James W. Wilson and

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.

If we set

for a= -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 a is changed. The parabola always passes through the same point on the y-axis ( the point (0,1) with this equation). For a < 0 the parabola will go down with two real roots. For a = 0, the equation become the linear equation. For a> 0, the parabola will go up.

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.

if we set

for c = -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 c is changed. The parabola always has the same shape and the graph go up as c increases and go down as c decreases.

Consider again the equation

If we take any particular value of a, say a = 1,-1 and 0.25, and overlay
this equation on the graph we add a line parallel to the x-axis. If it
intersects the curve in the xa plane the intersection points correspond
to the roots of the original equation for that value of a. We have the
following graph.

For each value of a we select, we get a horizontal line. It is clear
on a single graph that we get one negative and one positive real
roots of the original equation when a < 0, one negative real root -1
when a = 0, two negative real roots for 0 < a < 0.25, One negative
real root when a=0.25 , and no real roots when a > 0.25.

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 < -2.

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

For each value of b, we always get one positive and one negative real roots of the original 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 < 0.

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