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2.5  Solving Equations Reducible to Quadratic Form

Prerequisite knowledge and skills

       A working knowledge and understanding of solving quadratic equations by factoring

       A working knowledge and understanding of graphs of factoring expressions reducible to quadratic form

Terms to know:  

·         factor

·         factoring

·         index

·         polynomial

·         rational number

·         trinomial

·         whole number

·         x-intercepts

·         zero of a function

Skill prep:  Solving quadratic equations using factoring

Concept prep assignment:  Matching quadratic-type equation with a simple quadratic

Dynalab Activity 2

Recall the factoring we did in section 1.2.   In this section we will simply add on one extra step, i.e. we will set these “quadratics in disguise ” equal to zero and solve.

Example 1.  Solve  

First we will factor the left hand side as we did in section 1.2.

Notice that the binomial   is in two of the terms in this polynomial and is squared in one of them.   Substitute u for   and the result is much easier to work with:

Original polynomial:                       

Substitute u in for (x  1):                                                       

Now, we’ll factor    :

                                     =   .

Since the original polynomial was in (x  1), not u, we need to reverse the substitution and replace the u  with (x  1):

Now we take our factored form and set it equal to 0:

CHECK:

We can check by substituting these values back into the original equation:

We can also check by graphing.

Example 1b.    Some   people might take this same problem, notice that the left side is a “quadratic in disguise” and factor it directly, without using substitution:

and then set each factor to 0 and solve, just as in the above example. 

Example 2.  Solve     

First, we will factor the left side like we did in section 1.2.

Again, substitution will make this polynomial easier to factor.  Since the binomial

 appears twice  once as a linear term and once squared - we’ll substitute in u  for it:

This factors as    

Next, we replace the u  with :     

which equals        

Since the original problem was set equal to zero,   we take the factored form and set it equal to zero:

   so     

Now, solve:        

CHECK:

We can check by substituting these values back into the original equation:

:    

:                               

We can also check by graphing.

Example 2b.  .  As in Example 1, you may prefer to factor it directly without using substitution.

Then set each factor to 0 and solve, as we did above.                                        

      Checkpoint  1

Example 3.   Solve    

Again, we will factor the left hand side as we did in section 1.2 :

Look at the middle term (without the coefficient):         

 If we square it, we get the variable in the first term:         .

This is how we know that this is reducible to a quadratic,  a “quadratic in disguise.”

Use substitution, letting u  =   ,                

the variable in the middle term:                

Factor:                                                      

Substitute  back in  for u:                  

Set each factor to 0 and solve:            

To eliminate the  power, cube both sides:       

The checks are left for you. 

Example 3b.   Solve   

Again, you may choose to factor   directly:

Then set each factor to 0 and solve, as we did above.

      Checkpoint 2

More Worked Examples

Homework Problems

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