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: 
Outline fonts such as those in TrueType typically describe glyph shapes with quadratic Bézier curves. To render them on screen or on paper one typically needs to find where horizontal rays cross the glyph outline and this requires solving a quadratic equation.

In other words, what you reading right now is the outcome of many, many quadratic equations...
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.