Factoring quadratics with difference of squares
- [Voiceover] Moussa and Fatu were each asked to factor the quadratic expression 16 x-squared minus 64. Their responses are shown below. So Moussa factored it this way. Fatu factored it this way. Which student wrote an expression that is equivalent to 16 x-squared minus 64? So I encourage you to pause the video and figure that out. Which student wrote an expression that is equivalent to our original one, 16 x-squared minus 64? Well let's work through it together, so let's see if first we can factor this out somehow to get what Moussa got and it looks like Moussa first factored out a 16 and then he was left with a difference of squares. So let's see if we can do that. So, we can write our original expression. 16 x-squared minus 64, we can write that as 16 times x-squared minus 16 times four. And when you write it like that, it's very clear that you can factor out a 16. So this is going to be equal to 16 times what you have left over is x-squared minus four and then x-squared minus four, that's a difference of squares right over there. So, that part we can factor as, so we have our original 16 and then... this part right over here, we can write as x plus two times x minus two. x plus two times x minus two. If what I just did in this last step, going from x-squared minus four to x plus two times x minus two doesn't make any sense, I encourage you to watch some of the introductory videos on factoring and difference of squares. But the basic idea, I have a form here of a-squared minus b-squared, so it's going to have the form of a plus b times a minus b and in this case it's x-squared minus two squared. So it's going to be x plus two times x minus two. So that's exactly what Moussa got. So this one, so Moussa, did get an expression that is equivalent to 16 x-squared minus 64. Now let's think about Fatu. So Fatu didn't factor out a 16 from the get-go. It looks like he just immediately recognized that our original expression is itself a difference of squares even if we don't factor out a 16, and so let's re-write it. So our original expression, we could write as, so instead of writing, well I'm just going to write it like this, this is our original expression. 16 x-squared minus 64. That's the same thing as, 16 x-squared is the same thing as four x, the whole thing squared and then minus eight squared. So when you write it like this, it's clear that this is a difference of squares, so this is going to be four x plus eight times four x minus eight. Four x plus eight times four x minus eight. Once again, if this last step that I did doesn't make a lot of sense I encourage you to watch the video on factoring difference of squares where we go a lot more into the intuition of it. But when you see it this way you realize that Fatu also got an expression that is equivalent to 16 x-squared minus 64, so they both did.