Question

If $\alpha ,\beta$ are the roots of $a{x}^2+bx+c=0$ then find the values of
$\frac{{\alpha }^2+{\beta }^2}{{\alpha }^{-2}+{\beta }^{-2}}$

Answer 1

Given that,

$a{x}^2+bx+c=0$

roots are $\alpha$ and $\beta$

then,

sum fo roots$=-\frac{b}{a}$

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

Product of roots$=\frac{c}{a}$

$\alpha -\beta =\frac{c}{a}$

Now,

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

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

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

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

$={\alpha }^2{\beta }^2=(\alpha \beta {)}^2$

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

Hence, this is the answer.