Some Different Ways to Examine

ax^2 + bx + c =0

by

James W. Wilson and Rives Poe
University of Georgia

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

ax^2 + bx + c =0

and to overlay several graphs of

y = ax^2 + bx + c

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

ax^2 + bx + c =0

can be followed. There are three different outcomes for the roots of this problem: you will either have no real roots, one real root, or two real roots.

To understand how a, b, and c effect the roots of the equation above, we will use the quadratic formula.

and .

To obtain information about the roots, we need to look at the discriminant. The discriminant is the part of the formula under the radical sign, (b^2-4ac). The standard rule for determining roots is as follows: if the discriminant is positive, there will be two real roots, if the discriminant equals zero, there will be one real root and if the discriminant is negative there will be no real roots.

To help students visually understand this rule and see why it works, use a graphing tool, such as Graping Calculator. Graph the equation in the xb plane, where the x-axis represents the x value and the y-axis will represent the b value. Therefore we will actually use the equation:

So, let's begin our exploration to see if we can use the coefficients a and c to predict the roots of a parabola.

If a and c are both positive, let a =1 and c =2, then our equation and graph look like:

If you will remember earlier when we discussed the "root" rules of algebra and the discriminant, it seems that for the graph to have two real roots then, or . On the graph above, it looks like b must be greater thanapproximately 2.8 to have two real roots or less than -2.8(approximately) to have two real roots.

If we take any particular value of b, say b=6, and overlay this equation of 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 original equation for that value of b. The following graph shows this intersection.

b=6 intersects the curve in two places, corresponding to the two real roots at x= -5.64575 and x=-0.354249. There will also be roots if b is less than -4, as stated earlier, if the line crosses the curve in the xb plane, the intersection points correspond to the roots of the original equationfor that value of b. Take a look at b=-5.

The intersection points, correspond to the real roots for b=-5 . (x= 0.438447 and x = 4.56155.)

If we plug our values for a, b and c into the quadratic formula, we get that for the equation to jave two real roots, (=2.82843) and (=-2.82843), which are the values we determined from the graphs above!

Therefore let b =or b=to have one real root as we can see in the graph below:

And for there to be no real roots, and. Which we can see below that there is no intesection with the curve if we let b = 1 and b=-1.

What if both a and c are negative? Let's investigate this option. Let a = -1and c=-1.

(Notice that this graph is a reflection over the x-axis of the previous investigation!) Looking at the graph it seems that b must be greater than 2 or less than -2, for the equation to have two real roots.

b=3, b=-3

The real roots if b=3, are x=0.381966 and x = 2.61803. And if b=-3, the real roots are: x=-2.61803 and -0.381966.

According to the rules of roots, we know that when or when , we will only have one real root.

b=2, b=-2

And according to the third algebraic rule of roots, if and then there will be no real roots.

b=1, b=-1

Onve again, we are able to show the algebraic rules in graph form. So far, so good!

We need to investigate the third scenario that a and c have different signs. Let's allow a to equal 1 and c to equal -1.

The graph looks a bit different here, but it is actually easier to see that there are going to always be two roots in this case. For example, if we let b = 5 and b=-7:

b=5, b=-7

This can be proved by looking at the discriminant. If a and c have different signs, then -4ac will always be positive. Also, since b is squared, b will always be positive, so there will always be two solutions.

This ends the investigation of the xb plane and how a and c effect the roots of an equations such as

ax^2 + bx + c =0.

Using technology is an excellent way to grasp your students attention and SHOW them what is happening and how rules came to be!