Zoe,

I can give you a few places where quadratic are important. The graph of a quadratic is a parabola so the first two examples are about parabolas.

Satellite dishes have a parabolic cross section. There is a not in the Resource Room on Math Central titled Why are satellite dishes parabolic? that explains why the parabolic shape is important.

The shape of the cables supporting a suspension bridge are approximately parabolic. In the Math Beyond School section of Math Central there is a note on suspension bridges. The Wikipedia page on suspension bridges is another resource on this topic. (Do a search on this page for the word parabola.)

The third situation I want to mention is in describing the path of an object under a constant gravitational force, like a ball thrown on Earth or on the moon. Newton's laws of motion lead to a quadratic expression for the height of the ball in terms of time.

I hope this helps,
Harley