Exploring Quadratics


Tonya DeGeorge


In this investigation, we will be looking at quadratics again, but in a different way.  We will be looking at the following equation:

This graph will show the relationship in the (x, b) plane.

By adding a horizontal line in the (x, b) plane, it the intersection of the two graphs corresponds to the roots of the equation for a particular value of b. For example, suppose b = 2, we get the following graph:

If we were to solve the equation by hand, we would find:

This corresponds to the graph above because the two graphs intersect at the point (-1,0) where x = -1.  So how can we generalize this for different values of b?  There are a number of cases to consider:

Case (i): b > 2

Case (ii): b = 2

Case (iii): -2 < b < 2  

Case (iv): b = -2

Case (v): b < -2

LetÕs look at these different regions.  WeÕve already seen what happens when b = 2.  We found that there exists one double root: x = -1.  For Case (i), where b > 2, lets choose b = 5.  We find the following:


This implies that the equation has 2 negative roots.  LetÕs check this by looking at the equation:



Therefore, we see that there exists 2 negative real roots.

Below are summaries for the next three cases:

For Case (iii), where -2 < b < 2, letÕs choose the value b = 1.  We find:



There are two imaginary roots (hence, the line does not intersect the graph).

For Case (iv), where b = -2, we get:




There is one repeated root at x = 1 (where the graph intersects at one point).


For Case (v), where b < -2, letÕs choose b = -4.  We find:



There exists two positive real roots.

In conclusion, we find:

Values of b

Types of roots

b > 2

Two negative real roots

b = 2

One double root (x = -1)

-2 < b < 2

Imaginary roots

b = -2

One double root (x = 1)

b < -2

Two positive real roots






Now suppose we look at the value of c and change it to -1 and compare what happens to the graph when this change is done.



As we can see, when we change the value of c to -1, the graph changes into a hyperbola.  What happens when we use other values of c?  LetÕs use the following values:

c = -2, -1, 0, 1, 2 (all on the same graph):

From here, we can see that the value of c changes the graph in three different ways.  It could be a hyperbola (two different types) or a line.  The three cases seem to be:

Case 1: when c < 0: hyperbola (asymptotic to x = 0 and y = -x)

Case 2: when c = 0: a line (y = -x)

Case 3: when c > 0: a hyperbola (asymptotic to x = 0 and y = -x)

*Note that the hyperbola is restricted to different areas of the plane depending on the value of c.