Terminalogy Clarification:

The terms "root" and "zero" are encountered when working with equations and functions. In mathematics, the terms "root" and "zero" are often used interchangeably. There is, however, a subtle distinction between these terms. "Root" is typically used in reference to equations. "Zero" is typically used in reference to functions (graphs).

"Math" behind solving a quadratic equation:

A quadratic equation is a polynomial equation of degree two, which can be written in the form ax2 + bx + c = 0, where x is a variable and a, b and c are constants with a ≠ 0.

Consider: If a = 0, there would be no x² term and the equation would be "linear", not "quadratic".

If you know that a quadratic equation (degree 2) has exactly 2 roots, then you
already know (and have used) what is called 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.
• The theorem mentions "multiple" roots . . . so, if a polynomial has a "repeated root", each repetition of the root is counted.
• The theorem mentions "complex" roots . . . so, if a polynomial has a complex root, then the conjugate is also counted since complex roots come in "conjugate pairs".

According to the Fundamental Theorem of Algebra, a quadratic equation set = 0 has exactly two roots.

For quadratic equations, the two roots can appear as any one of three forms:
• 2 Real distinct roots
• one repeated Real root
• one Complex conjugate pair

Quadratic Function Graphs illustrate these roots:

The quadratic function graph may show 2 distinct real roots where the graph will . cross the x-axis in two locations. These graphs may open upward or downward. This graph has 2 real zeros.

It may appear that the quadratic function has only one real root. But, it actually has one repeated root. This graph is tangent to the x-axis in one location (touching once). This graph has 1 repeated zero.

The quadratic function may show two non-real complex roots called a complex conjugate pair. This graph will not cross or touch the x-axis, but it has two complex (conjugate) roots. This graph has 2 complex zeros.

Remember, the term "roots" generally refers to equations, while "zeros" refers to functions, but are interchangeable..

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

When solving a factorable quadratic equation, each factor can be set equal to zero to find the roots. This process is a fundamental concept known as the Zero Product Property.

Function:
f (x) = x² + 5x - 24
zeros (x-intercepts) at
(3,0) and (-8,0)