It has now become a rather standard exercise,
with availble 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.
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 parabola 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
intersets the x-axis twice to show two 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
= 5, 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. We can see that when b=5 our roots are approximately -0.2 and -4.8.
Graphs in 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 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. Same as in the xb plane!
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 zero when c = 0 and one negative and one positive when c<0.
Graphs in the xa plane.
Again, we look at the equation
and graph the relation in the xa plane. We get the following graph.
In this graph, the graph of a = 1 is shown. We get the same values in the xa plane as we did in the xc plane. That is, this graph also shows that the equation
will have two negative roots -- approximately -0.2 and -4.8.