 This video is called Solve Trinomials by Factoring 1. Now one thing to think about that we haven't talked about yet is when you have a trinomial, when you have a term of x squared, if you were to put this on a graph, you would end up with some sort of a parabola or a u-shaped figure. So keep in mind the x squared always means you'll have a parabola. And what we're going to do is when you have this trinomial with an equal zero at the end, we're not just going to factor it, but it says to solve by factoring. And what you're doing is trying to figure out if you have a parabola, you're figuring out where your parabola crosses the x-axis. Maybe it'll happen in two spots, like my example in red. Maybe it will, or my sketch really. Maybe it'll happen nowhere, like my sketch in blue because it doesn't cross the x-axis. Or maybe you'd have something like my sketch in green where you have one answer because it hits it only once. Let's see what happens here. We're going to go about this factoring how we're used to. Multiplying to give me five and adding to give me negative six. Well, when I make my list of factors, five times one is the only way to multiply to give me five. And a negative five plus a negative one gives me a negative six. And a negative five times a negative one gives me a positive five. So I split up my x's and have a negative five and a negative one. Now if we were just factoring and this equals zero wasn't there, we'd be done. But it is there and we can't forget about it. So when you see the word solve, we actually have to do some algebra. Now it's time to simply take my two binomials and set them both equal to zero. Now we've got two equations to solve. When you add five and add five, you get x equals five. So that's one answer. When you add one and add one, you get x equals one. So that's your second answer. So what that tells us is on a graph, our parabola or u-shaped curve is going to cross the graph at one and at x equals five. So if you're going to do just a sketch, it would look something like that. Now the reality is it'll be a lot more exact, but what we found is where it crosses the x-axis. So a sketch at this time is sufficient. Let's try a second example. To solve this trinomial by factoring, I know it's a trinomial because I've got three terms. I know it's a parabola or a u-shape because of the x squared. But before you can get started, I need to get all my terms on the left-hand side of the equal sign. So let's move that 36 over. So you get x squared plus 13x plus 36 equals zero. Now I'm ready to factor because all my terms are on the same side of the equal sign. You cannot factor from this stage. You'll get the answer wrong. It has to be from here. So what multiplies to give us a positive 36 and adds to give us a positive 13? Well, let's go ahead. We can make kind of the skeleton of our problem. I can break down the x squared to x and x. And now let's work on some factors of 36. Well, there's 36 and 1. Gosh, there's a lot. There's 6 and 6. There's 12 and 3. There's 9 times 4. Well, any of these add up to a positive 13. I think it's the positive 9 and the positive 4. So I'll put a plus 9 and a plus 4. Now if this was just a simple factoring problem that didn't have the equals zero, we would be done. This would be our answer. But because we're solving by factoring, we need that equals zero and we have to go one step further. x plus 9 equals zero and x plus 4 equals zero. So when you subtract the 9 from both sides, you get x equals negative 9. When you subtract the 4, you get x equals negative 4. So we have two answers. So that tells me if I was going to make a sketch of a graph, it tells me that my parabola is going to cross the x-axis twice. One time will cross at negative 4. And the other time will cross at negative 9. So again, this is a very kind of crude sketch of what my parabola looks like, but it gives me an idea and lets me know where it crosses the x-axis. The last problem in this video to solve trinomials by factoring, the equals zero tells me that I have to solve. I'll be doing some equations. We're looking to see where this crosses the x-axis. The x squared tells me that I've got a parabola, which is a U shape. It might be a really skinny U. It might be a really fat U. I don't know. This will help us get an idea of what it looks like. So first thing, ask yourself, are all my terms on the left-hand side of the equals? Yes. Second thing I want to ask myself is, can I take out a GCF? And I think I can divide out a 2, a 2, and a 2. So let's go ahead and fill in what's left over. 2x squared will get me back to 2x squared. 2 times a negative 7x gets me back to negative 14x and 2 times a plus 12 gets me back to 24. I like this because now the numbers are smaller. They're simpler than what I had up here. So what multiplies to give me 12 and adds to give me a negative 7? Well, factors of 12 are 12 and 1, 6 and 2, 4 and 3. Which combo will add to be a negative 7? It'll be a negative 4 plus a negative 3 gives me that negative 7 and a negative 4 times a negative 3 gets me that positive 12. So don't forget about the 2. Don't forget about the equals 0. Split up the x. Negative 4, negative 3. So now I've got 3 terms. I'm setting equal to 0. 2 equals 0, x minus 4 equals 0, and x minus 3 equals 0. Well, 2 does not equal 0. So I can get rid of that. So the 2 that I had here and here has kind of gone away. Now I add 4 and I add 4. So I have x equals 4. I add 3, I add 3. I have x equals 3. So these are my 2 answers. That is where, if I was going to graph this, don't leave much room to do that. But if I was going to graph it, my parabola would cross the x-axis at 3 and at 4. So this is going to be a skinny parabola. Remember, it's not perfect. It's not exact, but I'm just starting to get an idea of what 2x squared minus 14x plus 24 looks like on a graph.