\(\Pi=aQ-bQ^2-\alpha Q+\beta Q^2=\left(a-\alpha\right)Q+\left(\beta-b\right)Q^2\). \(\Pi=0\) iff \(Q\left[\left(a-\alpha\right)+\left(b-\beta\right)Q\right]=0\), that is: \(Q=0,\ \frac{\alpha-a}{b-\beta}\). Thus, \(\Pi\) is maximized at \(Q=\frac{1}{2}\frac{\alpha-a}{b-\beta}\)