2.09  Quadratic Equations by Factoring

 Now, factor the trinomial, using 9 and 3, with opposite signs.

                       

               Set each factor equal to zero.

                          or    

Final Answer:  

 

22.                       

Solution:  The first step is to set the equation equal to zero.  To do this, you can add  and subtract from each side:

                

              

Now, factor the trinomial, using 9 and 2, with opposite signs.

                       

Set each factor equal to zero.

                          or    

Final Answer:   

 

23.                           

Solution:  The first step will be to remove parentheses by the Distributive Property.

                            

Next, you must set the equation equal to zero.  There are two methods!  You decide which way you think is easier!!

 

METHOD I:  In order to set the equation equal to zero with a POSITIVE coefficient of , take everything to the RIGHT side of the equation!

                          

                               

Now, factor the trinomial, using 10 and 4, with opposite signs.

                       

Set each factor equal to zero.

                          or    

Final Answer:   

 

METHOD II:             

In order to set the equation equal to zero you may choose to add   to each side of the equation

                          

                               

In order to factor this, you may want to multiply both sides of the equation by (-1).

                          

                                        

Now, factor the trinomial, using 10 and 4, with opposite signs.

                                   

Set each factor equal to zero.

                           or    

Final Answer:    

 

25.                         

Solution:  Since this equation is already equal to zero, the first step will be to factor the left side.  The first step in any factoring problem is to try to take out a common factor.  In this case the common factor is

                       

Set each factor equal to zero:

                       

Final Answer: 

 

26.          

Solution:  Since this equation is already equal to zero, the first step will be to FACTOR the left side.  The first step in any factoring problem is to try to take out a common factor.  In this case the common factor is

                       

Set each factor equal to zero:

                       

Final Answer: 

 

27.           

              

Notice that this is really just a TRINOMIAL, and as such, it can be factored into the product of two binomials.  In this case, the FIRST times FIRST gives you  which is . 

           

Next, the LAST times LAST must give you , so try  or  , where the numbers are the SAME sign.  In order for the numbers to add and give you a  you will need to put the  in the first binomial and the  in the second binomial.   Then the OUTER times OUTER will be , and the INNER times INNER will be for a total of .

 It looks like this:

            

Now, set each factor equal to zero, and solve for x.  There are two solutions:

            

 

28.                            

Solution:  The first step is to multiply out the parentheses and set the equation equal to zero by subtracting 2 from each side of the equation. 

The next step is to factor the resulting trinomial.  There is no common factor, so it must begin like this:

Final Answer: 

30.                             

Solution:  The first step is to multiply out the parentheses and set the equation equal to zero by subtracting 5 from each side of the equation. 

The next step is to factor the resulting trinomial.  There is no common factor, so it must begin like this:

 Final Answer:    

34.    Since the equation is already set equal to zero, the first step is to FACTOR the left side of the equation.  There IS a common factor of 5x, so the first step in factoring must be to take out the common factor of 5x.

The next step is to factor the resulting trinomial.  There looks like this::

Final Answer:    

37.          

Solution:  Since this equation is already equal to zero, the first step will be to factor the left side.  The first step is to notice the  .  Unfortunately, you can’t take out a common factor, so it looks like an “advanced trinomial factoring” problem.  However, it might not be too bad!  Notice the perfect squares on each end of this trinomial.  Maybe it’s a perfect square trinomial!  Let’s try that first.  Anyway, set up the two sets of parentheses, and prepare to factor the trinomial however it works out!              

First times First will be 2x times 2x.

Last times Last will be 5 times 5.

                                   OR            

Set each factor equal to zero.

Final Answer:        Final Answer:                                      

38.               

Solution:  First, you must set the equation equal to zero.  Do this by adding  to each side. 

The first step will be to factor the left side.  The first step is to notice the .  Unfortunately, you can’t take out a common factor, so it looks like an “advanced trinomial factoring” problem.  However, it might not be too bad!  Notice the perfect squares on each end of this trinomial.  Maybe it’s a perfect square trinomial!  Let’s try that first.  Anyway, set up the two sets of parentheses, and prepare to factor the trinomial however it works out!           

First times First will be 2x times 2x.

Last times Last will be 3 times 3.

                                   OR            

Set each factor equal to zero.

Final Answer:            Final Answer:                                       

42.          

Solution:  Since this equation is already equal to zero, the first step will be to factor the left side.  Remember FCFF?  Factor the Common Factor First!!  Factor out the common factor of x.           

Next, notice that you have a difference of two squares!

Notice that there are 3 factors (in keeping with an  equation).  Set each factor equal to zero.

Final Answer:            

46.          

Solution:  Since this equation is already equal to zero, the first step will be to factor the left side.  Remember FCFF?  Factor the Common Factor First!!  Factor out the common factor of x².          

Notice that there are 3 factors (in keeping with an  equation).  Set each factor equal to zero.

Final Answer:      

NOTE:  The answer of       is repeated to illustrate the 3 solutions for this  equation. 

55.                                

Solution:  The first step in solving this equation is to square the binomial—i.e., remove the parentheses:  

The next step is to set the equation equal to zero, by subtracting from each side:

Notice that this is a trinomial which factors:

Therefore,

 56.          

Solution:  The first step in solving this equation is to square the binomial—i.e., remove the parentheses:  

The next step is to set the equation equal to zero, by subtracting and adding from each side:

Notice that this is a trinomial which factors:

Therefore,

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