• Equations of the form
 (1)

 (2)

gives the solutions.

• The in Eq. 2 is an important detail. In general, there are two solutions to a quadratic equation. The two solutions are also called the roots of the equation.

• Example

A ball is tossed directly upward from a height of 2.0 m above the ground with an initial velocity of 5.0 m/s. It is subject only to the force of gravity while in flight, so it has an acceleration of -9.8 m/s. When does the ball reach a height of 3.0 m? The position of the ball as a function of time is given by the equation

Solution

The equation with the given information is

which in the form of Eq. 1 becomes
 (3)

We can identify , , and and apply Eq. 2,

To understand the two solutions, a graph of the left side of Eq. 3 is helpful

• The extreme value of the quadratic occurs at . In the example, the extreme value is at 0.51 s.

• The roots of the quadratic lie to either side of . In the example, the two roots of the equation lie 0.24 s to either side of 0.51 s.

• A graph like this is often helpful in choosing appropriate solutions. In the example, both roots of the equation are appropriate. The ball reaches a height of 3.0 m twice, once going up, and once coming down.

• Special Cases

• Single solution

If , then , and there is only one solution, . In the above example, this corresponds to 0.51 s, the time at which the ball reaches its maximum height.

• Complex solutions

If , then is imaginary (involves ). The above example does not have a physically reasonable solution corresponding to this situation. (This situation corresponds to heights never reached by the ball.)

 Copyright © 2002-2004, Lewis A. Riley Updated Mon Jan 19 13:29:10 2004