Question

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$, find the value of $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$ in terms of $a,b,c$.

Answer 1

We have, $\alpha +\beta =-\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$

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

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