Question

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

• A. $\frac{b^2+4ac}{a^2}$
• B. $\frac{b^2-4ac}{a^2}$
• C. $\frac{b^2-4ac}{a}$
• D. $\frac{b^2+4ac}{a}$

Answer 1

The correct option is A $\frac{b^2+4ac}{a^2}$

Let the roots of the equation $a{x}^2-bx-c=0$ 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, $(\alpha -\beta {)}^2=(\alpha +\beta {)}^2-4\alpha \beta$
$=\frac{b^2}{a^2}-4\left(\frac{-c}{a}\right)=\frac{b^2+4ac}{a^2}$