Binomials: Difference of Two Squares

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Sometimes a difficult calculation can be simplified and a complicated math expression can be made smaller with factoring formulas. In this lesson we explore one of the factoring formulas called the difference of squares.

Differences of Squares

Have you ever had to find the difference between two things? For example, find the difference between Whiskers the cat and Fido the dog. Hmmm… different animal families: feline, canine. Still, most things aren't completely different. Both animals have a common pastime: sleeping… In math, 'different' and 'common' can be like 'minus' and 'plus', and we may use both when finding the difference.


cat and dog sleeping


What if, instead, we're trying to find the difference for a special kind of binomial, a mathematical expression with two terms? This special binomial is made up of the difference of two squares, like a squared and b squared. If we recognize it, we can use the difference of squares formula: (a2 - b2) = (a - b)(a + b). The minus part is a - b, and the plus part is a + b. Let's explore how this formula works while Whiskers and Fido sleep.

Using the Formula

First, here are some questions: what is 8 squared? Answer: 82 is 64. Okay, what about 6 squared? Answer: 62 is 36. Do you see how 64 - 36 is the same as 82 - 62 ?

What about the formula (a2 - b2) = (a - b)(a + b) ? Do you recognize 8 as the a and 6 as the b?

Let's use the difference of squares formula for 8 and 6:


8^2-6^2


No big deal! We could have easily subtracted 36 from 64 to get the same answer. But what if we had something like 552 - 452 ? The short path to the answer is recognizing a difference of squares and using the formula:


55^2-45^2


Are you impressed yet? Not sure if Whiskers and Fido are impressed because they are still asleep.

While we're waiting for them to wake up, let's explore why this formula works. If we expand (a - b)(a + b) we get a(a + b) - b(a + b) which is a2 +ab - ba - b2. The ab - ba cancel, leaving us with a2 - b2. And that's why the formula works!

The key to using the formula is spotting a difference of two terms where the terms are squares. Let's see how good we are at this. Which of the following binomials could be factored (something times something) using the difference of squares formula?

x2 - 9

x2 + 36

• 113 - 73

The first one is a difference of squares and can be factored as (x - 3)(x + 3).

In the second example, we have squares but not the difference so we won't use the formula. In the third example, we have the difference but not squared terms, so we won't use the difference of squares formula here either. (By the way, there is also a formula for the difference of cubes.)

Taking the Formula Further

What if we had 25x4 - 16y2? There is a difference, but are we dealing with squares?

Yes! Squaring 5x2 gives us 25x4. Squaring 4y gives us 16y2. Using the formula: 25x4 - 16y2 = (5x2 - 4y)(5x2 + 4y). We've come a long way from the simple looking a2 - b2 = (a - b)(a + b).

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