How did quadratics help find an object trajectory? And why only two outcomes?

Question: How did quadratics help find an object trajectory? And why only two outcomes?
- Keywords:
  quadratics
Asked by secretsaudience11 on 13 Jan 2020.
- William Pettersson answered on 13 Jan 2020:
  
  The first step would be experimentation – people tested how gravity worked by dropping things and timing how long they took to fall various distances. This gave equations for motion due to gravity, equations that are quadratic. The fact that there are (often) two outcomes is just down to the mathematics. If you try equations that have more outcomes, then the numbers won’t fit the real world data.

Diana commented on 13 Jan 2020:

the equations that predict motion when forces are quadratic, including when force is gravity. so solution requires solving quadratic equations

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Question: How did quadratics help find an object trajectory? And why only two outcomes?