If you know that a quadratic equation (degree 2) has exactly 2 roots, then you already know
(and have used) the Fundamental Theorem of Algebra.
While there are various statements of this theorem,
the interpretation that we are interested in for this course is the following:

The Fundamental Theorem of Algebra: If P(x) is a polynomial of degree n ≥ 1, then P(x) = 0 has exactly n roots, including multiple and complex roots. 

In plain English, this theorem says that the degree of a polynomial equation tells you how many roots the equation will have.
A linear equation (degree 1) will have one root.
A quadratic equation (degree 2) will have two roots.
A cubic equation (degree 3) will have three roots.
An n^{th} degree polynomial equation will have n roots.
Read carefully to fully understand what this theorem is saying.
• It mentions "multiple" roots . . . so, if a polynomial has a repeated root, each repetition of the root is counted.
• It mentions "complex" roots . . . so, if a polynomial has a complex root, then the conjugate is also counted since complex roots come in conjugate pairs.
This theorem deals with complex roots, both real and nonreal. Remember that all Real Numbers are a subset of the Complex Numbers.
Dealing with Quadratics:
Let's verify that the Fundamental Theorem of Algebra holds for quadratic polynomials.
A quadratic polynomial is a second degree polynomial. According to the Fundamental Theorem of Algebra, the quadratic set = 0 has exactly two roots.
As we have seen, factoring a quadratic equation will result in one of three possible situations.
The quadratic may have 2 distinct
real roots. This graph crosses the
xaxis in
two locations. These
graphs may open
upward
or downward.

It may appear that the quadratic
has only one real root. But, it
actually has one repeated root.
This graph
is
tangent to the xaxis
in one
location (touching once).

The quadratic may have two
nonreal complex roots called
a conjugate pair.
This graph will
not cross or
touch the xaxis, but
it will have two roots.

In all three situations, there are two solutions to the quadratic equations,
as stipulated by the Fundamental Theorem of Algebra.
