Question

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

Answer 1

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

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

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

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

$=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }$

$=\frac{{(\alpha +\beta )}^2-2\alpha \beta }{\alpha \beta }$

$=\frac{{(-1)}^2-2\times -15}{-15}$

$=\frac{1+30}{-15}$

$=\frac{31}{-15}$

$=\frac{-31}{15}$