 So how can we solve a quadratic equation? Well, one way is to use factoring. One way to solve quadratic equations relies on what's known as the zero product property of real numbers. Suppose A and B are expressions of real numbers. Then the product AB equals 0 if and only if A is 0 or B is equal to 0. In other words, if I have a product equal to 0, I know that one of the factors must be 0. So for example, suppose I want to solve the equation x minus 5 times x plus 8 is equal to 0. So again, a useful thing to be able to do is to identify what type of expression we have. And over here on the left-hand side, the last thing we have to take care of is this product. And that means that we have a product equal to 0. And since we have a product equal to 0, we know the zero product property applies. And so we know that either x minus 5 is equal to 0 or x plus 8 is equal to 0. And so this gives us two equations to solve. So solving these separately, first, x minus 5 equal to 0 can be solved as x equal to 5. And then x plus 8 equals 0 can be solved, giving us the solution x equals negative 8. And so our two solutions are x equals 5 or x equals negative 8. And this leads to the following idea. We can use this idea if we factor the quadratic expression. However, the important thing to remember is that factoring is the hardest easy problem in mathematics. It's easy to explain what we want to do, but generally very hard to do it. If you choose to solve a quadratic equation using factoring, you have chosen to solve it in the hardest way possible. So let's try to solve x squared minus 4x minus 12 equals 0. We want to write x squared minus 4x minus 12 as a product of two things. Now, if we look at this right-hand side, when we expand, we want those first terms to give us x squared. And so that means they have to be x and x. Since we don't actually know the factorization, we don't know what the second terms are, so we'll just write them as plus a plus b. But the important thing we know is that the product of these two last terms, a times b, must be negative 12. So now this factoring problem comes down to determining what a and b are. In order to do that, we go through all possible factors and see which one gives us the correct middle term minus 4x. Now, to keep ourselves organized, we'll list all these possible factors. So things that multiply to negative 12, 1 and negative 12, negative 1 and 12, 2 and negative 6, negative 2 and 6, 3 and negative 4, negative 3 and 4. And at this point, the only thing we can do is try them until we find a factorization or determine we don't have a factorization. So if we try 1 and negative 12, then maybe x squared minus 4x is x plus 1 times x minus 12. But the only way we can answer this question of whether or not this is a factorization is we have to expand the right-hand side and we find. And so this is not the factorization we're looking for. What if we try negative 1 and 12? Is x squared minus 4x minus 12 equal to x minus 1 times x plus 12? The only way we can determine this is to expand and check. And it isn't. What about 2 and negative 6? Is x squared minus 4x minus 12 equal to x plus 2 times x minus 6? We have to expand and check. And it is. So x squared minus 4x minus 12 factors as x plus 2 times x minus 6. And we're back at the point where we have a product equal to 0. So we know that one of the two factors is 0 and we can solve. One important thing to keep in mind is that it's vitally important to have our quadratic expression equal to 0. So for example, in solving x squared plus 2x minus 3 equals 21, perhaps you're walking along and out of the sky, the factorization x minus 1 times x plus 3 falls and hits you on the head. And we can apply our 0 product property. Since we have a product equal to 0, we know that one of the factors is 0. Wait, the product is 21. We don't have a 21 product property. Lots of things multiply out to 21. So even though we have a factorization, this factorization is completely useless. So that means we need to rearrange our equation a little bit. So we have a 21 over on the right-hand side. We don't want that there. We want it to be a 0. So we can subtract 21 from both sides. And that gives us a different quadratic expression to factor. So again, we'll try to factor x squared plus 2x minus 24. Well, since our first terms multiply out to x squared, that means they can both be x. And our second terms, we don't know what they are, but we do need them to multiply to our constant negative 24. And so whatever a and b are, we need a b to be minus 24. So again, we'll try every possibility and see which one gives us the correct middle term. To make sure we check every possibility, we'll list them. And again, the only way to determine what our factorization is is to try out every possibility until we find one that works or until we run out of possibilities. So let's try 1 and negative 24. The only way to check is to expand and no. How about negative 1 and 24? We expand and no. How about 2 and negative 12? We try it out and expand. How about negative 2 and 12? We expand. How about 3 and negative 8? We expand. Since we would never have to do a problem that takes a lot of time and effort, negative 3 and 8 has to work, right? How about 4 and negative 6? Ah, come on. Negative 4 and 6. Nope. Oh, wait, wait, wait, wait. That works, that works, that works. OK. So we know that x squared plus 2x minus 24 factors as x minus 4 times x plus 6. And once again, we have a product equal to 0. So we know that one of the factors is 0. Either x minus 4 is 0 or x plus 6 is equal to 0. So we'll solve these two equations and get our solutions. x equals 4, x equals negative 6.