 So let's take a look at a few more examples of solving quadratic equations by factoring, and, while you should be able to solve quadratic equations by factoring, one of the things to realize is that this is the most difficult and most painful way of trying to solve quadratic equations. What if we want to solve 4x squared minus 21x minus 18 equals 0? So since our expression is already equal to 0, it's useful to factor it. So 4x squared minus 21x minus 18 is going to be a product of two things where our first terms have to multiply out to 4x squared, and our last terms have to multiply to negative 18. So we factor this as ax plus b times cx plus d, where ac is 4 and bd is minus 18. So we'll try out every pair of numbers that we can find where one product is 4 and the other product is minus 18. So 4 and 1 multiplied to 4, 1 and negative 18 multiplied to negative 18, so maybe this factors as 4x plus 1 times x minus 18. So let's expand the right-hand side, and it doesn't work. How about a different set of numbers? Well again, 4 and 1 multiplied to 4, negative 1 and 18 multiplied to negative 18, so one possibility is 4x minus 1 times x plus 18. We'll expand, and it doesn't work. How about 2x plus 1 times 2x minus 18, 2x minus 1 times 2x plus 18, 4x plus 2 times x minus 9, 4x minus 2 times x plus 9. Nope. 2x plus 2 times 2x minus 9, 2x minus 2 times 2x plus 9. Really? 4x plus 3 times x minus 6. Finally, we have a factorization. Let's see what we're redoing. Oh yeah, we wanted to factor this expression, so we have our factorization. Since we have a product equal to 0, we know that one of the factors must be 0, either 4x plus 3 is 0, or x minus 6 is 0, and solving gives us our two solutions. Let's take a look at another problem, and remember, we did say this would be a painful set of examples. In fact, part of the purpose of this is to show you that you really don't want to solve quadratic equations by factoring unless you have to solve them that way. But suppose you had to solve a quadratic equation by factoring, well, since we need to make sure that we have quadratic expression equal to 0, we'll rearrange our equation, and now we have quadratic expression equal to 0, so if we can factor the left-hand side, we'll be able to solve. We want x squared minus 8x minus 36 to be something times something. Our first terms have to multiply out to x squared, so we don't have a lot of choice there. We'll make them x and x, but our constant terms could be anything as long as their product AB is minus 36. So we list all of our possibilities, and again, the only thing we can do with this point is to try every single one until we find one that works. So let's see if 1 and negative 36 works. Is x squared minus 8x minus 36 the same thing as x plus 1 times x minus 36? Will it expand? And no. How about x minus 1 times x plus 36? Will it expand? And no. x plus 2 times x minus 18? x minus 2 times x plus 18? x plus 3 times x minus 12? x minus 3 times x plus 12? Well, seventh time's a charm, so it's gotta be x plus 4 times x minus 9. x minus 4 times x plus 9? Well, there's only one pair left, 6 and negative 6, so it's gotta be x plus 6 times x minus 6. Trust me, I'm on the internet. Maybe we should expand just to make sure. And that doesn't work. And since these are all the possibilities for factors, no pair of factors works to factor this expression. So now what? Well, the good news is that the problem exists whether or not you know how to solve it. So even if we can't factor this, we still have to solve this problem. Wait a minute, that's not good news. That means we have a problem we don't know how to solve. Well, actually, that's not too big of a problem. We can solve this problem using other methods. And in fact, those other methods are things we should preferentially use when trying to solve quadratic equations. Again, solving quadratic equations by factoring is very painful, because even when it is possible, we have to go through a lot of trial and error. And in some cases, we have to go through all that trial and error and not find a factorization. So as a general rule, solving quadratic equations by factoring should be your last choice.