### 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.

**Figure out if each of the terms is a “perfect square.”****If so, take the square root of each term.****Put each square root in its proper place inside two ( ).****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.

**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.-
**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.** **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)****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)**