 Hi, this video is called Dimensions of a Shape. It's really important in problems to just take some time to read the instructions and figure out what you're supposed to do. This problem says the sides of a rectangle whose area is 40 units squared are x plus 2 and x plus 5. Find x, so that's the first thing we'll have to do, and the dimensions of the rectangle. So it looks like a problem like this, there's two parts to it, and I always like to start by drawing a picture. I think that's important. So drawing just a little rectangle, it says the side lengths are x plus 2 and x plus 5. It doesn't tell me which is the length or the width. In reality, it doesn't really matter. And it says that the area is 40 units squared. So they kind of gave us everything, but now they want us to figure out what x is. Okay, so think about it. What formula do you use for area of a rectangle? Well, you use base times height. So now what I can do is I can rewrite this formula, but instead of a, put in 40 instead of the base, put in x plus 2, and instead of the height, put in x plus 5. So let's go ahead and do that. All right, so here I am. All I did was instead of a for area, I replaced it with 40. The base was an x plus 2 and the height was an x plus 5. I put that into the formula. So now to solve for x, I've got to get the x alone. So it looks like I'm going to have to spend a little bit of time on the right-hand side of my equation to start simplifying it. When you've got two binomials that need to simplify together, we have to foil. Remember first, outside, inside, last. So we're going to have x squared plus 5x plus 2x plus 10. Which simplifies to x squared plus 7x plus 10. And all along, that equals 40. Now when I have this x squared and an x term, this trinomial, to solve for x, I'm going to have to factor. And when you factor, you've got to get everything to the same side of the equal sign. And when you do this, make sure that your x squared term stays positive. So the easiest thing to do here is to subtract 40 from both sides. So that gives us 0 equals x squared plus 7x minus 30. So at this stage, hopefully you're comfortable with factoring now. We want to figure out what multiplies to give us negative 30 and adds to give us 7. Well factors of 30 are, let's see, 30 in 1, 10 in 3, 6 in 5. Will any of these multiply to a negative 30? Sorry about that. It looks like to me that the 10 and the 3 will be our best bet because we have a positive 10 and a negative 3 add up to 7 and a positive 10 times a negative 3 is negative 30. So I'm going to do x plus 10 and x minus 3. So when I set both of these things equal to 0, x plus 10 equals 0 and x minus 3 equals 0. I end up with x equals negative 10 and x equals positive 3 for my answers. Now, since a picture is involved, we've got to analyze our answers a little bit. If I plugged in 3, I would get side lengths of 3 plus 5, which is 8 and 3 plus 2, which is 5. They're both positive and when I multiply 8 times 5, it gives me 40. So that tells me that 3 is a logical answer. What happens when we deal with negative 10? If you plug negative 10 in, you get negative 10 plus 5, which is negative 5 and negative 10 plus 2, which is negative 8. You are very well aware that side lengths can't be negative. So my answer of x equals negative 10 just isn't logical. It's not going to work. So our first answer for finding x is x equals 3. Then it wants me to give the dimensions of the rectangle. Well, I've already found that too. I took my x equals 3 and I plugged it in 3 plus 5 and I got 8, 3 plus 2 and I got 5. So my dimensions is we have a rectangle that's 8 units by 5 units. That would be the dimensions.