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 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 <
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 orignal 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 <