so... notice the picture below, that's P(x)
so [tex]\bf \qquad \qquad \textit{vertex of a parabola}\\ \quad \\
y = {{ a}}x^2{{ +b}}x{{ +c}}\qquad
\left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right)[/tex]
so.. when the quantity is [tex]\bf -\cfrac{{{ b}}}{2{{ a}}}[/tex] , is when the profit [tex]\bf {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}[/tex] is the highest
after that, as you can see from the graph, it goes back down
