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.
We can find the exact roots of any of these equations, real or imaginary, by plugging the values of a, b, and c into the quadratic equation.
We can see an interesting pattern developing by looking at the vertices of the parabolas of the form
To find the exact vertices of these parabolas, we need to take the derivatives of the equations, set them equal to zero, and solve for x. This value will give us the point where a horizontal line will be tangent to curve, which can only happen at the vertex.
After finding these points for a few of the curves (let b range from -4 to 4, for example), we can see that we are getting another curve, which seems to a parabola. It turns out that it is a parabola with the equation:
Graphs in the xb plane.
Consider again the equation
Now graph this relation in the xb plane. We get the following graph.
Now, we can take any equation for a line horizontal to the x-axis, such as b = 5 and look for intersection points with the original graph. 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, and there will be no real roots for -2 < b < 2, The reason there will be no real roots between -2 and 2 is because our horizontal line will not intersect our original graph at any points. Also, we will get one positive real root when b = -2 and two positive real roots when b < -2.
Consider the case when c = - 1 rather than + 1. The graph for the equation where c = -1 is shown in red.
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 line horizontal to the x-axis, 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.
This graph shows that the equation
will have two negative roots -- approximately -0.2 and -4.8 because those are the points of intersection for our two equations.
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 because the horizontal line will not intersect the parabola at all. For c < 6.25 the equation will have two roots. Both will be negative for 0 < c < 6.25, one negative and one 0 when c = 0 and one negative and one positive when c < 0.