MathChat

Factoring Trick: How to Flawlessly Factor any “Difference of Two Squares” Binomial

If you’re staring at two terms you need to factor, but feel like a deer looking at the headlights of an oncoming semi, here’s a way to leap to safety!

It’s called the “Difference of Two Squares” trick.

It requires four simple steps.

1. Figure out if each of the terms is a “perfect square.”
2. If so, take the square root of each term.
3. Put each square root in its proper place inside two (    ).
4. Put a + sign inside the first (   ), and put a – sign inside the second (   ).

Let’s do an easy example. Suppose the terms you’re looking at are these:
x^2  – 9

Let’s go through the 4 steps together.

1. Figure out if each term is a “perfect square.”

   So, what does it mean for a number or term to be a “perfect square”?  It means that you get the number or term by multiplying a number or term by itself. For example, 16 is a perfect square because you can get 16 by multiplying 4 by itself:  4 x 4 = 16.

   So when we look at our two terms, x^2 and 9, we notice that both
   are perfect squares.
   9 is just 3 times 3.
   And in the same way, x^2 is just x times x.

2.  Take the square root of each term.
   The square root of x^2 is just x.
   And the square root of 9 is just 3.

3. Put each square root in the proper place inside two sets of (    ).
   We put the square root of the term that was positive first, and the square root of the term that was negative second.Since the x^2 was the positive term, we put its square root, x, first inside each
   (   ).  So far, that gives us:  (x    ) (x     )

   Since the 9 was the negative term because it had the negative sign in front of it: – 9, we put its square root, 3, second inside each (   ). So our (   )s now look like this:  (x   3) (x   3)

4. Finally, we just need to put in signs that connect the terms inside
   the (    )s.
   That’s easy. We put a + sign inside one (    ), and we put a – sign
   inside the other (    ).
   I prefer to put the + inside the first (   ), but it really doesn’t matter.The final factored form, then, looks like this:  (x + 3) (x – 3)
   That’s all there is to it.

Now try these problems for practice.

           a)  x^2 – 16
           b)  x^2 – 100
           c)   x^2 – 121
           d)   x^4 –  16x^2
           e)   49x^8 – 144y^12

Answers:

           a)   x^2 – 16   =  (x + 4) (x – 4)
           b)  x^2 – 100  = (x + 10) (x – 10)
           c)   x^2 – 121  = (x + 11) ( x – 11)
           d)   x^4 –  16x^2  = (x^2 + 4x) (x^2 – 4x)
           e)   49x^8 – 144y^12  = (7x^4 + 12y^6)(7x^4 – 12y^6)