It has now become a rather standard exercise, with available technology, to construct graphs to consider the equation
and to overlay several graphs 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 this equation can be followed. For example, if we set b=-3, -2, -1, 0, 1, 2, 3, and overlay the graphs, the following picture is obtained.
We can discuss the "movement" of a parabola as b is changed. The parabola always passes through the same point on the y-axis (the point (0,1) with this equation). For b < -2 the parbola will intersect the x-axis in two points with positive x values (i.e. the original equation will have two real roots, both positive). For b = -2, the parabola is tangent to the x-axis and 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. Similarly for b=2 the parabola is tangent to the x-axis (one real negative root) and for b > 2, the parabola intersects the x-axis twice to show two negative real roots for each b.
Now consider the locus of the vertices of the set of parabolas graphed from this equation. Show that the locus is the parabola
Graphs in the xb plane
Consider again the equation
Now if we 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 we select, 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 = -2 rather than 1.
Graphs on the xc plane.
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 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 and 5.
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 at 0 when c = 0 and one negative and one positive when c < 0.