If equation : $ax^{2}+bx+c=0$ has roots $\left( \alpha \right) $ and $\left( \beta \right) $. & equation $Ax^{2}+Bx+C=0$ has roots $\left( \alpha +k \right) $ and $\left( \beta +k \right) $. then $\dfrac { { b }^{ 2 }-ac }{ { a }^{ 2 } } =\dfrac { { B }^{ 2 }-AC }{ { A }^{ 2 } } $

The difference of the roots of the first equation is sqrt(b^2 - 4ac)/a, which equals sqrt(b^2 - ac)/a. The second equation is formed by translating the roots by k, which preserves the difference between them, so its difference is sqrt(B^2 - 4AC)/A. Equating the identical differences gives sqrt(b^2 - ac)/a = sqrt(B^2 - AC)/A, and squaring both sides yields the statement (b^2 - ac)/a^2 = (B^2 - AC)/A^2.

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