 So let's talk about solving equations by factoring them, and this is based on a particular property of real numbers that's usually referred to as the zero product property, and that's the following. If I have two real numbers, a and b, and I know that their product is equal to zero, so that's our zero product, then the zero product property guarantees that either a is equal to zero or b is equal to zero. And in some very, very, very, very, very rare cases, in some occasions, this can actually be a useful way of solving equations. Generally speaking, this is actually very important, but as a practical way of solving equations, you'll find that it's very hard to implement this. So let's take a look at that. So here's an example. I want to solve this expression equal to zero, and a little analysis goes a long way. Again, remember that the type of expression is determined by the last thing we do. So let's read this parenthesis, say do stuff here, parenthesis, say do stuff here. Once we've done that, the last thing we do over on the left-hand side is multiply these two expressions, and so that tells me that on the left I have a product. So I have a product of things equal to zero, and so this is product equals zero, and I can use the zero product property. One of those two things is zero. Either x minus five is zero or x minus three is equal to zero. Well, that gives me two much simpler equations that I can solve. On the first case, I have adding five to both sides. I have x equal five. On the second case, I have adding three, and I have x equals three, and I have two solutions. Now, we do want to write these in a way that indicates we have two solutions. So typically, we say x equals five or x equals three. Well, let's take a look at another case for x cubed equals 12x squared, and again, a little analysis goes a long way. Order of operations says x put it down first, i cubed, i squared, then I multiply. So I have a product equal to a product. Okay, I don't have product equal to zero, so it doesn't do me any good to apply the zero product property. Well, I can do a little bit of algebra. I can at least get a zero by, well, I can subtract 12x squared from both sides. For example, that'll give me a zero over on the left, on the right-hand side, and something else on the right. So I subtract 12x squared, and I have my zero over on the right-hand side. I have a zero over on the right-hand side, and now I can go. Oh, wait a minute. Again, order of operations, a little analysis goes a long way. I have powers first, multiplication next, subtraction. The last thing I do over on the left-hand side is a subtraction. So this is a difference equal to zero, and I don't have any special properties about difference equal to zero. Well, here's a general rule in algebra. Anytime you get stuck on something, you can try one of two things, factoring or multiplying stuff out. And in this particular case, remember factoring is going to transform a sum or difference into a product. So over on the left-hand side, I will have a product equal to zero, and that's exactly what I want. So I take a look at these, and both of these terms have a factor of 4x squared. So I can remove that common factor. And again, a little analysis goes a long way. Parentheses are done first, multiplication, multiplication equal to zero. I have product equal to zero, and I can use these zero property. So one of these, in this case, three factors is zero. Either 4 is equal to zero, x squared is equal to zero, or x minus 3 is equal to zero. So I can set those factors up. So 4x squared is a product, again, and I know that one of those factors has to be zero. So let's see if this is 4 times x times x. Either 4 is zero, well, that can't be. Or x could be zero. So that first term, first factor there, gives me x equals zero as a solution. My second factor might be zero. Since x minus 3 might be equal to zero, and I can solve that x is equal to 3. And there's my second solution.