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Mathematical Expression Editor
We explore different methods for solving quadratic functions.
We have previously looked at different forms of quadratic functions. We’ve looked
at standard form and vertex form, where characteristics like -intercept and
vertex can be found easily by looking at the function. Another useful way
to look at quadratic functions is to have them written out as a product
of linear factors. This can help us to quickly determine the -intercepts of
a quadratic function and to get a good idea of the position and shape of
the graph. Not all quadratics can be written in factored form, so we will
begin by addressing those.
Irreducible quadratic factors are quadratic
factors that when set equal to zero only have complex roots. As a result they
cannot be reduced into factors containing only real numbers, hence the name
irreducible.
As seen in the graphs below, the graphs of the functions do not cross the -axis, so
they do not have -intercepts. The first graph, is entirely above the -axis and the
second graph, is entirely below the -axis. Since neither of them cross the -axis, they
have no -intercepts and are irreducible.
Factored Form (or Root Form)
Factored (Root) Form of a Quadratic Function A quadratic function whose graph has -intercepts (called roots) at the points and
can be written as:
This form is called Root Form because the roots of the quadratic can be easily read
off from this form. It is also sometimes called Factored Form because the quadratic
is factored into a product of linear terms. We often call quadratics written as a
factored quadratic. This is not quite the same as Factored Form (Root
Form) though because the leading constant is not pulled out to the front. To
minimize this confusion, we will typically but not exclusively use the name Root
Form.
Factoring from Standard Form when
When , putting a quadratic in Root Form is the same as factoring a quadratic. In
general, factoring refers to writing as a product of linear terms, but does
not necessarily imply that the term is pulled out front like it is in Root
Form.
Factor the following quadratic into a product of linear factors:
For us to begin factoring this quadratic, we have to look at the and terms. We are
looking for numbers that muliply to (or ) and add up to (or ). By going through
the factors of we can see that the only numbers that satisfy these conditions are and
. This means that we can factor the quadratic the following way: The quadratic is
now written as a product of linear factors and because , these are also our -intercepts
(or roots) for our function.
Factoring from Standard Form when
Rewrite the following quadratic in Root Form.
First we will start by pulling out a
from every term. We now have to find factors that add up to and multiply to . In
this particular case, we can see that the difference between the numerators ( and ) is ,
and since our job will be a little easier. This leads us to the following factors: So,
our factored form is as follows:
Note that in the previous example, it is not necessary to pull at the as the first
step. Instead, we could pull out the as the last step and still have the root
form.
This way, we start with We now have to find numbers such that:
This means that we need
because the only way two quadratics in standard form can be equal is if they have
the same coefficients for each term.
Through a little trial and error, we find that:
will work.
We now have the equation written as a product of linear components Now,
to write our answer in Root Form, we just need to factor out both and .
Since in this example, , we don’t actually have to do any thing for that one.
Now, we have our quadratic in Root Form and can read off our roots as and
.
Quadratic Formula
When there appears to be no easy way to factor a quadratic, our best option is to use
the Quadratic Formula. Let’s try the previous example with the Quadratic Formula.
when
Find the solutions to the following quadratic equation:
We still get and as our roots. This can be a very useful tool especially with more
complicated quadratic equations.
Now that we know the roots, we can use the -value we see in standard form, and the
two roots and to plug into our Root Form formula , so again we get as a final
answer that
Factoring when missing a term
We’ve talked about factoring when all terms are present, but what do we do when
one of the terms is missing? Since there are three terms we have three different cases
to address.
The first case also happens to be the easiest to solve. How do you a solve a quadratic
that is missing the term? This is a bit of a trick question, because without an
term, we are no longer dealing with a quadratic.
Solve the following equation
We can see here that we are only dealing with a linear term and there are no
quadratic () terms. This means we do not have to factor and we can solve for
directly.
The second case is when the middle term is missing.
Factor the quadratic into
linear components.
This quadratic is a special case called “difference of squares.” There is no “middle”
term and the remaining two terms are both perfect squares, so we can use a shortcut
when factoring.
Difference of Squares When , are non zero.
In our case, we can see that is a perfect square and is also a perfect square because .
This means that our original quadratic will be factored like this: We can also think
of it in the same way as factoring other quadratics. Since there is no middle term, we
can look at factors of that add up to . and add up to and multiply out to . The
difference of squares is just a useful pattern that helps to speed up our factoring
process.
The last case is when there is no constant or term.
Factor the quadratic into linear
components.
Since there is a common factor in both terms we can pull out that factor and we are
left with a product of linear components.
Again, our factoring is already simplified. We do not have go through the
whole process of factoring. If we have a quadratic function with only the
and term, then we will always be able to pull out at least an term when
factoring.