# Simon’s Favorite Factoring Trick

When I was quite younger, one of my hobbies was joining internet forums (fora?) on problem solving. I was not really good at it, so my role was only to ask questions. One of the internet forums I joined was the Art of Problem Solving math forum.

Art of Problem Solving (AOPS) is a community of problem solvers dedicated for math competitions – probably the best place on the web to ask hard (and very hard) math questions. One of the tricks I learned there was Simon’s Favorite Factoring Trick (SFFT), a factorization technique popularized by one AOPS member. The general strategy (see example 3) of SFFT is to add a constant or variable to an expression to make it factorable. This strategy can also be named as “completing rectangle” in analogy with “completing the square.”

Let’s have a few examples.

**Example 1:** Find all positive integer such that .

*Solution*: Using SFFT, we add 1 to both sides of the equation giving us . This gives us which is equivalent to . Therefore, and .

**Example 2:** Find the length and the width of a rectangle whose area is equal to its perimeter.

*Solution*: Let and be the length and width of the rectangle. Since its perimeter is equal to its area, it follows that , which is equivalent to . Adding to both sides of the equation, we have . Factoring the left hand side, we have . This gives us . Therefore and , or and . The latter satisfies the condition above, so and .

**Example 3:** SFFT can be used in general for equations of the form . This simplify to . This simplifies to which is equal to .

Art of Problem Solving is good place for gifted math students. I bought two of their books a year ago, The Art of Problem Solving: The Basics and The Art of Problem Solving: And Beyond, and until now, I have not yet finished solving all the problems. They are excellent books if you are preparing for math competitions.

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