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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,
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
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.
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.
Let’s do a couple of examples for just completing the square
before looking at how we use this to solve quadratic equations.
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Example 1 Complete
the square on each of the following.
(a) [Solution]
(b) [Solution]
Solution
(a)
Here’s the number that we’ll add to the equation.
Notice that we kept the minus sign here even though it
will always drop out after we square things.
The reason for this will be apparent in a second. Let’s now complete the square.
Now, this is a quadratic that hopefully you can factor
fairly quickly. However notice that it
will always factor as x plus the
blue number we computed above that is in the parenthesis (in our case that is
-8). This is the reason for leaving
the minus sign. It makes sure that we
don’t make any mistakes in the factoring process.
[Return to Problems]
(b)
Here’s the number we’ll need this time.
It’s a fraction and that will happen fairly often with
these so don’t get excited about it.
Also, leave it as a fraction.
Don’t convert to a decimal. Now
complete the square.
This one is not so easy to factor. However, if you again recall that this will
ALWAYS factor as y plus the blue
number above we don’t have to worry about the factoring process.
[Return to Problems]
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It’s now time to see how we use completing the square to
solve a quadratic equation. The process
is best seen as we work an example so let’s do that.
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Example 2 Use
completing the square to solve each of the following quadratic equations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
We will do the first problem in detail explicitly giving
each step. In the remaining problems
we will just do the work without as much explanation.
(a)
So, let’s get started.
Step 1 : Divide the equation by the
coefficient of the x2
term. Recall that completing the
square required a coefficient of one on this term and this will guarantee
that we will get that. We don’t need
to do that for this equation however.
Step 2 :
Set the equation up so that the x’s
are on the left side and the constant is on the right side.
Step 3 :
Complete the square on the left side.
However, this time we will need to add the number to both sides of the
equal sign instead of just the left side.
This is because we have to remember the rule that what we do to one
side of an equation we need to do to the other side of the equation.
First, here is the number we add to both sides.
Now, complete the square.
Step 4 :
Now, at this point notice that we can use the square root property on this
equation. That was the purpose of the
first three steps. Doing this will
give us the solution to the equation.
And that is the process.
Let’s do the remaining parts now.
[Return to Problems]
(b)
We will not explicitly put in the steps this time nor will
we put in a lot of explanation for this equation. This that being said, notice that we will
have to do the first step this time.
We don’t have a coefficient of one on the x2 term and so we will need to divide the equation by
that first.
Here is the work for this equation.
Don’t forget to convert square roots of negative numbers
to complex numbers!
[Return to Problems]
(c)
Again, we won’t put a lot of explanation for this problem.
At this point we should be careful about computing the
number for completing the square since b
is now a fraction for the first time.
Now finish the problem.
In this case notice that we can actually do the arithmetic
here to get two integer and/or fractional solutions. We should always do this when there are
only integers and/or fractions in our solution. Here are the two solutions.
[Return to Problems]
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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.
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,
and that’s very important, then the solution to any
quadratic equation is,
Let’s work a couple of examples.
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Example 3 Use
the quadratic formula to solve each of the following equations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
The important part here is to make sure that before we
start using the quadratic formula that we have the equation in standard form
first.
(a)
So, the first thing that we need to do here is to put the
equation in standard form.
At this point we can identify the values for use in the
quadratic formula. For this equation
we have.
Notice the “-” with c. It is important to make sure that we carry
any minus signs along with the constants.
At this point there really isn’t anything more to do other
than plug into the formula.
There are the two solutions for this equation. There is also some simplification that we
can do. We need to be careful
however. One of the larger mistakes at
this point is to “cancel” to 2’s in the numerator and denominator. Remember that in order to cancel anything
from the numerator or denominator then it must be multiplied by the whole
numerator or denominator. Since the 2
in the numerator isn’t multiplied by the whole denominator it can’t be
canceled.
In order to do any simplification here we will first need
to reduce the square root. At that
point we can do some canceling.
That’s a much nicer answer to deal with and so we will
almost always do this kind of simplification when it can be done.
[Return to Problems]
(b)
Now, in this case don’t get excited about the fact that
the variable isn’t an x. Everything works the same regardless of the
letter used for the variable. So,
let’s first get the equation into standard form.
Now, this isn’t quite in the typical standard form. However, we need to make a point here so
that we don’t make a very common mistake that many student make when first
learning the quadratic formula.
Many students will just get everything on one side as
we’ve done here and then get the values of a, b, and c based upon position. In other words, often students will just
let a be the first number listed, b be the second number listed and then
c be the final number listed. This is not correct however. For the quadratic formula a is the coefficient of the squared
term, b is the coefficient of the
term with just the variable in it (not squared) and c is the constant term.
So, to avoid making this mistake we should always put the quadratic
equation into the official standard form.
Now we can identify the value of a, b, and c.
Again, be careful with minus signs. They need to get carried along with the
values.
Finally, plug into the quadratic formula to get the
solution.
As with all the other methods we’ve looked at for solving
quadratic equations, don’t forget to convert square roots of negative numbers
into complex numbers. Also, when b is negative be very careful with the
substitution. This is particularly
true for the squared portion under the radical. Remember that when you square a negative
number it will become positive. One of
the more common mistakes here is to get in a hurry and forget to drop the
minus sign after you square b, so
be careful.
[Return to Problems]
(c) |