 Welcome! We are going to be doing some more quadratic equations today and we're going to be using the calculator quite a bit, so make sure you have yours out and ready to go. So there's a couple different ways you can solve quadratics with the calculator. If your equations in the form f of x equals f of x equals zero, as I wrote here, then you're going to want the points on your graph where y is zero because remember y is the same thing as f of x. So these are known as your x intercepts, which your calculator also calls your zeros. So I'm going to go in and highlight that word. I'm using that one a lot today. If your equation is not in that form, then you can just graph them in your y1 and y2 and then you can figure out where they intersect at. So I'm going to show you both ways here today just so you can practice with both of them. Alright, so the first one, let's solve the equation 2x squared plus 11x minus 63 equals zero. Okay, so I'm going to go to my y equals, clear anything out that I have in there, clear both of these out, then I'm going to type in my equation 2x squared plus 11x minus 63. Now, if you want to put a zero in your y2 and then do the intersection, that would be perfectly fine too, but I'm going to show you a little bit different way to do it today. Alright, let's graph this. Oops, we're going to have to fix our window for sure. That doesn't look much like a parabola to me. So let's do a zoom and let's start with number six, our standard window. Alright, that's looking more parabola-like. We can't see the vertex, but that's probably down here. But remember what we're looking for. We're looking for the x-intercepts. So that's going to be about right here, maybe at negative nine-ish, and then somewhere in here. So that would be like 3.5-ish. Okay, so to find those exactly then, what you're going to do is you're going to go to second calc. Actually, let's take a look at the table first. Change my mind. So let's see, I said what about negative nine-ish? So let's scroll backwards. So looking at the table, yes. So here at three, we have negative 12, but at four we have positive 13. So because this one's negative, this one's positive, there has to be a zero in between there. So that'll be somewhere between three and four. But then let's see if at negative nine, it's exactly a zero, because then we don't have to worry about finding that one. And it sure is. So our y is zero when our x is a negative nine. So that is one of our solutions. Now to find the other one, we're going to go to second calc, which is right above your trace key, and you're going to look for number two, called a zero. That's why I wanted to use that word a few different ways. So let's press option number two. This is very similar to finding the minimum and the maximum. So it's going to want a left bound and a right bound. Right now my left bound is at zero negative 63. So I want to get somewhere close here to, I think we said around 3.5. So I'm going to go ahead and toggle to the right until I get close to 3.5, or at least until I can finally see my cursor. Oh, there it is. Okay, that looks pretty good. So I'm going to hit enter. That's on my left side. Now it wants a right side. So I'm going to toggle this time above my axis. That looks pretty good. Okay, hit enter. Now it wants a guess. We're going to have to sit to go in there and hit enter. And sure enough, my guesstimate was pretty darn good. It's at 3.5 zero. Okay, so what we're going to want to do over here then is on our paper, we'll go ahead and draw in our parabola. So then we can say here this is negative 9 comma 0. And this point was 3.5 comma 0. So x equals negative 9 and x equals 3.5. Alright, number two, this one says solve the equation 2x squared plus 27x equals, or sorry, plus 27 equals 21x. So let's go to the calculator. If you wanted to, you could move this 21x over, solve this equal to 0 and do it exactly like we just did. But I would like to show you another method on this one. So here's where we're going to use our y1 and our y2. Okay, so in our y1, we're going to put 2x squared plus 27. And then in our y2, I'm going to put the other side of the equation, which is 21x. So what I'm going to be looking for then is where these two functions intersect at. So when I hit graph, I'm not going to see my, oh that doesn't really help us very much. So in this case, instead of monkeying around with the window, if we go to zoom and let's toggle down to the option, I believe at zero, that says zoom fit. I don't like to use this one very often because it messes with your window and it makes it really hard to see what's going on. But sometimes it's kind of necessary and I know what the outputs of this one is going to be. So we definitely need it. Okay, so there's our parabola that represents the left side of the equation because that was a quadratic. And here's our line that represents the right side of our equation because that's linear. And they intersect, it looks like around two-ish and maybe nine-ish if I had to guesstimate. Okay, so let's go ahead and calculate that. We've done these before where we want to use option number five, that's the intersection because these two graphs are going to cross. Okay, oops, I don't think I hit the right thing there. Try that again. Let's do second calc. Option number five, I think I hit graph for some reason. Option number five, there we go. Okay, so once it's to be close to the first curve, so we want to get close to that first point, that's pretty darn good, hit enter, hit enter again, hit enter again. And so that gave us an intersection of 1.5, 31.5. So again, let me go ahead and draw a