Factoring quadratics with a common factor | Algebra 1 | Khan Academy

Avril was trying to factor 6x squared minus 18x plus 12. She found that the greatest common factor of these terms was 6 and made an area model. What is the width of Avril'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 6x squared minus 18x plus 12, and she figured out that the greatest common factor was 6. So one way she could think about it is this could be rewritten as six times something else.

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 6 and the width, let's just call that the width for now. So this is the width right over here. If you multiply 6 times the width, and maybe I could write width right over here, if you multiply 6 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, 6x squared minus 18x plus 12. And that's exactly what's drawn here. Now, what's interesting is that they broke up the area into three sections. This pink section is the 6x squared, this blue section is the negative 18x, and this peak section is the 12.

Of course, these aren't drawn to scale because we don't even know how wide each of these are because we don't know what x is. So this is all a little bit abstract, but this is to show that we can break our bigger area into three smaller areas.

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, 6 times what is 6x squared? Well, 6 times x squared is 6x squared, so the width here is x squared.

Now, what about this blue area? A height of 6 times what width is equal to negative 18x? So let's see, if I take 6 times negative 3, I get negative 18, but then I have to multiply it times an x as well to get negative 18x. So 6 times negative 3x is negative 18x.

And then last but not least, 6, our height of 6, times what is going to be equal to 12? Well, 6 times 2 is equal to 12. So we figured out the widths of each of these sub-regions, and now we know what the total width is.

The total width is going to be our x squared plus our negative 3x plus our two. So the width is going to be x squared, and I can just write that as minus 3x plus 2. So we have answered the question, and you could substitute that back in for this, and you could see if you multiplied 6 times all of this. If you distributed the 6, you would indeed get 6x squared minus 18x plus 12.