Question

If the equation $x^2+kx+1=0$ has the roots $\alpha$ and $\beta$, then what is the value of $(\alpha +\beta )\times ({\alpha }^{-1}+{\beta }^{-1})$?

• A. $k^2$
• B. $\frac{1}{k^2}$
• C. $2k^2$
• D. $\frac{1}{\left(2k^2\right)}$

Answer 1

The correct option is A $k^2$

$\therefore$ The roots of the equations $x^2+kx+1=0$ are $\alpha and\beta$.
$\therefore \alpha +\beta =-kand\alpha \beta =1$
$Now,(\alpha +\beta )\times ({\alpha }^{-1}+{\beta }^{-1})$ = $\alpha +\beta )(\frac{1}{\alpha }+\frac{1}{\beta })$.
$=(\alpha +\beta )\left(\frac{\alpha +\beta }{\alpha \beta }\right)=\frac{(\alpha +\beta {)}^2}{\alpha \beta }=\frac{(-k^2)}{1}=k^2$