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A quadratic equation can be used to describe the arc that a ball travels in the air, with x being the distance it moves and y as its height. You can also relate the ball's height, y, to the amount of time it's been in the air, x. And it doesn't have to be a ball—it could be a spherical cow, or a chunk of frictionless ice, or a pendulum with a massless spring that experiences no air resistance.
Quadratic equations also have a practical application in statistics. As we mentioned when talking about linear regression, sometimes the dots in a scatter plot end up looking like a line, while other times they look like a curve. Just as we can use linear regression to fit a line to the dots, there are various forms of non-linear regression that can fit different curves. Quadratic regression is one of the more common types of non-linear regression.
Polynomials make some of the simplest functions around, and quadratic polynomials are the simplest kind of polynomial after a straight line. Wherever you find a reason to use functions—in Algebra, Calculus, and beyond—you'll find quadratics not far behind. When you're being introduced to some new concept, it's comforting to have an old friend along to show you the ropes, instead of some new, scary function that would rather beat you up and take your lunch money.