 Let's take a look at some more general problems that involve completing the square. So in general, if I want to solve an equation of the form ax squared plus bx plus c equals zero, what I can do is I can do the following. First of all, I can rearrange things so that all of my x terms are in one site and then everything else is on the other, and then I'll complete the square. And I can do this in one of two ways. Ultimately, you want to be able to do this algebraically, but that is the hard way at first. The other possibility is you can do this geometrically, which is really the easier way. Eventually, you'll want to switch over to the algebraic method, but geometrically is a good way to start. So for example, let's take the problem solve 3x squared plus 6x minus 10 equals zero. And one of the nice things about this is that in some ways, you've already learned that if you want to solve an equation, first thing you do is you get all of your x terms onto one side. And so we've done that. So I have 3x squared plus 6x equals 10, and I'll draw the picture. So again, we could still represent our two sides of this equation as the areas of geometric figures. So this 3x squared plus 6x, I can represent that as a rectangle. And again, my first term, 3x squared, maybe that's x by 3x. My second term, 6x, that's 6 by x. They have the same width, so I can attach them to each other. And so there's my 3x squared plus 6x rectangle. And that has area equal to 10. Now, if I don't think about what I'm doing, then I might try to complete the square as follows. I am ignoring the picture for a second. I'll take half of this 6x, I'll split it in half, and I'll slide that extension over and complete the square at that point. Well, the problem is that we don't see it if we just look at the algebra. But if we look at the geometry, the problem is when I slide that piece over, it's x by 3y, so it fits about here on our picture. It does not fit this entire area here. So if I want to complete the square, first of all, I have a problem. I don't have things with equal width. This is x plus 3, this is 3x plus 3. This is not even a partial square. This is something that is a nomen that I have to add a bunch of pieces together to make into a square. So what can we do? Well, the problem is that this portion isn't a square. So let's go ahead and make it into one. And there's a couple of ways we might do this, but multiplication is easier than division. So let's just take three of these entire figures, 1, 2, 3. And now I have a portion here that's 3x by 3x and then whatever the leftover is. And now I'm back at the same place I was with the other completing the square problems. I have a square and an extension. So if I want to complete the square, I'll split the extension. I'll slide these pieces over and attach them down here. And again, in this case, these pieces here, remember this rectangular block here is x 3x by 3. And so it's 3x, it'll fit on this side by 3. And now I have a square that's 3x. I have a partial square that's 3x plus 3 by 3x plus 3. And I'm just missing this little piece here to make it into a perfect square. So let's figure out what our areas are so far. Now remember 3x squared plus 6x was something with an area of 10. So here's our 3x squared plus 3x plus 3x. If I take a look at the blue pieces, those together have an area of 10. Likewise, 3x squared plus 3x plus 3x, that's 3x squared plus 6x. The red pieces have area 10. And the green pieces also have area 10. And that little byte that we're missing here, this is a 3 by 3 region. That's going to have area 9. And so my entire figure has area 10, 20, 39. And that's 3x plus 3 squared has area 39. And that tells me algebraically that the square root 3x plus 3 is plus or minus square root 39. And that breaks into the two possible solutions for my original equation.