Industrial Economics I am currently struggling trying to find the short-run equilibrium price, output per firm, and profit per firm if $190$ firms supply the market. I am given $p=102-1/2Q$ and $C(q)=5q-6q^2+3q^3$. 

First I started off with $P=MC$. 
So $P=9q^2-12q+5$, thus $9q^2-12q+(5-P)=0$. 

Then, I Used the quadratic equation and came out with $q=2/3+1/3\sqrt{p-1}$.

Then I did $Q=nq=190\left(2/3+1/3\sqrt{p-1}\right)$. 

Equating demand and supply I obtained $P\approx 2.32$ and then plugged back into $q\approx2/3+1/3\sqrt{2.32-1}$ to get $q\approx1.05$ 

Finally, I plugged $q$ and $P$ into $\pi=q\cdot P-C(q)$ and came up with $\pi\approx 0.33$. This just seems crazy to me! I have tried for the past few days over and over again to understand, but I am completely lost. 

Any help would be greatly appreciated!
I went over your calculations and they seem to be right. In principle, this is a valid answer, but I understand that the numbers seem suspicious. Are you sure this is the right problem? 

Update: everything checks. I think your answer is correct, although it looks "weird" for a textbook problem.