Quadratic equation from geometric context

0:00- [Instructor] We're told, "A photo for a yearbook is 8 inches by 10 inches. The photo is placed in a frame so that there is a uniform border of thickness x inches all the way around. The total area of the photo and the visible frame border is 168 square inches.

0:18Write an equation relating x to the total area. And what does a solution to this equation represent?" So pause this video, have a go at that before we do it together. Okay, now let's do it together. And I'm just going to draw it out to visualize what's going on.

0:32So we have a photo that's 8 inches by 10 inches. So let's say that that right over there is 10. I'm gonna assume everything is in inches. This over here is 8. Let me make it so we can see the photo. So the photo would look something like that. It's a yearbook photo.

0:53So maybe I'll put a little face here. All right. (laughing) So then they are, there's a frame with a uniform border thickness of x inches all the way around. So the frame, let me draw the frame here, it's uniform all the way around. So it has the same thickness all the way around.

1:12I'm just obviously hands-drawing it. I'm able to draw it straight because of a little help from the computer. Actually, let me make it a little bit wider, so it looks like it's uniform all the way around. That's pretty close. So what they're saying is this distance right here is x, this distance right here is x, that distance right over there is x, and that this distance right over there is x.

1:35Now to figure out or to have an equation for the total area, we want to know what this dimension is, the height, and what the width is. Well, the way I've set it up, what is the height here? You're gonna have x plus 10 plus another x. So this entire distance is going to be 2xs plus 10, 2x plus 10.

1:58And this entire width is going to be this x plus 8, plus this x. So 2x plus 8. And the entire area is gonna be the product of these two things. So if I take the product, if I get 2x plus 10 times 2x plus 8, that's gonna be the area, the total area. But they told us what the total area is.

2:23It is 168 square inches. So this is going to be equal to 168. So that's the first part. Now, what does the solution to this equation represent? Well, if we were to solve this for x, what essentially we would figure out is what are X values that would make this, that would make all of these statements up here true.

2:47So what could actually be the width of the frame? Because remember, that is what x is. It is the width of the frame. So width. So I'll just write width of frame. And we're done.