**Two
Different Ways to Examine Roots Of**

**by**

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**University of Georgia**

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

and to overlay several graphs of

for different values of **b.**

From these graphs discussion of the patterns for the roots of

can be followed.

For example, if we overlay the graphs of

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

We can now discuss the
movement of the roots of a
parabola as **b** is changed. For

**b
< -2** the parabolas will each
intersect the **x-axis
**in two points with both having
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 equation has one real root at the point of tangency. That root is positive. For **–2 < b < 2, **the parabolas do not intersect the **x-axis**, therefore, the equations have no real
roots. For **b = 2, **the parabola is tangent to the **x-axis** , thus, having one real, negative root. Finally, for **b > 2**, the parabolas will each intersect the **x-axis** twice to show each as having two negative real
roots.

Consider the equation

,

keeping in mind
that we want to determine the roots of this equation for a given **b**.

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 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 for each equation when **b > 2**, one negative real root when **b = 2**, no real roots for the equations
when **–2 < b < 2**, one
positive real root when **b = -2**. Notice that this is the same
conclusion we obtained by graphing

for **b = -3, -2, -1, 0, 1, 2, **and **3** and analyzing the individual graph for each
value of **b**.

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Plotting

gives a graph in the **xb **plane. This
single graph, is the locus of the roots of

as **b**** **changes. Therefore, this is a graph of the movement
of these roots as **b** changes.

This is one example of
how the technology available today gives students the opportunity to easily explore
different approaches to solving the same problem. The insight gained by exploring problems in this manner and
the insight gained by taking the time to understand why these different
approaches work yields a better understanding of the content being explored.

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