rough sketch of this. So I've got our parabola like that and our line like that to pretend like they actually intersect at reasonable points. So at this point here, we saw was 1.5 comma 31.5. So you see why I didn't want to mess with the window this time. These y values are huge. Okay, let's go to the other intersection. So do second calc again. Pick option number five for intersect. Let's toggle to the right to our other intersection point. Keep toggling over until you get close. I think that's pretty darn close. Okay, hit enter, hit enter, hit enter. And it spits out the point nine hundred and 89. Yes, I'm glad we didn't try to figure out that window ourselves. So that's going to be the point nine hundred and 89. So that means my x values because I'm solving the equation x is going to equal 1.5 and x is going to equal nine. And those are our solutions. Okay, graphing quadratics and looking for x interstops or zeros are also connected with the factors of a quadratic equation that we talked about in the last chapter. Okay, so looking at the graph of, let's bring the calculator back up. Clear these equations out. Okay, so we have parentheses x plus four, parentheses, parentheses x minus one. And before I hit graph, I'm going to do zoom and let's go back to our standard window again because remember that window before was crazy with that hundred and 89 on our Y. Okay, so here's our graph. So I asked the question, what do you notice? Well, okay, it's a parabola. It opens up. But looking at these factors, so what do you notice here? Well, where are my zeros at? Where are my x interstops? It looks like it's at one and one, two, three, negative four. Let's take a look at the table to confirm that. And sure enough, negative four, zero. There's a zero there. Let's see if there's one at one. Absolutely. So sometimes I use my graph and do my zero options. Sometimes I use my table. It just depends on which one I think might be easier for that problem. So x is going to equal negative four and x equals one are the zeros. Now I say set each factor equal to zero. What do you get? So my first factor here is x minus four. Let me highlight that one. It's my first factor. My second factor is x minus one. And you notice I didn't multiply these out. I just left them alone. So x plus four equals zero gives me, oh, x equals negative four. Interesting. x minus one equals zero when I move that one over gives me x equals one. Interesting. Okay, so what you want to see here then is you notice your factor was x plus four, your zero's negative four. Your factor was x equals, or x minus one, your zero is x equals one. Okay, so there certainly is a relationship between these two things. And it makes it pretty easy then to go back and forth between factors and zeros. You just have to remember to change your signs. We highlight those phrases, change your signs when you go back and forth between the two forms. And that's because you're moving over zero. So you're going to have to do that. All right, so this one, I gave you a picture. So find a possible, and here's the keyword possible quadratic function for this graph. Okay, so I look, I'm looking at this graph and I see a zero here at negative three. And I see a zero here at positive three. Well, that's nice and symmetric for us. Okay, I want to find a function though. So let me start with my function notation f of x equals, so if x equals negative three is a zero, I have to change the sign. So that would be x plus, oopsie, not four, three is a factor. I don't know where that four came from. Okay, so these two pieces go together. So this one gave us that one. If we know x equals three is a fat is a zero, then that means x minus three is a factor. So this one gave us that one. And that's a possible quadratic function. You could multiply it out if you want to, but I think I'm just going to leave it like that. It doesn't say what kind of form it has to be in. So you might as well be an easy on yourself as possible. Okay, so I put here then the good news is if you're not very good at factoring, you can always look at your graph, find your zero, work backwards. Okay, this especially works well when your zeros are whole numbers. If they're not, then you have to do a little bit more finacling, but that's okay. And I put on here, if you are good at factoring though, you can always set your quadratic equation equal to zero, factor it, then set each factor equal to zero and solve. Okay, so last problem here is d plus seven times d minus four equals negative 24. So you may think, well, these are our factors, but you notice it's not set equal to zero. It's set equal to negative 24. So unfortunately, that's not going to work. All right, we can do this one by graphing. We can do this one by factoring. It just depends on how you would like to prefer or prefer to attack it. But I think I'm going to go ahead and graph this. So let's clear this off. And then I'm just going to do it all with a table this time. So x plus seven, hopefully out with a table times x minus four. Put that in my y one. I'm going to put negative 24 in my y two. Let's take a look at our table and see if we can come up with values that match. And sure enough, right here. So at negative four, you notice they both spit out a value of negative 24. And here at positive one, they both spit out a value of negative 24. So from our table, oopsie, x, missed my variable there, d. See from our table, d equals, I got myself all confused. All right, we have negative four and positive one. So d equals negative four, d equals positive one. Okay, so keep practicing these and we'll do some more equations next. Thank you.