Question

If $\alpha ,\beta$ are the roots of the equation $x^2+x-15=0$ then find $\frac{\alpha +\beta }{{\alpha }^{-1}+{\beta }^{-1}}$

Answer 1

The given equation is $x^2+x-15=0$

Here $a=1,b=1,c=-15$

Sum of the roots$=\alpha +\beta =\frac{-b}{a}=\frac{-1}{1}=-1$

Product of the roots$=\alpha \beta =\frac{c}{a}=\frac{-15}{1}=-15$

$\frac{\alpha +\beta }{{\alpha }^{-1}+{\beta }^{-1}}$

$=\frac{\alpha +\beta }{\frac{1}{\alpha }+\frac{1}{\beta }}$

$=\frac{\alpha +\beta }{\frac{\alpha +\beta }{\alpha \beta }}$

$=\frac{\alpha \beta (\alpha +\beta )}{\alpha +\beta }$

$=\alpha \beta$

$=-15$

Hence, the required value is $-15$