 4,000 years ago in Mesopotamia students were giving canal problems. These translate into quadratic equations, and we can solve them using the quadratic formula. So, if we translate one of these problems into modern language and modern terms, we might get something like the following, the length of a rectangle is 25 meters more than its width, the area is 300 square meters, find the length and width. Now, since this is a geometry problem, because it involves length, width, areas and rectangles, let's go ahead and draw a picture. Now, since this is an extremely complicated picture, we should be sure to label all the sides. We have a length and width, and because the length is longer, we'll make the longer side the length and the other side the width. And we can translate the given information into an equation using L for length and W for width, the length, 25 meters more than the width, can be written as, and we know the area, which is the product of length and width. So we know that LW is equal to 300. So remember, equations with one variable are easier to work with. So let's see if we can eliminate one of these variables. And remember, equals means replaceable. So here, we have a way of replacing L with W. So we can substitute into our other equation. Now, our next step might not be obvious, but remember, if in doubt, expand or factor. And it's worth keeping in mind that factoring is hard, so let's do the easy thing and expand this product. And now, we have an equation with a square, so we can use the quadratic formula as long as our equation has the correct form. So we'll need to rewrite our equation so that everything is equal to zero. So now, this is in the form, we can apply the quadratic formula. So let's write that down. And our equation has A equals 1, B equals 25, and C equals negative 300. So filling those in gives us. And remember, this plus or minus really means that we have two solutions. So let's split this and write them as two solutions and evaluate. And since W is a width, only the positive solution makes sense. So let's go ahead and write that down. Now, it's important to remember we should always answer the question in the same language it was asked. So our answer should be, no, not that. In English, we need the length and the width, and we only have the width. If only we knew of some relationship between the length and the width. Oh, we do. We do have the relationship between the length and width. And remember, equals means replaceable. Since we know the width, we can replace it and compute the length.