Question

If the roots of $a{x}^2-bx-c=0$ change by the same quantity then the expression in $a,b,$ that does not change is

• A. $\frac{b^2-4ac}{a^2}$
• B. $\frac{b-4c}{a}$
• C. $\frac{b^2+4ac}{a^2}$
• D. None of these

Answer 1

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

Given equation are $a{x}^2-bx-c=0$

$\alpha ,\beta$ are the roots

$\alpha +\beta =\frac{b}{a}$ and $\alpha \beta =c/a$

when $\alpha ,\beta$ are roots

$\frac{b^2+4ac}{a^2}$

$\Rightarrow {\left(\frac{b}{a}\right)}^2+4\left(c/a\right)$

$\frac{b^2+4ac}{a^2}=(\alpha +\beta {)}^2-4\alpha \beta$ ......$\left(1\right)$

Roots: $\alpha -n,\beta -n$

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

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

$(\alpha -n)(\beta -n)=-\frac{c}{a}$

$\frac{b^2+4ac}{a^2}$

$\Rightarrow {\left(\frac{b}{a}\right)}^2+4\left(c/a\right)$

$\Rightarrow [\alpha +\beta -2n{]}^2+4[(\alpha -n)(\beta -n)]$

$\Rightarrow {\alpha }^2+{\beta }^2+4n^2+2\alpha \beta -4\alpha n-4\beta n-4\alpha \beta +4\alpha n+4\beta n-4n^2$

$\Rightarrow {\alpha }^2+{\beta }^2+2\alpha \beta -4\alpha \beta$

$\frac{b^2+4ac}{a^2}$

$\Rightarrow (\alpha +\beta {)}^2-4\alpha \beta$ .......$\left(2\right)$

From $\left(1\right)$ & $\left(2\right)$

$\frac{b^2+4ac}{a^2}$ is same in both cases.

$\frac{b^2+4ac}{a^2}$ remains unchanged even when roots are changed by same quantity.

The correct answer is option C.