Simon's Favorite Factoring Trick
Contents
The General Statement
The general statement of SFFT is: 
. Two special common cases are: 
and 
.
The act of adding 
to 
in order to be able to factor it could be called "completing the rectangle" in analogy to the more familiar "completing the square."
Applications
This factorization frequently shows up on contest problems, especially those heavy on algebraic manipulation. Usually 
and 
are variables and 
are known constants. Also, it is typically necessary to add the 
term to both sides to perform the factorization.
Fun Practice Problems
Introductory
- Two different prime numbers between
and 
are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
and 
are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?
(Source)
Intermediate
are integers such that 
. Find 
.
are integers such that 
. Find 
.(Source)
Olympiad
- The integer
is positive. There are exactly 2005 ordered pairs 
of positive integers satisfying:
of positive integers satisfying:![\[\frac 1x +\frac 1y = \frac 1N\]](http://latex.artofproblemsolving.com/6/6/d/66dc47078d1e109634e02a0d0e05c3c5bfb9d1c6.png)
Prove that 
is a perfect square.
Source: (British Mathematical Olympiad Round 3, 2005)
