So this might be a little confusing

Hey, so my teacher had once talked about solving something for projectile motions while using the quadratic equation, I was wondering when that would be used?. In my guide, it doesn't teach to find time from the equation of displacement without final velocity using the quadratic formula. Which, I have a feeling that is where the quadratic formula is used.

Here is a step by step solution for one of the questions:

Practice Question 3 An object is launched at 15 m/s [370 above the horizontal]. What is its range? Start by breaking the v, down into its horizontal and vertical components. V1=15-(cos 370)=12 % and v1, =15 (sin 370)=9.0m Then find the object's hang time At using the information about the object's vertical motion. Givens: Ad,=0 Vly= 9.0mls Unknown: At=? Equation: Ad, = Vly At-58 At - At=- 2vly for Ad=0 g Substitute and Solve: At=2(9.0mls) 1.837s or 1.8 s to 2 sig digs 9.8 mls2 Finally, find the range (the object's horizontal displacement). Givens: v=12mls At= 1.837s Unknown: Ad =? Ad Equation: V.= - Ad =vAt At Substitute and Solve: Ad, =v,At=(12 ") (1.837 s)=22m