Content - Solution by Graphing

Can you visualize what the path of a projectile (ball) looks like?

Objective: to visualize when a quadratic equation will have two solutions, only one solution, or none at all.

Solution Using Graphing

Graphing can be used to solve equations of the form ax² + bx + c = 0. The solutions (also called the roots) of the equation are the values of x that make ax² + bx + c = 0 true. In other words, the solutions are the values of x where the graph y = ax² + bx + c = 0 crosses the x-axis (y = 0). These x values are the x-intercepts of the graph. The following specific example shows how this method works:

x² - x - 2 = 0

The graph of y = x² - x - 2 can be drawn by plotting a number of points (x, y) as x takes a series of different values. Since the graph is continuous, we can join these points with a smooth curve.

x	y
:	:
-1	0
0	-2
0.5	-2.25
1	-2
2	0
:	:

As the graph shows, it crosses the x-axis two times. This means that there are two x values that make ax² + bx + c = 0. In other words, ax² + bx + c = 0 has two solutions. Careful examination will reveal where the graph crosses the x-axis. One is where x = -1 and the other is x = 2. Generally, we write the solutions as follows:

x₁ = -1
x₂ = 2

There are two more cases in which the equation can either cross the x-axis once or not at all, as illustrated by the following two specific equations:

Conclusion: a quadratic equation can have two solutions, only one solution, or no solutions at all, when the graph crosses the x-axis twice, only once, or not at all.

[Experiment] [Exercise]