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Online Notes / Algebra / Solving Equations and Inequalities / Quadratic Equations - Part I
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Equations With More Than One Varaible E-Book   Chapter   Section Quadratic Equations - Part II

 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

 

 

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 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. 

 

 

 

 

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.

 

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,

                                                    

Tto 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]

 

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.

 

Example 2  Solve each of the following equations.

(a)    [Solution]

(b)    [Solution]

Solution

Okay, just like with the linear equations the first thing that we’re going to need to do here is to clear the denominators out by multiplying by the LCD.  Recall that we will also need to note value(s) of x that will give division by zero so that we can make sure that these aren’t included in the solution.

 

(a)  

The LCD for this problem is  and we will need to avoid  and  to make sure we don’t get division by zero.  Here is the work for this equation.

                                

So, it looks like the two solutions to this equation are,

                                                      

Notice as well that neither of these are the values of x that we needed to avoid and so both are solutions.

[Return to Problems]

 

(b)  

In this case the LCD is  and we will need to avoid  so we don’t get division by zero.  Here is the work for this problem.

                                            

So, the quadratic that we factored and solved has two solutions,  and .  However, when we found the LCD we also saw that we needed to avoid  so we didn’t get division by zero.  Therefore, this equation has a single solution,

 

[Return to Problems]