Question

If $\alpha$ and $\beta$ the zeros of the quadratic polynomial $a{x}^2+bx+c=0$ then find the value $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$

Answer 1

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

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

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

$=\frac{{(-\frac{b}{a})}^2-2\left(\frac{c}{a}\right)}{{\left(\frac{c}{a}\right)}^2}$

$=\frac{b^2-2ac}{c^2}$