Factoring quadratics with a common factor

- [Instructor] Averil was trying to factor six x squared minus 18x plus 12. She found that the greatest common factor of these terms was six and made an area model. What is the width of Averil's area model? So pause this video and see if you can figure that out, and then we'll work through this together. All right, so there's a couple of ways to think about it. She's trying to factor six x squared minus 18x plus 12, and she figured out that the greatest common factor was six. So one way you could think about it is this could be rewritten as six times something else. And to help her think about it, she thought about an area model, where if you had a rectangle, if you had a rectangle like this, and if the height is six and the width, let's just call that the width for now, so this is the width right over here. If you multiply six times the width, maybe I could write width right over here, if you multiply six times the width, you multiply the height times the width, you're going to get the area. So imagine that the area of this rectangle was our original expression, six x squared minus 18x plus 12. And that's exactly what's drawn here. Now, what's interesting is is that they broke up the area into three sections. This pink section is the six x squared, this blue section is the negative 18x, and this peach section is the 12. And, of course, these aren't drawn to scale, 'cause we don't even know how wide each of these are 'cause we don't know what x is. So this is all a little bit abstract, but it's to show that we can break our bigger area into three smaller areas. And what's useful about this is we could think about the width of each of these sub-areas, and then we can add them together to figure out the total width. So what is the width of this pink section right over here? Well, six times what is six x squared? Well, six times x squared is six x squared, so the width here is x squared. Now, what about this blue area? A height of six times what width is equal to negative 18x? So let's see, if I take six times negative three, I get negative 18, then I have to multiply it times an x as well to get negative 18x. So six times negative three x is negative 18x. And then, last but not least, six, our height of six, times what is going to be equal to 12? Well, six times two is equal to 12. So we figured out the widths of each of these subregions, and now we know what the total width is. The total width is going to be our x squared plus our negative three x, plus our two. So the width is going to be x squared, and I can just write that as, minus three x, plus two. So we have answered the question. And you could substitute that back in for this, and you could see, if you multiplied six times all of this, if you distributed the six, you would indeed get six x squared minus 18x plus 12.