AC Method for
Factoring Trinomials

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Consider this unconventional method of factoring trinomials:

ax2 + bx +
where
 

The AC Method for Factoring Trinomials. Factor:  
1.  Multiply the leading coefficient (the number in front of x2 ) times the constant term (the last term).  Remove the leading coefficient.
(Obviously an illegal algebraic move!)


 
2.  Factor this new trinomial.
3.  Replace x with the original leading coefficient times x.  In this example, the leading coefficient is 2.
4.  Factor.
5.  Divide by the original leading coefficient.

 

How can an "illegal" move result in a correct answer?  Is this really a valid method of factoring?  And if so, why does it work?

 

Why it works:

   The secret to understanding this method is to realize that these steps are actually a shorthand for a more complex process of multiplication and replacement.       

Let's take one more look at the whole process.  The AC Method is simply a shorthand version of the following procedure.

Multiply the entire expression times the leading coefficient:

Distribute:

Re-write:

Replace 2x with another variable, such as m,
where 2
x = m:

Factor:

Replace m with 2x:

Factor:

These factors now equal what we started with, which is TWICE the original problem:  =

Divide both sides by 2:                  =