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
The only requirement here is that we have an in the equation. We guarantee that this term will be present
in the equation by requiring . 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 they
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.
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.
In this case there is no reason to believe that either a or b
will be 6. We could have and 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.
Let’s take a look at a couple of examples. Note that it is assumed that you can do the
factoring at this point and so we won’t be giving any details on the
factoring. If you need a review of
factoring you should go back and take a look at the Factoring
section of the previous chapter.
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Example 1 Solve
each of the following equations by factoring.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
(f) [Solution]
(g) [Solution]
Solution
Now, as noted earlier, we won’t be putting any detail into
the factoring process, so make sure that you can do the factoring here.
(a)
First, get everything on side of the equation and then
factor.
Now at this point we’ve got a product of two terms that is
equal to zero. This means that at
least one of the following must be true.
Note that each of these is a linear equation that is easy
enough to solve. What this tell us is
that we have two solutions to the equation, and . As with linear equations we can always
check our solutions by plugging the solution back into the equation. We will check and leave the other to you to check.
So, this was in fact a solution.
[Return to Problems]
(b)
As with the first one we first get everything on side of
the equal sign and then factor.
Now, we once again have a product of two terms that equals
zero so we know that one or both of them have to be zero. So, technically we need to set each one
equal to zero and solve. However, this
is usually easy enough to do in our heads and so from now on we will be doing
this solving in our head.
The solutions to this equation are,
To save space we won’t be checking any more of the
solutions here, but you should do so to make sure we didn’t make any
mistakes.
[Return to Problems]
(c)
In this case we already have zero on one side and so we
don’t need to do any manipulation to the equation all that we need to do is
factor. Also, don’t get excited about
the fact that we now have y’s in
the equation. We won’t always be
dealing with x’s so don’t expect to
always see them.
So, let’s factor this equation.
In this case we’ve got a perfect square. We broke up the square to denote that we
really do have an application of the zero factor property. However, we usually don’t do that. We usually will go straight to the answer
from the squared part.
The solution to the equation in this case is,
We only have a single value here as opposed to the two
solutions we’ve been getting to this point.
We will often call this solution a double root or say that it has multiplicity of 2 because it came from a term that was squared.
[Return to Problems]
(d)
As always let’s first factor the equation.
Now apply the zero factor property. The zero factor property tells us that,
Again, we will typically solve these in our head, but we
needed to do at least one in complete detail.
So we have two solutions to the equation.
[Return to Problems]
(e)
Now that we’ve done quite a few of these, we won’t be
putting in as much detail for the next two problems. Here is the work for this equation.
[Return to Problems]
(f)
Again, factor and use the zero factor property for this
one.
[Return to Problems]
(g)
This one always seems to cause trouble for students even
though it’s really not too bad.
First off. DO NOT
CANCEL AN x FROM BOTH
SIDES!!!! Do you get the idea that
might be bad? It is. If you cancel an x from both sides, you WILL miss a solution so don’t do it. Remember we are solving by factoring here
so let’s first get everything on one side of the equal sign.
Now, notice that all we can do for factoring is to factor
an x out of everything. Doing this gives,
From the first factor we get that and from the second we get that . These are the two solutions to this
equation. Note that is we’d canceled
an x in the first step we would NOT
have gotten as an answer!
[Return to Problems]
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Let’s work another type of problem here. We saw some of these back in the Solving Linear Equations section and since they
can also occur with quadratic equations we should go ahead and work on to make
sure that we can do them here as well.
Before proceeding to the next topic we should address that
this idea of factoring can be used to solve equations with degree larger than
two as well. Consider the following
example.
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Example 3 Solve
.
Solution
The first thing to do is factor this equation as much as
possible. In this case that means
factoring out the greatest common factor first. Here is the factored form of this equation.
Now, the zero factor property will still hold here. In this case we have a product of three
terms that is zero. The only way this
product can be zero is if one of the terms is zero. This means that,
So, we have three solutions to this equation.
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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,
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 and the other is . 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. Let’s see some examples of
this property.
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Example 4 Solve
each of the following equations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
There really isn’t all that much to these problems. In order to use the square root property
all that we need to do is get the squared quantity on the left side by itself
with a coefficient of 1 and the number on the other side. Once this is done we can use the square
root property.
(a)
This is a fairly simple problem so here is the work for
this equation.
So, there are two solutions to this equation, . Remember this means that there are really
two solutions here, and .
[Return to Problems]
(b)
Okay, the main difference between this one and the
previous one is the 25 in front of the squared term. The square root property wants a
coefficient of one there. That’s easy
enough to deal with however; we’ll just divide both sides by 25. Here is the work for this equation.
In this case the solutions are a little messy, but many of
these will do so don’t worry about that.
Also note that since we knew what the square root of 25 was we went
ahead and split the square root of the fraction up as shown. Again, remember that there are really two
solutions here, one positive and one negative.
[Return to Problems]
(c)
This one is nearly identical to the previous part with one
difference that we’ll see at the end of the example. Here is the work for this equation.
So, there are two solutions to this equation : . Notice as well that they are complex
solutions. This will happen with the
solution to many quadratic equations so make sure that you can deal with
them.
[Return to Problems]
(d)
This one looks different from the previous parts, however
it works the same way. The square root
property can be used anytime we have something
squared equals a number. That is what
we have here. The main difference of
course is that the something that is squared isn’t a single variable it is
something else. So, here is the
application of the square root property for this equation.
Now, we just need to solve for t and despite the “plus or minus” in the equation it works the same
way we would solve any linear equation.
We will add 9 to both sides and then divide by a 2.
Note that we multiplied the fraction through the
parenthesis for the final answer. We
will usually do this in these problems.
Also, do NOT convert these to decimals unless you are asked to. This is the standard form for these
answers. With that being said we
should convert them to decimals just to make sure that you can. Here are the decimal values of the two
solutions.
[Return to Problems]
(e)
In this final part we’ll not put much in the way of
details into the work.
So we got two complex solutions again and notice as well
that with both of the previous part we put the “plus or minus” part
last. This is usually the way these
are written.
[Return to Problems]
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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.