Question

If the roots of $ax^2 -bx-c = 0$ are increased by same quantity, then which of the following expressions does not change?

Answer 1

Let the roots of the equation $ax^2 -bx-c = 0$ be $\alpha, \beta.$
Assuming that the roots are increased by $k$, so new roots are $\alpha +k, \beta +k .$
By observation, we can see that difference of roots $\alpha - \beta$ remains same.
So, $(\alpha - \beta)^2 = (\alpha+\beta )^2 - 4 \alpha \beta$
$~~~~~~~~~~~~~~~~~~~~= \dfrac{b^2}{a^2} - 4\left(\dfrac{-c}{a}\right)\\
~~~~~~~~~~~~~~~~~~~~= \dfrac{b^2+4ac}{a^2}$