# How to Graphically Interpret the Complex Roots of a Quadratic Equation

As a secondary math teacher I have taught my students to
find the roots of a

quadratic equation in several ways. One of these ways is to graphically look at
the

quadratic and see were it crosses the x-axis. For example, the equation of y =
x^{2} – x – 2,

as shown in Figure 1, has roots at x = -1 and x = 2. These are the two places in
which the

sketched graph crosses the x-axis.

Figure 1

y = x^{2} – x – 2

The process of uses the quadratic formula will always find
the real roots of a

quadratic equation. We could have also used the quadratic formula to find the
roots of the

this equation, y = x^{2} – x – 2.

We can think of the first term (½) as a starting place for
finding the two roots.

Then we see that the roots are located 3/2 from the starting point in both
directions.

This leads us to roots of a quadratic equation that does
not cross the x-axis. These

roots are known as complex (imaginary) roots. An example of a quadratic drawn on
a

coordinate plane with complex roots is shown in Figure 3. Notice that the vertex
lies

above the x-axis, and the end behavior on both sides of the graph is approaching
positive

infinity. The complex roots to can be found by using the quadratic formula, but
it is

beneficial to students to visualize a graphical connection.

Figure 3

y = x^{2} + x + 1

GRAPHICAL INTERPRETATION OF COMPLEX ROOTS

We know that any quadratic can be represented by y = ax^{2} + bx + c. We also

know the roots of quadratic equations can be derived from the well-known
quadratic

formula:

If the roots are real we visually interpret them to cross the x-axis as shown in Figure 4a.

Figure 4a

y = −x^{2} − x + 1/2

But, we are interested in graphically interpreting the
roots of a graph that does not cross

the x-axis, as in Figure 4b.

Figure 4b

y = x^{2} + x +1

Let’s use what we graphically know about quadratics with
real roots (Fig. 4a) to

explain what we don’t know graphically about quadratics with complex roots (Fig.
4b).

Now there are infinitely many quadratics with real roots and infinitely many
quadratics

with complex roots. But, when comparing one to another, it would be helpful if
the two

quadratics were related in some way.

If y = ax^{2} + bx + c produces real roots (bold line in
Figure 4c), a reflection of this

graph upward would yield a new quadratic equation that would produce complex
roots.

(See Figure 4c)

Figure 4c

To see how this is configured analytically, we will start
with the general equation

of a quadratic. (Remember: The quadratic that we are starting with (bold line)
is known

to have real roots.) First we complete the square. Then to create the flipped
quadratic

(which is a different equation), a negative is applied to a. Then I simplified
the equation

by multiplying.

y = ax^{2} + bx + c

--------------Complete the square----------------

------------------Completed the square---------------

WE ARE CREATING A NEW QUADRATIC AT THIS POINT.

-------------------Reflect the Quadratic-------------

-------------------Reflected the Quadratic-------------

----------------------------Simplify------------------------

---------------------------Simplified------------------------

Now to be able to compare these complex roots to the real
roots that we started

with, plug the coefficients of the equation above into the quadratic formula.

-------------------------Simplify-----------------------

Because we are adding and subtracting from
it is unnecessary to write the

negative in the second denominator.

-------------------------Simplified-----------------------

Since we know that we the roots are complex, we can show
that by extracting a

negative out of the radical.

Complex Roots of the Flipped Quadratic

Real Roots of the Quadratic We Started With

Notice that the real roots and the complex roots of
different quadratic equations

yielded very similar answers. They are actually the same except for the i in the
complex

roots.

The next step is to use figure out how to use these
similarities to find the complex

roots graphically. First, let’s review how to graph the complex number plane.
Horizontal

movement on the graph denotes the real part of the complex number, while
vertical

movement represents the imaginary part of the complex number. (See Figure 5)

Figure 5

Now, working in three-space, imagine that these complex
coordinate plane is the

“floor”. It is represented by the (i, x) coordinate plane in Figure 6.

Figure 6

If we now use the (x,y) coordinate plane to draw a
quadratic with complex roots

we could get something that looks like Figure 7. Notice the quadratic does not
cross the

x-axis.

Figure 7

We currently can not see the complex roots graphically.
But if we flip the

quadratic horizontally over the vertex, from the proof about we should get roots
that

differ only by a number i. In order to graphically see the complex roots we need
to rotate

the reflected image 90 degrees to place the quadratic into the complex number
plane.

Figure 8

Notice that the two points indicated can be found by
starting at x units in

the real direction and i units in both the positive and negative direction. We
are

then able to graphically see the complex roots of a quadratic equation.

References

Weeks, Audrey. Connecting Complex Roots to a Parabola’s
Graph. June 20, 2007.

Vest, Floyd. The College Mathematics Journal, Vol. 16, No. 4. (Sep., 1985), pp.
257-

261.