Question

$f\left(x\right)=2x^2-5x+7$ find

$i)\frac{1}{\alpha }+\frac{1}{\beta }$
$ii)\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$

Answer 1

$f\left(x\right)=2x^2-5x+7$

$\alpha +\beta =\frac{5}{2}$

$\alpha \cdot \beta =\frac{7}{2}$

i) $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }=\frac{5}{7}$

ii) $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }=\frac{(\alpha +\beta {)}^2-2\alpha \beta }{\alpha \beta }=\frac{\frac{25}{4}-\frac{2\left(7\right)}{2}}{\frac{7}{2}}=\frac{11}{14}$