Hello. I'm Professor von Schmohawk and welcome to Why U.
So far we have introduced quadratic functions
and quadratic equations.
We have seen that the graph of a quadratic function of x is a parabola
and by setting that function equal to zero and solving the resulting quadratic equation
we can find the values of x where the parabola intersects the x-axis
in other words, the parabola's "x-intercepts".
These values graphically correspond to the "zeros" or "roots" of the function.
So let's take a look at the different ways that a quadratic function of x
can intersect the x-axis.
We'll start with the simplest quadratic function of x, "x-squared".
Setting this function equal to zero
gives us the quadratic equation for this function.
The only value of x which when squared is equal to zero, is zero.
So the quadratic function "x-squared" has a single x-intercept
at the point where x equals zero.
Now let's do the same thing with a slightly more complicated quadratic function
"x-squared - 4".
Setting this function equal to zero
gives us the quadratic equation "x-squared - 4" equals zero
or "x-squared equals 4".
There are two values of x which when squared are equal to 4
"positive 2" and "negative 2".
These two solutions can be written as "x equals plus or minus 2"
which corresponds to the two points where the function's graph intersects the x-axis.
So when the graph of a quadratic function of x intersects the x-axis
it can have either one or two x-intercepts
and thus one or two roots.
It is also possible for a quadratic function of a real variable to have no roots.
This situation occurs when its graph does not intersect the x-axis.
However, this is true when only real numbers are considered.
Several hundred years ago
a new type of number called an "imaginary number" was created by mathematicians
specifically because of functions which have no real roots.
When imaginary numbers
and "complex numbers" which include imaginary and real numbers
are included in our number system
it can be said that all quadratic functions have two roots
although these roots may or may not be real
and may or may not be distinct.
Later in these lectures, we will introduce imaginary and complex numbers
and see how these numbers opened up a whole new world of mathematics.
As we have just seen
solving a quadratic equation of the form "x-squared minus c equals zero" is easy
since as long as "c" has a positive value
this simply amounts to finding the square root of c.
The same is true for equations of the form "a x-squared minus c equals zero "
since if we divide the equation by a
we get the same type of equation as before
except that we take the square root of "c over a"
instead of the square root of c.
On the other hand, quadratic equations that include an x term
can be more difficult to solve.
There are a number of methods for solving quadratic equations like this.
The methods we will examine in the next few lectures
are based on the "zero product property".
So what exactly is the zero product property?
Let's say that we have a number or expression
which is the product of two or more factors.
For example, let's assume that the product is composed of two factors
which we will call A and B.
The zero product property states that if any factor is equal to zero
then the product of those factors must also be zero.
This same logic applies to the linear factors of a quadratic expression.
As we showed in the lecture "Creating Quadratic Expressions Using the FOIL Method"
a quadratic expression can be created by multiplying two linear expressions together.
Those linear expressions will therefore be factors of the resulting quadratic expression.
So when x has a value that causes any factor of a quadratic expression to be zero
then the quadratic expression will also have a value of zero.
This seems obvious since the product of anything multiplied by zero must be zero.
But the zero property also states that the converse is true.
If a product of factors is zero
then one or more of the factors must be zero.
So every zero of a quadratic function
must also be a zero of one or more of its linear factors.
Therefore, we can find the zeros of a quadratic function
by finding the zeros of the simpler linear functions that are its factors.
This means that instead of having to solve a quadratic equation
we can solve linear equations which are easier.
So to find the zeros of a quadratic function, we factor it into linear functions
and then identify the zeros of those functions.
But is it always possible to factor a quadratic function into linear functions?
If the quadratic's graph intersects the x-axis, the answer is "yes"
although when there is only a single intersection point
the two linear factors will be identical
However, if we are only considering real numbers
it will not be possible to factor quadratic functions
whose graph does not intersect the x-axis.
But any quadratic function whose graph does intersect the x-axis
and therefore has real zeros
can always be factored into two real linear functions
which may either be identical
or distinct.
So to factor a quadratic function
we must find a pair of linear expressions which when multiplied together
produce the quadratic expression defining the function.
Fortunately, there are several easily recognized cases of quadratic expressions
called "special products"
whose factors can be readily identified.
In the next two lectures we will examine two kinds of special products
including quadratic expressions which are the "difference of squares"
and quadratic expressions which are "perfect squares".