Question

This is a physics question that requires a lot of math. Hoping on of you genius could help me.

Answer 1

I don't know why you need the quadratic formula.  

If it takes 0.5 seconds before stepping on the breaks while going 16 m/s, then you travel 8 meters before anything happens. That means you are 44-8 = 36 meters away from the deer now. 
 v^2 = v_initial^2 + 2(acceleration)(distance)
 0 = (16 m/s)^2 + 2(-10m/s^2)(distance)
 256 m^2/s^2 / (2(10 m/s^2) = distance = 12.8 meters (13 meters after significant figures are taken into account). 
 The maximum speed would be: 
 0 = v_initial^2 + 2(-10m/s^2)(36m)
 v_initial^2 = 720m^2/s^2
 v_initial = 27 mph