In the previous two sections we’ve talked quite a bit about
solving quadratic equations. A logical
question to ask at this point is which method should we use to solve a given
quadratic equation? Unfortunately, the
answer is, it depends.
If your instructor has specified the method to use then
that, of course, is the method you should use.
However, if your instructor had NOT specified the method to use then we
will have to make the decision ourselves.
Here is a general set of guidelines that may be helpful in determining which method to use.
- Is it
clearly a square root property problem?
In other words, does the equation consist ONLY of something squared
and a constant. If this is true
then the square root property is probably the easiest method for use.
- Does
it factor? If so, that is probably
the way to go. Note that you
shouldn’t spend a lot of time trying to determine if the quadratic
equation factors. Look at the
equation and if you can quickly determine that it factors then go with
that. If you can’t quickly
determine that it factors then don’t worry about it.
- If
you’ve reached this point then you’ve determined that the equation is not
in the correct form for the square root property and that it doesn’t factor (or
that you can’t quickly see that it factors). So, at this point you’re only real
option is the quadratic formula.
Once you’ve solve enough quadratic equations the above set
of guidelines will become almost second nature to you and you will find yourself
going through them almost without thinking.
Notice as well that nowhere in the set of guidelines was
completing the square mentioned. The
reason for this is simply that it’s a long method that is prone to mistakes
when you get in a hurry. The quadratic
formula will also always work and is much shorter of a method to use. In general, you should only use completing
the square if your instructor has required you to use it.
As a solving technique completing the square should always
be your last choice. This doesn’t mean
however that it isn’t an important method.
We will see the completing the square process arise in several sections
in later chapters. Interestingly enough
when we do see this process in later sections we won’t be solving equations! This process is very useful in many situations
of which solving is only one.
Before leaving this section we have one more topic to
discuss. In the previous couple of
sections we saw that solving a quadratic equation in standard form,
we will get one of the following three possible solution
sets.
- Two
real distinct (i.e. not equal)
solutions.
- A
double root. Recall this arises
when we can factor the equation into a perfect square.
- Two
complex solutions.
These are the ONLY possibilities for solving quadratic
equations in standard form. Note
however, that if we start with rational expression in the equation we may get
different solution sets because we may need avoid one of the possible solutions
so we don’t get division by zero errors.
Now, it turns out that all we need to do is look at the
quadratic equation (in standard form of course) to determine which of the three
cases that we’ll get. To see how this
works let’s start off by recalling the quadratic formula.
The quantity 
in the quadratic formula is called the discriminant. It is the value of the discriminant that
will determine which solution set we will get.
Let’s go through the cases one at a time.
- Two
real distinct solutions. We will
get this solution set if . In this case we will be taking the
square root of a positive number and so the square root will be a real
number. Therefore the numerator in
the quadratic formula will be b plus or minus a real number. This means that the numerator will be
two different real numbers.
Dividing either one by 2a
won’t change the fact that they are real, nor will it change the fact that
they are different.
- A
double root. We will get this
solution set if . Here we will be taking the square root
of zero, which is zero. However,
this means that the “plus or minus” part of the numerator will be zero and
so the numerator in the quadratic formula will be b.
In other words, we will get a single real number out of the
quadratic formula, which is what we get when we get a double root.
- Two
complex solutions. We will get this
solution set if . If the discriminant is negative we will
be taking the square root of negative numbers in the quadratic formula
which means that we will get complex solutions. Also, we will get two since they have a
“plus or minus” in front of the square root.
So, let’s summarize up the results here.
- If then we will get two real solutions to
the quadratic equation.
- If then we will get a double root to the
quadratic equation.
- If then we will get two complex solutions to
the quadratic equation.
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Example 1 Using
the discriminant determine which solution set we get for each of the
following quadratic equations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
All we need to do here is make sure the equation is in
standard form, determine the value of a,
b, and c, then plug them into the discriminant.
(a)
First get the equation in standard form.
We then have,
Plugging into the discriminant gives,
The discriminant is negative and so we will have two
complex solutions. For reference
purposes the actual solutions are,
[Return to Problems]
(b)
Again, we first need to get the equation in standard form.
This gives,
The discriminant is then,
The discriminant is positive we will get two real distinct
solutions. Here they are,
[Return to Problems]
(c)
This equation is already in standard form so let’s jump
straight in.
The discriminant is then,
In this case we’ll get a double root since the
discriminant is zero. Here it is,
[Return to Problems]
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For practice you should verify the solutions in each of
these examples.