Question

If $\alpha$ and $\beta$ are the zeros of the quadratic polynomial $f\left(x\right)=x^2-x-4$, find the value of $\frac{1}{\alpha }+\frac{1}{\beta }-\alpha \beta$.

Answer 1

$f\left(x\right)=x^2-x-4$

Roots are $\alpha$ and $\beta$

$\alpha +\beta =1$

$\alpha \beta =-4$

$\frac{1}{\alpha }+\frac{1}{\beta }-\alpha \beta =\frac{(\alpha +\beta )-{\left(\alpha \beta \right)}^2}{\alpha \beta }$

$\frac{1-(-4{)}^2}{-4}$

$\frac{1-16}{-4}$

$\frac{15}{4}$