Question

For the quadratic equation, $a{x}^2+bx+c=0$ which has non-zero roots $\alpha \&\beta$, the value of $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}$ is

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

Answer 1

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

Since $\alpha \&\beta$ are the non-zero roots of $a{x}^2+bx+c=0.$
$\Rightarrow \alpha +\beta =-\frac{b}{a}\&\alpha \cdot \beta =\frac{c}{a}$

To find: $\frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}=\frac{{\alpha }^2+{\beta }^2}{{\alpha }^2{\beta }^2}=\frac{(\alpha +\beta {)}^2-2\alpha .\beta }{(\alpha \beta {)}^2}$

$=\frac{{(-\frac{b}{a})}^2-2\left(\frac{c}{a}\right)}{{\left(\frac{c}{a}\right)}^2}$

$=\frac{\frac{b^2}{a^2}-\frac{2c}{a}}{\frac{c^2}{a^2}}=\frac{b^2-2ac}{c^2}$

$\Rightarrow \frac{1}{{\alpha }^2}+\frac{1}{{\beta }^2}=\frac{b^2-2ac}{c^2}$