Difference of squares intro | Mathematics II | High School Math | Khan Academy

We're now going to explore factoring a type of expression called a difference of squares. The reason why it's called a difference of squares is because it's expressions like x² - 9. This is a difference; we're subtracting between two quantities that are each squares. This is literally x squared. Let me do that a different color. This is x squared minus 3 squared. It's the difference between two quantities that have been squared.

And it turns out that this is pretty straightforward to factor. To see how it can be factored, let me pause there for a second and get a little bit of a review of multiplying binomials. So put this on the back burner a little bit before I give you the answer of how do you factor this. Let's do a little bit of an exercise.

Let’s multiply x + a * x - a, where a is some number. We can use that. We could do that using either the FOIL method, but I like just thinking of this as a distributive property twice. We could take x plus a and distribute it onto the x and onto the a. So, when we multiply it by x, we would get x * x, which is x². a * x is plus a x.

And then when we multiply it by negative a, well, it'll become -a². So these middle two terms cancel out, and you are left with x² - a². You're left with a difference of squares, x² - a². So we have an interesting result right over here that x² - a² is equal to (x + a) * (x - a).

We could use this pattern now to factor this here. What is our a? Our a is 3. This is x² - 3², or we could say minus a² if we say 3 is a. So to factor it, this is just going to be equal to (x + a), which is 3 * (x - a), which is 3. So, x + 3 * x - 3.

Now let's do some examples to really reinforce this idea of factoring differences of squares. So let’s say we want to factor y² - 25. It has to be a difference of squares; it doesn’t work with a sum of squares. Well, in this case, this is going to be y, and you have to confirm okay, yeah, 25 is 5² and y² is well, y².

So it’s going to be (y + 5) * (y - 5). The variable doesn’t have to come first; we could write 121 - b². Well, this is a difference of squares because 121 is 11². So, this is going to be (11 + b) * (11 - b).

So, in general, if you see a difference of squares, one square being subtracted from another square, and it could be a numeric perfect square or it could be a variable that has been squared that you could take the square root of, well then you could say all right, well that's just going to be the first thing that's squared plus the second thing that has been squared times the first thing that was squared minus the second thing that was squared.

Now, some common mistakes that I've seen people do, including my son when he first learned this, is they say okay, it’s easy to recognize the difference of squares, but then they say, “Oh, is this y² + 25 * y² - 25?” No, the important thing to realize is that what is getting squared over here, y, is the thing getting squared, and over here it is 5 that is getting squared. Those are the things that are getting squared in this difference of squares.

So it’s going to be (y + 5) * (y - 5). I encourage you to just try this out. We have a whole practice section on Khan Academy where you can do many, many more of these to become familiar.