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.
Headlights in motor vehicles: The parabolic shape is the unique shape such that all rays emanating from the focal point reflect off the parabola and travel parallel to each other outwards. Note that many headlights do not actually use a smooth parabola, since lights heading straight outwards would not spread enough to provide good visibility. However, the theory of the parabolic mirror is what influences how the mirrors are shaped.

In this vein, the mirror telescope also focuses all received light using a parabola.

In any acceleration scenario (including approximating the curve due to gravity), the curve of position vs. time is quadratic. Among the many possible applications, one that is practical is to consider a car moving at constant speed, and a second car currently moving faster and approaching from behind but also applying a constant deceleration (i.e., braking). The point(s) at which the positions of the two cars would be the same are the solutions of a specific quadratic.