A quadratic is graphed as a parabola, but often you don't see rollercoasters shaped like that. Interestingly enough, if a rollercoaster would follow a certain parabolic arc, it would make people feel as if they were weightless - but there are problems with that! Most people think that the shape of the track is how quadratics would be used, however, quadratics (and other equations) model (or help predict) how certain variables would behave on the roller coaster.
Easiest to understand: Brakes: How much track do we need to allow for a safe slow down to a stop at the end of a ride? This can be modeled with a simple quadratic equation: Where ∆x is the track length, v is the initial speed, and a is the average acceleration (a<0 opposing the motion of the object) of the brakes.
This is a simple quadratic where v is the variable input and ∆x or a are the ouputs.
The other question that this formula could answer is: We have a certain length of track, how much do we have to accelerate the cart to a stop?
This could be rearranged to be a different question: how fast would we be going from zero if the track was a certain length and we accelerated at a certain pace? (however, that is a squareroot function - the inverse quadratic).
Instead of slowing down, you can also speed up. From rest, the above equation loses the negative sign. Acceleration is positive (speeding up in the direction of travel) and v is the speed acheived at the end of a certain track length (∆x).
Lastly, the centripetal acceleration around a loop or turn in a track can be found using ` a_c = v^2/r` where r is the radius of the turn. This value must be balanced such that the cart stays on the track in either case. Too little speed, and you end up stalled on a loop; too much speed, and you end up hurting your passengers.
Hope that helps. Take a look at roller-coaster physics to get more ideas to completely answer the question.