Question

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=6x^2+x-2$, find the value of $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$.

Answer 1

$f\left(x\right)=6x^2+x-2$

Roots are $\alpha$ and $\beta$

$\alpha +\beta =-\frac{1}{6}$

$\alpha \beta =-\frac{2}{6}$

$\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }=\frac{{(\alpha +\beta )}^2-2\left(\alpha \beta \right)}{\alpha \beta }$

$=\frac{{(-\frac{1}{6})}^2-2(-\frac{2}{6})}{(-\frac{2}{6})}$

$=\frac{\frac{1}{36}+\frac{4}{6}}{-\frac{2}{6}}$

$=\frac{\frac{1+24}{36}}{-\frac{2}{6}}$

$=-\frac{25}{36}\times \frac{6}{2}$

$\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=-\frac{25}{12}$