I am in the final stages of a math project and I need to interview an expert for the last part. Please try to answer at least a few of these questions.
The main place where I use quadratics is in my teaching. I teach mathematics at a university and, especially in my calculus classes we use quadratic equations. Problems are usually designed that way because quadratic equations are easy to solve. Many practical problems result in more difficult equations to solve than quadratics but using quadratics with students I can show them how to approach the problem and worry about the details later.
I am not sure about the most important function of quadratic equations but I can show you some places where quadratics naturally arise. The graph of a quadratic equation is a parabola and you see many parabolas. Satellite dishes have a parabolic shape. You can read why in a note I called Why are satellite dishes parabolic?. The laws of motion that come from Newton are quadratic. If you throw a ball then the path it takes is approximately a parabola. (Air friction distorts the path slightly from a parabola.) Also some arches in bridges are built in the shape of parabolas. There is a note called Bridges and parabolas that has some information on this topic.
I decided to become a mathematician because I love the subject. I have the best job in the world. I can't imagine a better job than one that pays you to come to work and do what you love to do.Cheers,