Re-examining Second Degree Equations

James W. Wilson and Pat Hartney

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.

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 intersects 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.

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

It is clear that for any value of b, we get two real roots, one negative and one positive.

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.

In the following example the equation

is considered. If we graph this in the xa plane, we get the following:

For each value of a considered, its graph will be a line crossing
the curve in 0, 1, or 2 points -- the intersections being at the
roots of the original equation at that value of a. In the following
graph, the graph of a = 1 is overlayed.

We see that the equation has two negative real roots when a = 1. When a > 2, the equation has no real roots. When a = 2, the equation has one negative real root. When 0 < a < 2, the equation has two negative real roots. When a = 0, the equation has one negative real root. When a < 0, the equation has one negative real root and one positive real root.

By graphing the equation

in the xb, xc, or xa planes, we can quickly analyze roots for
various values of a, b, and c without layering numerous graphs
on top of each other.