How to find out number of real solutions to this without graphing? $$
3^x = 4x^2
$$
My prof's notes just says to look at the graph and you'll see that they intersect in 3 points. But is there another way of doing this if you can't graph it?

The Intermediate Value Theorem migh be usefull here.
Let $f(x) = 3^x - 4x^2$. 
Then $f(-1) <0$, $f(0) >0$, $f(1)<0$ and $f(4)>0$
As $f$ is continuous, there is at least one root in ]-1,0[, one root in ]0,1[ and one root in ]1,4[
