 Let's solve the quadratic equation x squared plus five x equals 24. We're gonna do this by factoring. Now to factored, we have to set the right-hand side equal to zero, which means we're gonna put this thing in standard form. So subtract 24 from both sides and we get x squared plus five x minus 24 equals zero. If I check the quick factoring techniques, right, because this is a trinomial, we can't do a difference of squares. It's not a perfect square trinomial. 24 is not a perfect square. And there's also no common divisors I can see amongst the three terms. So we're now gonna result to using this reverse foil method we learned before. So we're gonna take the first coefficient, which is really just a one, and times it by negative 24. You're gonna see that itself is just negative 24. And you wanna find factors of negative 24 that add up to be five. 24 has a couple of factorizations, like one and 24, two and 12, four and six, eight and three. Because it's a negative 24, one of the factors will have to be negative, the other one has to be positive. And after you search around for a while, you can see that eight and negative three is our magic pair. Because eight times negative three is equal to negative 24. And then eight minus three is equal to five. So we're gonna break up our magic pair. As we're gonna break up five into the magic pair, eight and negative three, we get x squared plus eight x. And then we're gonna get for the next group, a negative three x minus 24. Factor out GCDs in the first group. The only thing we can take out is an x, that leads behind x plus eight. And then the second group, if the first coefficient is negative, always take out a negative there. And you're gonna take out a negative three, that leads behind x plus eight. You're gonna see that the x plus eights match up because we do in fact have a magic pair. Factoring out the x plus eights, we're gonna get x plus eight. And we're gonna get x minus three, which is equal to zero. Setting each of these two, we're gonna set each factor equal to zero, you know, in accordance with the zero product property, zip zappities up. And therefore we see that x would equal negative eight and three as the solutions here. You know, when you look at that number, it's like, hmm, negative eight and three, I feel like I've seen those numbers somewhere before. Well, that almost was our magic pair. Just the negatives were in the wrong spot. But actually, when you look at here, x plus eight and x minus three, the magic pair is the factorization. If you just put an x in front of it, right? You got x plus eight and x minus three. And then when you set x plus eight equal to zero, you'll switch the signs when you move eight to the other side. Same thing with the negative three. And so what happened here? Like this, the magic pair essentially gave you the answers except that you have to switch the signs. We had a negative eight and a three. What happened here? Well, it turns out that this example is a very special type of quadratic equations. What we call a monic quadratic where the leading coefficient here is a one. If the coefficient of x squared is a one, it turns out this factoring technique is dramatically simplified. Once you find the magic pair, you can immediately jump to this factorization, x plus the first member times x plus the second member of the magic pair. That is, once you have the magic pair, you essentially have the factorization. We can skip over the factoring by group process because we can predict what's gonna happen when we have this monic polynomial. Let's look at another example. If we take x squared plus seven x plus 10, notice how the leading coefficient here is one. So I claim if we can find factors of 10 that add to be seven, then we have the factorization. And we can do five and two. Notice if you take x plus five and x plus two, that'll be a factorization here. x times x is x squared, five times two is 10. You're gonna get a two x and a five x, which adds together to give a seven x. That's the correct factorization. And then by the zero product property, the solutions would be negative five and negative two. If your leading coefficient is one, that's all there is to factoring. It's really, really, really nice. How about this next one right here? x squared plus x minus six right here. So we need factors of negative six that add up to be, factors of negative six that add up to be one. Well, six factors as two and three right. So those are a difference of one. So make sure you get the sign in the right spot. We're gonna take a plus three and a minus two and double check that three x minus two x is an x. So this is our factorization. Therefore the solutions when you switch the signs will be x is negative three and two. So when you have a monic quadratic equation, the leading coefficients one, we can factor these things super, super, super quickly and get the answers there very quickly. But this only works when the leading coefficient is one. If the leading coefficient's not one, we do need to go through the full blown factoring by group techniques. We'd still have to find a magic pair, which is a little bit more time consuming. But this is a very nice shortcut when it is applicable.