Question

If $\alpha$ and $\beta$ are the zeroes of quadratic polynomial $x^2+3x+6$, find the values of
i) $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$
ii) ${\alpha }^2+{\beta }^2$

Answer 1

Sum of the roots$=\alpha +\beta =\frac{-3}{1}=-3$

Product of the roots$=\alpha \beta =\frac{6}{1}=6$

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

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

$=\frac{{(-3)}^2-2\times 6}{6}$

$=\frac{9-12}{6}=\frac{-3}{6}=\frac{-1}{2}$

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

$={(-3)}^2-2\times 6$

$=9-12=-3$