 A useful property of the real numbers is known as the zero product property. If a and b are real numbers with a times b equal to zero, then we know with absolute certainty that either a is zero or b is zero. This helps us with solving equations in the following way. Suppose I want to solve an equation like x minus 5 times 3x plus 2 equals zero. Since this is a product equal to zero, we know that either x minus 5 is zero or 3x plus 2 is equal to zero. But we can solve those separately. So solving x minus 5 equals zero gives us the solution x equals 5. Solving 3x plus 2 equals zero gives us the solution x equals minus 2 thirds. So either x is equal to 5 or x is equal to minus 2 thirds. Now it's important to remember that theorems in mathematics are power tools, and like any power tool, if you don't use it correctly, the result could be catastrophic. Well, safety instructions are for wimps, so we'll set each factor equal to zero and solve, so x minus 2 is zero or x minus 3 is zero, and so x equals 2 or x equals 3. But since the product is not equal to zero, these answers are completely wrong. And the problem is this is not a product equal to zero, so we can't use the zero product property. And since this is not a product equal to zero, we should do a little algebra to try and rewrite this as a product equal to zero. So we can expand. We'd like it equal to zero, so let's subtract 2 for both sides. We want a product equal to zero, so that requires us to factor, and now we have a product equal to zero, so one of the factors must be zero, either x minus 4 is zero or x minus 1 is zero. So we'll solve and get our two solutions. Or how about something like this? A useful thing to remember is you can always use algebra to eliminate fractions. So here I have this fraction one-third. If I multiply everything by 3, I'll get rid of the fraction. So let's do that. And we could factor, but since this is not a product equal to zero, it's a waste of time. And remember, factoring is the hardest easy problem in mathematics. You're going to end up spending a lot of time factoring when you need to factor. You really don't want to waste a lot of time factoring when it doesn't do you any good. So remember, don't waste time factoring unless you know why you're factoring. We want to get product equal to zero. We have to have something equal to zero to begin with. So we'll rewrite our equation. And now if we factor, we'll get product equal to zero, which will help us solve the problem. So now we can try to factor. And so now we have a product equal to zero. So one of the factors must be zero. Either x minus 3 is zero or x plus 9 is zero. So solving these two equations gives us our solutions, x equals 3 or x equals minus 9. Or this horrifying expression. Well, we know we want it equal to zero, so we'll subtract 84 from both sides. Then factor. And remember, it's always helpful to remove common factors. So we might begin by removing a common factor if we can find it. Since we have a product equal to zero, one of the factors must be zero. Either 12 is equal to zero or x squared minus 6x minus 7 is equal to zero. Now despite what this guy says, and even if his friends say it, 12 is not equal to zero. And that means we have to solve this equation x squared minus 6x minus 7 is equal to zero. So let's try to factor that. And again, we have a product equal to zero. So one of the factors must be zero. So either x minus 7 is zero or x plus 1 is equal to zero. So solving gives us our two solutions, x equals 7 or x equals negative 1.