Experienced teacher shows how to factor difference of squares!

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Before we learn how to factor a difference of squares, let's talk about the meaning of these three words: difference of squares. So what is the difference of squares? The three words here actually already tell us the exact meaning. The difference of squares is the difference of who, of squares. That's it. So for example, the difference of x square and y square. Or the difference of x square and 5 squared. Or, difference of a square and b square. That's it. Okay, now we know the meaning of 'difference of squares, ' let's talk about how to factor a difference of two squares. For example, to factor the difference of and b square, this is what I want you guys to do. Write out two brackets. And inside the first bracket, I want you to put a positive sign. And in the second bracket, I want you to put a minus sign. This is what you do, and it's very cool, and it's very very easy. So guys, here we have the difference of square of the first guys, and square of the second guy. Who is the first guy? A. And this is what I want you to do: inside each bracket, put a as the first guy. Now here we have the square of the second guys. Who is the second guy right here? B. Inside each bracket, let's put b as the second value. That's it. That's the formula for factoring a difference of squares. A square minus b square can be factored as a plus b times a minus b. So for example, now if I change the first guy, instead of a to 3a, now the the first guy is 3a, no problem. The first guy in each bracket now must be 3a, just like that, very easy. Now what is the second value, now, is, say 5b. Easy! Inside each bracket, make sure the second guy is 5b. So guys, here we go, 3a squared minus 5b squared is 3a plus 5b and 3a minus 5b, right. So now, let me show you a typical question you will see on your test: factor this guy, 9a squared minus 16. Well, let's set it up. If we can somehow rewrite this question as the difference of two squares. And now, to factor the difference of two squares, we can write it as the product to two brackets. Now remember inside the first bracket we have to put plus, right, and inside the second bracket we have to put minus. Right? Cool. Now, guys, the question you have to ask yourself is, with the first guy here, 9a squared, can we write this as the square of someone? Sure. Because 9a squared is the square of 3a, right? Let's double check. What's 3 square? 9. What's a square? A square. That's it! Okay, next. Now, how about the second term here, 16? Can we write 16 as the square of somebody? Of course, 16 is the square of 4. So now, if we rewrite this question as the difference of two squares, now we know how to factor this, it's super easy, right? So here we go. Who is the first guy, here? 3a. So we want to make sure that in each bracket the first guy is 3a, 3a. Now, who is the second guy right here? 4. So we just have to make sure that inside each bracket, the second value is 4. And we are done! So to factor 9a squared minus 16 first of all we rewrite the question as the square of 3a minus the square of 4. And once we write it this way, we can do factoring very, very easily. So 9a squared minus 16 equals 3a plus four times 3a minus 4. We're done. I know I've always told you guys you should never memorize math. You should try to understand it, and hopefully after you understand it, it sticks right here, and you don't have to memorize anything. So now we know how the formula of difference of squares works, and let me prove the formula. It's actually very easy to prove the formula. All we have to do is just multiply everything out by doing FOIL. Do you still remember FOIL, F-O-I-L? FOIL stands for what? F stands for First, O stands for Outer, I stands for Inner, and L stands for Last. So whenever we have two terms multiplied by two terms, this is what we do: FOIL. Okay. So now, here it is: first thing we do is what? Multiply the first terms inside each bracket, right? So guys, this is the first. And guys, if we multiply the first terms in each bracket, we have a times a, we have a squared, right.

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