}{27}}$

$=\frac{\frac{-16}{3}}{\frac{125}{27}}$

$=\frac{-16}{3}\times \frac{27}{125}$

$=\frac{-16\times 9}{125}$

$=\frac{-144}{125}$