Question: Anyone ever come across a good situation where

a) a situation is modeled by a quadratic equation $y=ax^2+bx+c$ and/or

b) you've even needed to find where $y=0$ (roots, $x$-intercept, etc)

Answer: 
The quintessential example surely must be that the position of a thrown object under the action of gravity depends quadratically on time since the motion began.

I have however no idea where one could find an example where the dependence would be cubic.
I was writing a graphics program, and had to find where a ray intersected a sphere. The ray represented a line drawn from a pixel on the screen. That uses a quadratic equation to solve.

Also the projection of a sphere onto the screen. If you know the quadratic formula for a line-sphere intersection, then the edge of the sphere on the screen is where the discriminate goes to zero.

But for students simple things like line-circle intersections is going to be a lot easier.

HMM.As abiessu pointed out, classical mechanics is where knowing quadratics is absolutely essential.