hello and welcome today we are looking at exercise 5.1.2 a related area investigation in this exercise we're going to be using areas of squares and rectangles to create a visual representation of one of the factoring patterns that we saw in the previous section so let's start by looking at this 5x5 grid the area of which is 25 which we get by multiplying 5 by five or simply by counting the number of squares next let's imagine that we took the four squares that we have highlighted here and slid them over here so that they're along the side of the rectangle and then we'll use that to form a new rectangle that has four rows and six columns which would have an area of 24 which again we get by multiplying the four and the six when we compare that to the original area we'll notice that there was one square that got left out when we switched from the blue square to the yellow rectangle at the same time we'll notice that the area of the blue square is one more than the area of the yellow rectangle so an alternate way to calculate the area of the rectangle would be to take the area of the Blue Square subtract out the one square that was left over which would leave us with 24 squares of course we got the 25 from doing 5 * 5 and we could take the area of this singular Square by doing 1 * 1 and if we go back to the way that we originally got the area of the rectangle the blue rectangle had five rows and we took one away to get the four and then we added a column by sliding those rectangles over so the four and the six were actually 5 minus one and 5 + one so we've visually demonstrated the difference of squares that we saw when we were factoring that indeed 5 - 1 * 5 + 1 is the same as 5^ 2 - 1^ 2 and then of course that reduced down to 2 four so to demonstrate this in more general terms we'll go ahead and let a lowercase a represent the length of the original Square the lowercase b will represent the length of the square that was left over after we rearranged our grid and then the a will represent the area of the newly formed rectangle our newly formed rectangle will always have a length of a plus b and a width of a minus B there the area of the newly formed rectangle will always be A+ B * a minus B which is the same as a^ 2 minus b^ 2 and that's all we have for this one I will see you guys next time