Quadratic Equations – Part I
.
Before
proceeding with this section we should note that the topic of solving quadratic
equations will be covered in two sections. This is done for the benefit of those
viewing the material on the web. This is a long topic and to keep page load
times down to a minimum the material was split into two sections.
So,
we are now going to solve quadratic equations. First, the standard form of
a quadratic equation
is:
ax2 + bx + c = 0 a ≠ 0
The
only requirement here is that we have an x2 in the equation.
We guarantee that this term will
be present in the equation by requiring a ≠ 0 . Note however, that it is okay if b and/or c are zero.
There
are many ways to solve quadratic equations. We will look at four of them over
the course of the next two sections. The first two methods won’t always work,
yet are probably a little simpler to use when the work. This section will cover
these two methods. The last two methods will always work, but often require a
little more work or attention to get correct. We will cover these methods in
the next section.
So,
let’s get started.
Solving
by Factoring
As the
heading suggests we will be solving quadratic equations here by factoring them.
To do this
we will need the following fact.
If ab = 0 then either a
= 0 and/or b = 0
This
fact is called the zero factor property or zero factor principle.
All the fact says is that if a product of two terms is zero then at least one
of the terms had to be zero to start off with.
Notice
that this fact will ONLY work if the product is equal to zero. Consider the
following product.
ab = 6
In
this case there is no reason to believe that either a or b will
be 6. We could have a = 2 and b = 3 for instance.
So, do not misuse this fact!
To
solve a quadratic equation by factoring we first must move all the terms over
to one side of the equation. Doing this serves two purposes. First, it puts the
quadratics into a form that can be factored.
Secondly, and probably more
importantly, in order to use the zero factor property we MUST have a zero on
one side of the equation. If we don’t have a zero on one side of the equation
we won’t be able to use the zero factor property.
So,
provided we can factor a polynomial we can always use this as a solution
technique. The problem
is, of course, that it is sometimes not easy to do the factoring.
Square
Root Property
The
second method of solving quadratics we’ll be looking at uses the square root
property,
If p2 = d then p = ± √d
There
is a (potentially) new symbol here that we should define first in case you
haven’t seen it yet.
The symbol “ ± ” is read as : “plus or minus” and that is exactly what it tells
us. This symbol is shorthand that tells us that we really have two numbers
here. One is p = d and
the other is p = − d.
Get used to this notation as it will be used frequently in the next couple of
sections as we discuss the remaining solution techniques. It will also arise in
other sections of this chapter and even in other chapters.
This
is a fairly simple property to use, however it can only be used on a small
portion of the equations
that we’re ever likely to encounter.
As
mentioned at the start of this section we are going to break this topic up into
two sections for the benefit of those viewing this on the web. The next two
methods of solving quadratic equations,
completing the square and quadratic formula, are given in the next section.
Quadratic
Equations – Part II
The
topic of solving quadratic equations has been broken into two sections for the
benefit of those viewing this on the web. As a single section the load time for
the page would have been quite long. This is the second section on solving
quadratic equations.
In
the previous section we looked at using factoring and the square root property
to solve quadratic equations. The problem is that both of these solution
methods will not always work.
Not
every quadratic is factorable and not every quadratic is in the form required
for the square root property.
It
is now time to start looking into methods that will work for all quadratic
equations. So, in this section we will look at completing the square and the
quadratic formula for solving the quadratic equation,
ax2 + bx + c = 0 a ≠ 0
Completing
the Square
The
first method we’ll look at in this section is completing the square. It is
called this because it uses a process called completing the square in the
solution process. So, we should first define just what completing the square
is.
Let’s
start with
x2 + bx
and
notice that the x2 has a coefficient of one. That is required in order
to do this. Now, to this lets add
. Doing this gives the following factorable quadratic
equation.
. Doing this gives the following factorable quadratic
equation.
This
process is called completing the square and if we do all the arithmetic
correctly we can guarantee
that the quadratic will factor as a perfect square.
It’s now time
to see how we use completing the square to solve a quadratic equation.
A quick comment
about the last equation that we solved in the previous example is in order.
Since
we received integer and factions as solutions, we could have just factored this
equation from the start rather than used completing the square. In cases like
this we could use either method and we will get the same result.
Now,
the reality is that completing the square is a fairly long process and it’s
easy to make mistakes. So, we rarely actually use it to solve equations. That
doesn’t mean that it isn’t important to know the process however. We will be
using it in several sections in later chapters and is often used in other
classes.
Quadratic
Formula
This
is the final method for solving quadratic equations and will always work. Not
only that, but if you can remember the formula it’s a fairly simple process as
well.
We
can derive the quadratic formula be completing the square on the general
quadratic formula in standard form. Let’s do that and we’ll take it kind of
slow to make sure all the steps are clear.
First,
we MUST have the quadratic equation in standard form as already noted. Next, we
need to divide both sides by a to get a coefficient of one on the x2
term.
ax2 + bx + c = 0
Next,
move the constant to the right side of the equation.
Now,
we need to compute the number we’ll need to complete the square. Again, this is
one-half the coefficient of x, squared.
Now,
add this to both sides, complete the square and get common denominators on the
right side to simplify things up a little.
Now
we can use the square root property on this.
Solve
for x and we’ll also simplify the square root a little.
As a
last step we will notice that we’ve got common denominators on the two terms
and so we’ll add them up. Doing this gives,
So,
summarizing up, provided that we start off in standard form,
ax2 + bx + c = 0
and
that’s very important, then the solution to any quadratic equation is,
.
Over the course of the last two sections we’ve
done quite a bit of solving. It is important that you understand most, if not
all, of what we did in these sections as you will be asked to do this kind of work
in some later sections.
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