You'll probably only use
complexes in the context of solving
quadratics for
their zeroes. (There are many other practical uses for complexes,
but you'll have to wait for more interesting classes like "Engineering
201" to get to the "good stuff".)

Remember that the Quadratic
Formula solves
"ax^{2}
+ bx + c = 0"
for the values of x.
Also remember that this means that you are trying to find the x-intercepts
of the graph. When the Formula gives you a negative inside the square
root, you can now simplify that zero by using complex numbers. The answer
you come up with is a valid "zero" or "root" or "solution"
for "ax^{2}
+ bx + c = 0",
because, if you plug it back into the quadratic, you'll get zero after
you simplify. But you cannot graph a complex number on the x,y-plane.
So this "solution to the equation" is not an x-intercept.
You can make this connection between the Quadratic Formula, complex numbers,
and graphing:

x^{2}
– 2x – 3

x^{2}
– 6x + 9

x^{2}
+ 3x + 3

a
positive number inside the square root

zero
inside the square root

a
negative number inside the square root

two
real solutions

one
(repeated) real solution

two
complex solutions

two
distinct x-intercepts

one
(repeated) x-intercept

no
x-intercepts

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As an aside, you can graph
complexes, but not in the x,y-plane.
You need the "complex" plane. For the complex plane, the x-axis
is where you plot the real part, and the y-axis
is where you graph the imaginary part. For instance, you would plot the
complex number 3
– 2i like this:

Note that all points at
this distance from the origin have the same modulus. All the points on
the circle with radius sqrt(13)
are viewed as being complex numbers having the same "size" as
3 – 2i.