Question

If $\alpha ,\beta$ are the roots of the equation $x^2-x-4=0$, find the value of $\frac{1}{\alpha }+\frac{1}{\beta }-\alpha \beta$.

• A. $\frac{4}{15}$
• B. $\frac{15}{4}$
• C. $\frac{4}{5}$
• D. $\frac{5}{4}$

Answer 1

The correct option is B $\frac{15}{4}$

Given equation is: $x^2-x-4=0$
Sum of the roots $=1$
Product of the roots $=-4$
$\alpha +\beta =1$
$\alpha \beta =-4$
$\frac{1}{\alpha }+\frac{1}{\beta }-\alpha \beta =\frac{\beta +\alpha -{\left(\alpha \beta \right)}^2}{\alpha \beta }$
$=\frac{\alpha +\beta -{\left(\alpha \beta \right)}^2}{\alpha \beta }$
$=\frac{1-{(-4)}^2}{-4}=\frac{1-16}{4}$
$=\frac{-15}{-4}=\frac{15}{4}$