Thomas Earl Ricks

Mathematics Education

 

Assignment # 3

 

“Roots of a Quadratic Equation”

 


 

We will investigate in this webpage the roots of a quadratic equation.

 

To begin, we start with the standard quadratic equation, with a, b, and c as coefficients:

 

 

Drawing on our past knowledge, we remember the famed quadratic formula, which allows us to find the roots of the quadratic equation.  The roots of the quadratic equation are the solutions of the equation:

 

 

or in other words, the values of x which yield a y value of zero.  The quadratic formula allows us to solve to find the two x values that make

 

 

true.

 

The quadratic formula is really two equations:

 

 

and

 

 

 

Solving these two equations will give the values of x that “satisfy” or make true the mathematical sentence:

 

 

It is really quite remarkable that just knowing the coefficients a, b, and c, and then plugging them into the quadratic formula can find the solutions!

 

There are actually a variety of alternative ways to find the solutions of

 

 

by factoring (if possible), drawing the graph of

 

 

and observing where it has a height of zero (i.e., crosses the x-axis), trial and error guessing, etc.

 

The beauty of the quadratic formula is that it yields the solutions EVERY time, with exactness.

 

We will now look at the quadratic formula in more depth as it compares to the quadratic equation.

 


 

For simplicity, we will let a = 1 and c = 1.  Thus the quadratic equation we will look at is:

 

 

By setting this equation equal to zero:

 

 

and graphing in the x-b plane, we obtain an unusual picture:

 

 

 

What does this graph mean?  How can we interpret this graph?

 

If we graph the line b = -2 then we get a horizontal line that crosses the blue graph at one place, at x = 1.  This means that for the equation

 

 

when b = -2 then x must equal 1.

And this is indeed true!

 

 

 

Therefore, the blue graph represents all the solutions of the equation

 

 

for different values of b.

 

As another example, if b = 3, we observe that the line b = 3 intersects the blue graph twice, somewhere around negative -2.6 and –0.4.  We graph the equation

 

 

in green and observe that indeed the green equation does cross the x-axis at the precise x values for which the red line intersects the blue graph!  I have drawn vertical purple lines at these two x values.

 

 

Now how do we connect this to the quadratic formula?

 

We observe that each of the two quadratic formulas can be broken down into two pieces:

 

 

In the case of the quadratic equation

 

 

these two forms of the quadratic formula simplify to:

 

 

where b = 3.

 

In the graph below we draw the additional graph

 

 

 

 

You may notice that the x value for where the light blue line and the red line intersect is exactly half-way in between the roots of the green quadratic equation (which means it is halfway in between where the red line intersects the blue graph as well!)

 

What is going on here?  What does this mean?

The x-value where the red and blue line intersect is the piece of the quadratic formula

 

 

or since the red line is b = -3, then the x value is

 

 

which you can see on the x-axis.  Since the place on the x-axis where the green graph intersects is given by the quadratic formula

then we can see that distance from the x-value where the red and blue graph intersect, which is

 

 

is given by

 

Adding these two pieces together gives us:

 

 

which is one of the roots of the green equation.

 

The second root is found by subtracting

 

 

instead of adding it, which is the second quadratic formula:

 

 

So instead of adding the thick red distance in the graphic above, which distance was

 

 

we subtract that distance from

 

 

 to find the second root of the green graph, as shown in the graphic below by thick purple lines:

 

Thus by graphing

 

 

in the x - b plane, along with b = -3 and

 

 

we are able to find the roots of the original quadratic equation

 

 

which is superimposed on the x - b plane to show where its roots would be on the x-axis.  This is done by observing that we find the intersection of the line b = -3 with

 

 

and add and subtract the distance

 

 

to find the solutions to

 

 

with b = -3, or the roots of

 

 

which is the same as looking at the two pieces of the quadratic formula, which would give us the same result

 

 

Or we could look at the intersection of the red and blue lines to determine the roots of

 

 

I hope this has shed a little more light on the quadratic formula!

 


 

Click here for more information about the discriminant.

 


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