If α and β are the zeroes of $4x^2-9x-6$, the value of 1/α + 1/β is:

$\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }$. Here $\alpha +\beta =-\frac{b}{a}=\frac{9}{4}$ and $\alpha \beta =\frac{c}{a}=-\frac{6}{4}=-\frac{3}{2}$, so the value is $\frac{9/4}{-3/2}=\frac{9}{4}\times \left(-\frac{2}{3}\right)=-\frac{3}{2}.\frac{3}{2}$ loses the minus sign carried by the product; $-\frac{2}{3}$ divides the wrong way round (product ÷ sum); $-\frac{3}{4}$ uses $\frac{9}{4}÷(-3)$, dropping the a underneath $c$.