**The ABC's of Quadratic Equations**

**An Exploration of the xa, xb, and xc Planes**

by

**Sarah Major**

This write-up is based on Exploration 3 in which we are prompt to look the various relationships of *x* with the values of* b *and *c* from the following equation:

In particular, we are asked to view graphs in the *xb* and *xc* planes and note any relationships. In this write-up, I will first analyze graphs in the *xb* plane following by the *xc* plane. Afterwards, I will add the coefficient *a* to produce the following equation:

Using this equation, I will then analyze graphs in the *xa *plane.

First, I will analyze graphs in the *xb* plane. For the sake of focusing on values of *b*, I will pick an arbitrary value for *c* so that *b* is the only unknown. After analyzing the graphs, I will then pick another value for *c* to make sure the results are repeatable for multiple values of *c*. For simplicity in the first analysis, I will choose *c* = 1 to produce the following equation:

Before delving into graphs in terms of *b* and* x*, let's graph this initial equation, set equal to *y* instead of 0, and note what the roots of this equation are for particular values of* b* (in essence, where the equation is equal to 0). I have chose to use *b = *-5, -3, 3, and 5. My choice for these values will become clear later on in the analysis. Here are the graphs:

We can find where the graphs equal 0 by finding the roots of each corresponding equation (note that values are approximate to two decimal places):

For *b* = -5, the roots are *x = *0.21 and 4.79.

For *b* = -3, the roots are *x = *0.38 and 2.62.

For *b* = 3, the roots are *x = *-2.62 and -0.38.

For *b* = 5, the roots are *x = *-4.79 and -0.21.

These values will come into play later on in the analysis.

I then need to solve the previous equation for* b, *since I will be graphing in the *xb* plane:

I then graph this equation in the *xb* plane:

What does this graph tell us? In essence, this graph tells us what the values for *x* and *b* we can have for these particular equations when *c* is 1. Because the domain goes from negative infinity to 0 and 0 to infinity (not including 0), *x* can be any value but 0 because this would cause the equation to be inconsistent (a statement that would say 1 = 0, which is not true). Since the range is negative infinity to -2 and 2 to infinity, *b* can be any values except the values appearing between -2 and 2. This makes sense because having these values would result in the graph of the equation not crossing the *x*-axis, and thus, the observations occurring later on in this analysis. This will become clear in the next section.

Let's graph the values of *b* that we used previously along with the equation we just graphed:

Notice that the graphs of each line intersect the original graph in two places. These are the values at which they intersect:

For *b* = -5, the graphs intersect at* x = *0.21 and 4.79.

For *b* = -3, the graphs intersect at *x = *0.38 and 2.62.

For *b* = 3, the graphs intersect at *x = *-2.62 and -0.38.

For* b* = 5, the graphs intersect at *x = *-4.79 and -0.21.

These are the exact values of the roots from the original equation.

Will this work if we change the value of *c* from 1 to another value? Let's do a test to find out.

Say *c* = 5 and *b* = 5. Let's graph the equation then:

The roots of this equation are *x* = -3.62 and -1.38.

Let's graph in the *xb* plane along with our value of* b*:

Where do the graphs intersect? *x* = -3.62 and -1.38.

Therefore, this does work for different values of *c*.

Next, we will analyze graphs in the *xc* plane. We once again start with the following original equation:

Once again, we will analyze the graphs of the original equations before we go into an analysis of *c*. Similar to the previous analysis, we will have *b* = 1, but for the values of *c, *we will use -5, -3, and -1. Once again, this decision will be clear momentarily:

What are the roots for these equations?

For *c* = -5, the roots are *x* = -2.79 and 1.79.

For *c* = -3, the roots are *x* = -2.30 and 1.30.

For *c* = -1, the roots are *x* = -1.62 and 0.62.

We then solve our original equation for *c*:

And then graph this on the *xc* plane along with our values of *c*:

This graph shows us that for these particular types of quadratic equations where *b* is 1, *x* can be any value from negative infinity to infinity. However, the values of *c* are restricted to the range of negative infinity to 0.25 for the equations to have roots.

What are the values at the intersections?

For *c* = -5, the graphs intersect at *x* = -2.79 and 1.79.

For *c* = -3, the graphs intersect at *x* = -2.30 and 1.30.

For *c* = -1, the graphs intersect at *x* = -1.62 and 0.62.

So, like for the *xb* plane, the intersections in the *xc* plane are an indicator of the roots for the original equations.

Now, we will add an *a* to the equation and then make observations about the *xa* plane. We begin with the following equation:

Similar to the previous two analyses, I will set the two values that are not in question, *b* and *c*, equal to 1. I will also use the values of -5, -3, and -1 for *a*. My choice for these values will be clear later. Let's graph these equations:

What are the roots of these equations?

For *a* = -5, the roots are *x* = -0.36 and 0.56.

For *a* = -3, the roots are *x* = -0.43 and 0.77.

For *a* = -1, the roots are *x* = -0.62 and 1.62.

We then solve our original equation for *a*:

And then graph this equation on our *xa* plane along with our values of *a*:

The graph shows that for these types of equations, when *b* = 1 and *c* = 1, *x* can equal any value but 0, as there is an asymptote where *x* = 0. As for the range of *a*, it can equal any value from negative infinity to 0.25, the range for which the equations would have roots.

What are the values at the intersections?

For *a* = -5, the graphs intersect at *x* = -0.36 and 0.56.

For *a* = -3, the graphs intersect at *x* = -0.43 and 0.77.