My guess is that f(x) is the number of toothpicks, because if f(x)=3, we have 1.5x^2+1.5x=3, x^2+x-2=0=(x-1)(x+2)=0 so x=1, implying that the side length is 1. With 3 toothpicks the side length would indeed be 1 (toothpick).
So if f(x)=300, 1.5x^2+1.5x-300=0=x^2+x-200. From this x=13.65 toothpicks (approx). But if only whole toothpicks are allowed, the side would be 13 toothpicks. 273 toothpicks would make the largest triangle with 13 a side. (1.5x^2+1.5x-273=x^2+x-182=(x-13)(x+14).)
The largest triangle is equilateral and has the largest area.