Difference of Two Squares
So if we start off with a square, that is, dimensions 𝑎 by 𝑎, we know the area of this square will be 𝑎 squared. This is the difference of two squares. So let’s take away another square from this square. So, say that we want to find the area of the white shape. We can see that we’re taking away the area of the blue shape. So we’ve got 𝑎 squared and we’re taking away 𝑏 squared. Or we know that by cutting down here, that gives us different dimensions on the sides for different parts. This part is of course 𝑎 minus 𝑏, because the whole length is 𝑎 and we can see just that part at the bottom is 𝑏, so that part is 𝑎 minus 𝑏. And then, that’s of course is 𝑏. But if we take that shape and move it up on the side, we know that we can because our dotted length is as well 𝑎 minus 𝑏, we get a different shape. We’ll end up with this rectangle. So we can see the side length, again, is 𝑎 minus 𝑏 and the top has a length of 𝑎. And then we’re adding on the length of 𝑏. So it’s 𝑎 minus 𝑏 is one dimension, and 𝑎 plus 𝑏 is the other. We know that the area of our original shape was 𝑎 squared minus 𝑏 squared, because it was the big square take away the smaller square. So our new shape would be 𝑎 minus 𝑏 all multiplied by 𝑎 plus 𝑏, as it’s a rectangle. And putting these two together, gives us the difference of two squares.
Let’s have a look at it algebraically as well. So let’s expand these two brackets using FOIL. We’ll use the first terms multiplied together, which will be 𝑎 multiplied by 𝑎, gives us 𝑎 squared. Then 𝑎 multiplied by minus 𝑏 gives us minus 𝑎𝑏. 𝑏 multiplied by 𝑎 gives us plus 𝑎𝑏. And finally, 𝑏 multiplied by minus 𝑏 gives us minus 𝑏 squared. So we can see our middle terms cancel out, and we’re left with 𝑎 squared minus 𝑏 squared. So we’ve seen it geometrically and also algebraically. Let’s actually apply this to a problem.
So, we need to find what 𝑎 is. So what is being squared to give us sixteen 𝑥 squared. Well we know that four squared is sixteen, and 𝑥 squared is 𝑥 squared. So that’ll become four 𝑥 all squared. So four 𝑥 is the 𝑎 value. Now looking at our 𝑏. Well what squared give us nine, three. So that’ll become three 𝑦 all squared is nine 𝑦 squared. And that is 𝑏. So substituting it in for the relation, we’ve got 𝑎 plus 𝑏 all multiplied by 𝑎 minus 𝑏. So we have four 𝑥 plus three 𝑦 all multiplied by four 𝑥 minus three 𝑦. Let’s look at one more example.
So looking at this example, again, you can say what squared gives us thirty-six 𝑥 squared. Well that’ll be six 𝑥 all squared. And then, what squared gives us thirty-six 𝑦 squared. Again, that’ll be six 𝑦 all squared. So we can just apply the formula putting 𝑎 as six 𝑥 and 𝑏 as six 𝑦, and we’re done. But in this case, there’s actually another way we could’ve done it. If we look back to the first part of the question, thirty-six 𝑥 squared minus thirty-six 𝑦 squared, we can see that they’ve both got a common factor, the greatest common factor, of thirty-six. So we can factor the thirty-six first. And that would’ve given us this. Well then in this case, our 𝑎 is 𝑥 and our 𝑏 is 𝑦. So we could substitute that into the relation, and we would get this. Now these are exactly the same. So the answer we got using the first method and the second method are exactly the same. Because if in the first method we factored six out of each individual bracket, we would then have to do six multiplied by six, which we of course know is thirty-six. So we get the same thing either way but this second method is a fully factored method. So in summary, the only thing we really need to remember, in the difference of two squares, is the relation, and just working out how to spot it when we’re given an option. So when we see a square number subtracted by another square number, obviously with variables that squared as well, then we can just go straightaway and use this relation.