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Chapter 7

Polynomial Equations that can be Solved with the Introduction
of an Auxiliary Variable

 
Example 1

Solve the equation : 

Solution

Let's say that . Then (1) becomes:

.

Example 2

Solve the equation : 

Solution

Let's say that . Then (2) becomes:


Example 3

Solve the equation : ( 2x + 3 )( 4x + 3 )( x -1 )( 4x -1 ) = 9  (3)

Solution



 
Let's say that :  then the above equation becomes:



Exercise 1

Solve the equation : 
 

 

The equations below this point are of the form 
and belong to a category of equations known as inverse equations.

Example 4

Solve the equation: 

Solution

I observe that , since for x = 0 the equation becomes 1=0, which is not true.
I divide the equation (4) with  and we get :

But, according to the known formula we can conclude the following:
.
Thus, equation (5) becomes:

Now let's say that . The above equation becomes:


 
Exercise 2