Give an example of a real-world system that would be described by a quadratic equation. Explain the significance of the two real roots, a repeated root, and undefined roots. Is there any significance to a positive or a negative answer in your example?

For a ball that has been thrown, it's height above the ground can be described by a quadratic equation of the time since it was thrown and gives a parabola.

The two real roots of the equation are the times when it is at a particular height above the ground, and there are two because the ball goes up and then goes down. A repeated root would give the time at which the ball is at the top of its arc, as this happens at only one time. Undefined roots are given when trying to find the time at which it reaches a height greater than the top of its arc (i.e. a height is will never reach).

In this example, positive roots describe heights on the arc after the ball was thrown (the actual, real-life scenario). Negative roots are points in time before the ball was thrown on the arc if it was extrapolated backwards behind the thrower.