Some Different Ways to Examine

by

James W. Wilson and Nancy B. Williams
University of Georgia

It has now become a rather standard exercise, with available technology, to construct graphs to consider the equation

and to overlay several graphs of

for different values of a, b, or c as the other two are held constant. From these graphs discussion of the patterns for the roots of

can be followed. For example, if we set

for b = -3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.

There are several points of interest with the graph. First, we note that the parabola always passes through the same point on the y-axis - the point (0,1) with this equation.

Second, we note the number of times the parabola intersects the x- axis in relation to the value of b.

• For b < -2, the parabola will intersect the x- axis in two points. Since the original equation will have two real roots, both positive, these points will have positive x values.
  
  
• For b = -2, the parabola is tangent to the x - axis. So, the original equation has one real and positive root at the point of tangency.
  
  
• For -2 < b < 2, the parabola does not intersect the x- axis -- the original equation has no real roots.
  
  
• For b = 2, the parabola is tangent to the x - axis -- one real negative root.
  
  
• For b > 2, the parabola intersects the x- axis twice to show tow negative real roots for each b.

Consider again the equation

.

Now graph this relation in the xb plane. We get the following graph.

If we take any particular value of b, say b = 3, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the xb plane, the intersection points correspond to the roots of the original equation for that value of b. We have the following graph.

For each value of b selected, we get a horizontal line.

It is clear on a single graph that:
we get two negative real roots of the original equation when b > 2,
one negative real root when b = 2,
no real roots for -2 < b < 2,
one positive real root when b = -2,
and two positive real roots when b < -2.


Consider the case when c = (- 1 ) rather than (+ 1).

In the following example, the equation

is considered. If the equation is graphed in the xc plane, it is easy to see that the curve will be a parabola. For each value of c considered, its graph will be a line crossing the parabola in 0, 1, or 2 points -- the intersections being at the roots of the original equation at that value of c. In the graph, the graph of c = 1 is shown. The equation

will have two negative roots -- approximately -0.2 and -4.8.

There is one value of c where the equation will have only 1 real root -- at c = 6.25. For c > 6.25, the equation will have no real roots and for c < 6.25, the equation will have two roots, both negative for 0 < c < 6.25, one negative and one 0 when c = 0, and one negative and one positive when c < 0.