 So we've learned some special factorizations like the difference of squares and a perfect square trinomial. We've seen that you can factor out GCDs, but it turns out that factoring quadratics can be a little bit more difficult than that. Because after all, when you multiply out a binomial, if you have something like x minus 2 times x plus 3, when you multiply these things out, you're going to get an x squared, which tells us we're going to have a quadratic here, right? You're going to get x times 3, you're going to get a negative 2x, and then you're going to get a negative 6 right here. Combining these middle terms, you're going to end up with x squared plus x minus 6. And this is what we typically refer to as the FOIL method, right? Well, the thing is if I gave you x squared plus x minus 6 and I had to factor that, we have to essentially reverse this FOIL method to get back the factorization that's up here. How does one do that? Well, if we do this process backwards, the idea is this middle term was actually the combination of two like terms. So we have to replace the x with a 3x minus a 2x, and then we factor from there, right? But how do you decide what to break these things apart from? What clues do we have about these numbers 3 and negative 2 right here? Well, you'll notice that the 3 and 2 came about from taking the product of the outside terms and the product of the inside terms, for which you take the first term times, the first term times the last term, that's where the 3x came from, right? And then you take the inside terms, we'll talk together to get negative 2x right there. So, okay, this is the combination of the outside and the inside terms, but can we find any evidence about them? Well, what we have in front of us are the first terms and the last terms, the negative 6 right here. And so it turns out that we can kind of work this thing backwards in the following way. What we're going to do is we're going to take these numbers right here, the first terms and the last terms. We're going to multiply them together. You take the coefficient there. You take 1 times negative 6, and that's going to give you a negative 6. Taking this negative 6, we're going to look for another factorization of negative 6, such as 3 and negative 2, right? Notice that's negative 6 factors as 3 and negative 2. The significance of this though, is that when you combine 3 and negative 2 together using addition, 3 minus 2 equals 1. So, when it comes to this reverse foil process, what we want to do is we're going to take the product of the leading coefficient in the constant term. In this situation, this would give us a 4, a negative 4, 2 times negative 2 equals negative 4. We then have to look for factors of negative 4 that add up to be this middle number right here, the 3. And so this takes a little bit of experimentation, but notice that 4 times negative 1 multiplies to give you negative 4, but it adds together to give us 3. And so such a pair, 4 and negative 1, we're going to refer to this as a magic pair. And I like to think of, like Hercules is going to search for the magic pair upon the tree of Poseidon or something like that. I know it's not the right type of pair, but you get the idea, it makes it much more majestic that way. But we're looking for this magic pair, a pair of numbers which will multiply together to give us the first and last term, but adds together to give us the middle term. Because if we do that, we're going to replace the middle term with this magic pair. So we're going to see that 2x squared plus 4x plus a negative x right here minus 2. We're going to break it up here because after all, if you put these things back together, 4x and negative x gives you a 3x. So we've just broken the number up like that. Well, why are we going to do that? What do we do next? The thing to do next is we use a technique which is called factoring by groups. If you have four terms in your polynomial, you're going to put the first two into a group and you're going to put the second two into a group. That's all there is, the first two and the last two. And then amongst those groups, you look for the GCD. So we couldn't find a GCD for the whole thing, right? 2x squared 3x minus 2. There's no factor of x that's common to all of them and there's no common divisors 2, 3, 2, right? We can't pull a GCD out of everything, but if you look at the groups, right? The first group, 2x squared plus 4x, you could factor out a 2x. That would leave behind x plus 2. And then looking at the second group right here, we could factor out a negative one. That's common to both of them. Take out the negative one, you're left with an x plus 2. And then notice what we have here is that what's left over is the x plus 2 and the x plus 2. Even though it's a binomial, this is now a common factor which we can then factor out from those. And this then gives us the 2x minus 1 and then this also gives us an x plus 2. And this gives us the factorization. This will be mean by this reverse foil method. You look for a magic pair and then you factor by groups to finish this thing off. And if you have any doubt if this is the right factorization, work it out again. 2x times x is an x squared. Negative one times two is a negative two. And then you're going to get a 2x times two, which is a 4x, a negative x, which adds together to give us a 3x. And Bob's your uncle, that is exactly the thing we started off with. We have the correct factorization. And so when we set this thing equal to zero, we can now apply the zero product property. To see that 2x minus one equals zero or x plus two equals zero. Solving the first one, we add one divided by two. We see that x would equal one half. And then the other one, just to track two, we're going to get that x is negative two. And so we've now found the correct solutions to this quadratic equation. Alright, so that's kind of a, you know, it can be a challenging concept at first. Let's do another example to illustrate this. Let's solve the quadratic equation 2x squared equals x plus three. When it comes to quadratic equations, we always put in the standard form first. We didn't move all of the terms to the other side of the equation. So we get 2x squared minus x minus three equals zero. You don't have to put things on the left-hand side, but I really like the leading coefficient to be positive. That is the coefficient of the x squared. So that's why I moved everything to the left-hand side. So now I do a quick run-through. I don't see any GCDs to factor out. There's no special factorization. This is not a perfect square trinomial. Difference of squares only applies when you have two terms, a binomial. So that doesn't apply in this situation. So we're going to have to do this reverse foil process. So look at the first coefficient and the last coefficient. This is going to give us two times negative three, which is negative six. We need to find factors that add up to B, factors of negative six that add up to negative one. Now this is really close to what we did earlier, right? You'll actually notice that, wait a second, we already have a magic pair, right? Notice that negative three plus two is equal to negative one. Oh, sometimes the magic pair is just staring in front of you. You just have to pluck it. It's this proverbial low-hanging fruit, horrible pun I know, but it'll leave a message in your mind there. So we're going to break the negative one up to be negative three X with a two X. So you're going to get two X squared minus three X. That's the first group. And then the second group is going to be a two X minus three. This is equal to zero. Now sometimes people ask, well, which of the numbers do you put first? So I do negative three and then two or two and then negative three. It turns out it doesn't matter. You'll get the correct factorization, whichever way you put them. The intermediate steps will look a little bit different, but in the end you'll get the correct thing. So when you look at the first one, what's your common divisor? The biggest thing you can pull out is an X. So we factor out an X that leaves behind two X minus three. When you look at the second one, what's the biggest thing you can take out? Well, we're tempted to say, oh, there's nothing common to both. I can't take out anything. But that's not exactly true. We can always take out a one because again, when you factor out a one, it doesn't really do anything. So when you say like there's nothing in common, what you really mean is that one is the greatest common divisor. And as we must factor out something, we're going to factor out a one. But this is fortuitous because since if you've, if you chose your magic pair correctly, then these terms right here should be identical. And if we factor out the two X minus three as that common divisor, you're going to get two X minus three and then X plus one equals zero. And so then by the zero product properties, zip zappities up, we're going to get that two X minus three equals zero or X plus one equals zero. And solving these two linear equations, the first one will add three divided by two. We're going to get that X equals three halves. And then for the second one, you're just going to minus one on both sides and see that X equals negative one. And this is going to give us the solution here. And so this idea of reverse foiling, that is, you find a magic pair and then use the factoring by group techniques. This can help us factor many, many, many quadratic polynomials and help us solve many, many quadratic equations. It does depend on the fact that we have to be able to find a magic pair, which is not always an easy thing to do, which is why we'll present some alternative methods shortly. But when a magic pair is easy to find, then it turns out this is a very, very effective way of solving a quadratic equation.