Quadratic equations govern many real world situations such as throwing a ball, calculating certain prices, construction, certain motions and electronics. They are most often used to describe motion of some sort.
An equation is quadratic if it is of "order 2;" that is, an equation is quadratic if the highest power in the equation is 2. Therefore, x^2 and any variations of it are quadratic equations. For example, if a ball is thrown straight from 3 meters above the ground at a velocity of 14 meters per second, this allows the construction of an equation. The equation is quadratic because half of the gravitational velocity, 5t^2, is subtracted from the other constants. By setting the equation equal to zero, two solutions can be acquired. The interesting part is that these solutions show when height is equal to zero. In other words, solving this yields what time the ball was on the ground after being thrown. This information can then be used to acquire even more information, such as how long the ball was in the air, when it reached its highest point and where the ball is at any time after being thrown. The same method is employed in other situations where quadratics are involved.