 Since we're going to be solving quadratic equations using factorizations, there are some special factorizations that I want you to know. So one that's very common here is what's called the difference of squares. If you have two perfect squares and you take their subtraction, A squared minus B squared, this factors as A minus B times A plus B. Which people often remember and you can verify this by foiling, right? A times A is A squared. You're going to get a plus AB. You're going to get a minus AB. You're going to minus B squared. The ABs cancel out so you get A squared minus B squared. This difference of squares we're going to use all the time. The next one is what we call a perfect square trinomial because it's a trinomial because there's three terms in the polynomial. And it's a perfect square trinomial because it factors to be a perfect square A plus B squared. This will have the form that you start off with a perfect square, you end with a perfect square, and then the number in the middle is just double the square roots of these two numbers put together. If this number's a plus, then this will be a plus. If there's a minus sign right here, you get a minus sign right here. Let's look at such an example. Notice nine x squared minus six x plus two. Nine x squared is a perfect square. It looks like three x squared, right? Plus one is also a perfect square. You're going to get one squared. So this is my candidate for A and B right here. Now if you multiply this together, A times B, this should look like three x times one, which is equal to three x. Well, if you slap a two in front of that then, two, two, two, this should look like six x, which is exactly what we have right here. Now there is a negative sign, so when we factor this, we're going to get the following factorization. So nine x squared minus six x plus one, this factors as three x minus one squared is equal to zero. Now when you square something, that just means you repeat it in terms of multiplication. So you get three x minus one times three x minus one equals zero. Now the zero product property, zip zappity zoop comes into play. The only way this product would equal zero is that either three x minus one equals zero or three x minus one equals zero. You can see that the answer is kind of redundant here, that both possibilities are actually one of the same thing. So as we solve this, we're going to add one to both sides. We end up with three x equals one. We divide both sides by three and we get our solution is going to be x equals one third. Now in this example, this is actually what we refer to as a repeated solution or a repeated route because the factor in play here showed up twice. We only have to consider it once, but because it did show up twice, we call this a repeated route. As another example, you take two x squared minus 50 right here. This doesn't look like it has a special factorization, but if you factor out the two, then you get this difference of squares, five x squared minus 25, which that's going to factor as x minus five and x plus five. And then by the zero product property, you're going to get that x equals five or negative five. You can ignore the two because two can't equal zero. So the solutions must have come from those that involve the variable five and negative five.