If $\alpha ,\beta$ are the roots of $a{x}^2+2bx+c=0$  and  are the roots of$A{x}^2+2Bx+C=0$ , such that $\alpha ,\beta ,\gamma ,\delta$  are in A.P, then $\frac{b^2-ac}{B^2-AC}$  is

$\alpha ,\beta$ roots of  $a{x}^2+2bx+c=0$

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

$\gamma ,\delta$ are roots of  $A{x}^2+2Bx+C=0$

$\gamma +\delta =\frac{-2B}{A},\gamma \delta =\frac{C}{A}$

$\alpha ,\beta ,\gamma ,\delta$ are in A.P

$\beta -\alpha =\delta -\gamma$

$\alpha -\beta =\gamma -\delta$

$\Rightarrow \sqrt{{(\alpha +\beta )}^2-4\alpha \beta }=\sqrt{{(\gamma +\delta )}^2-4\gamma \delta }$

$\frac{b^2-ac}{a^2}=\frac{B^2-AC}{A^2}$

$\Rightarrow \frac{b^2-ac}{B^2-AC}=\frac{a^2}{A^2}$