Question

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=a{x}^2+bx+c,$ then evaluate:
$\frac{1}{\alpha }+\frac{1}{\beta }-2\alpha \beta$

Answer 1

$f\left(x\right)=a{x}^2+bx+c$

$\alpha ,\beta$ are the zeroes.

So $\alpha +\beta =\frac{-b}{a}\&\alpha \beta =\frac{c}{a}$

$\frac{1}{\alpha }+\frac{1}{\beta }-2\alpha \beta$

$\Rightarrow \frac{\alpha +\beta }{\alpha \beta }-2\alpha \beta =\frac{-b/a}{c/a}-\frac{2c}{a}$

$\Rightarrow \frac{-b}{c}-\frac{2c}{a}=-\frac{(ab+2c^2)}{ac}$