Common Core Algebra II.Unit 8.Lesson 6.The Quadratic Formula
Jan 3, 2017
Hello, I'm Kirk weiler, and this is common core algebra two. By E math instruction. Today, we're going to be doing unit 8 lesson number 6 on the quadratic formula. Now, you saw the quadratic formula back in common core algebra one. And you may have even had some teachers have you used it in common core geometry when you were trying to solve systems of equations algebraically. But it may have also been an entire year since you've seen it. So we're going to use this lesson to review the quadratic formula. How it works and what it can do for us. So let's get right into it. All right. Exercise one asks us to solve the following quadratic equation for all values of X by first completing the square on the quadratic expression. Express your answers in simplest radical form. All right. So we've got a bunch of different things here. We've got complete in the square. We've got simplest radical form, et cetera. So I'm going to rewrite the problem right here. X squared -6 X plus one equals zero. Now I can try to use the zero product law, but it would be very, very difficult in this case. Plus, they said to do it by completing the square. So remember how that's going to work? What I'm going to do is I'm going to take half of that negative 6, I know we just did this a little while ago. And then I'm going to square it. I'm going to get 9. So I'm going to add 9 on. All right. I also have the plus one. Now, what I'm going to also do is I'm going to add 9 to the other side. So what I'm really doing is just adding the same quantity to both sides. I'm then going to group these first three terms together. And leave the one out. It is the loneliest number. I can now factor this because it is a perfect square as X minus three squared. Plus one equals 9. Now I want to emphasize that these two equations are equivalent. All right, they're the same equation. But the written in very different forms and the beautiful thing about this form is that I can now solve for X by simply undoing everything that's been done to it, right? I've subtracted three. I've squared, and I've added one. So the first thing I'm going to do is subtract one from both sides. That's going to leave me with X minus three. Quantity squared equals 8. I'm now going to undo the squaring by taking the square root. But recall that if I have something like X squared equals 16 and I take the square root, I'm going to get X equals plus or minus four. So I'm going to have the same thing here. I'm going to get plus or minus the square root of 8. Now notice the simplest radical form business. I could at this point simplify the square root of 8, or I could wait until after I add the three. I think I'm going to simplify it right now. And this is what we did in the last lesson. We're going to break up the square root of 8 is the square root of four square root of two. Bring this over here. Then I'll have X minus three equals plus or minus. We don't want to lose that two root two. I'll add three to both sides. Now, here's where a lot of students can get thrown off. Some students will think, oh, I should add the three to the two. Or add three to the negative two. But I actually am just going to leave it like this. Three plus or minus two times the square root of two. You see, I can't combine this number. With this number. I can't combine those. For a couple of reasons. Number one, it's almost like combining. These two expressions. They're not like terms. They just aren't. All right. Second, the three is a rational number. The two Times Square root of two is an irrational number. All right? So again, I can't combine them to make them into 5 root two or anything like that. All right, but we'll just leave our answer like that. Pause the video now and write down anything you need to. All right, let's clear out the text. And let's review the quadratic formula. Here it is. So if we have a quadratic equation, rearranged so that it's in this form. AX squared plus BX plus C equals zero. Then that equation will always be solved using this formula, the quadratic formula. Negative B plus or minus the square root of B squared minus four AC divided by two a now it's important that you memorize this formula. Memorize it. And that you have it all correct. It can't be positive B plus or minus. It can't be B minus four AC it has to be B squared. It certainly can't be B squared plus four AC that would have huge ramifications. So even though it's kind of an abstract formula, it's important that you do what you need to do to memorize it. I'm going to clear the writing out, though. And let's do a problem with the quadratic formula. All right, and I'm going to keep having it on each page just so that we see it. We substitute, et cetera, but at some point, you got to memorize it. Exercise two says, using the quadratic formula shown above, solve the equation from exercise number one, state your answers in simplest radical form. So we're literally solving exactly the same one that we had before. And remember before we have the solution three plus or minus two root two. Let's see if we get the same thing. Well, we have to know what a, B and C are. A is always the coefficient multiplying X squared, and it's often one, but not always a is one here. B on the other hand is negative 6, and C is also one. All right, so a, B, C let's do it. X equals negative B negative negative 6. Plus or minus the square root, I'm going to put that negative 6 in parentheses. Negative 6 squared. Minus four times a times C, all divided by two times a so this is always your first step, just faithfully. And carefully, inserting all of those coefficients into where they go in the equation in the formula, sorry. Now let's do some simplify. Okay, first thing negative negative 6 is positive 6. Plus or minus. The number one place people will mess up the quadratic formula more than anywhere else is right here. Okay? Because they'll maybe type this into their calculator and the calculator will tell them negative 36, which is crazy. Negative 6 times negative 6 is positive 36. So if you absolutely positively must use your calculator to square a negative number, if you absolutely have to put that negative in parentheses. If the numbers are small enough, I would encourage you to do it yourself. So that's 36, four times one times one is four, and of course two times one is two. Let's keep going. Obviously, what's underneath the square root simplifies to be 32. Divided by two. Now, not simplest radical form, and we're fairly sure we should be getting the answer. So at this point, what I'm going to do is I'm going to simplify that radical. I can simplify that into the square root of 16. Times the square root of two. All divided by two, and now I can make that square root of 16 into four. So we're getting pretty close, but one last little piece, right? I can now distribute that division. So I can distribute the division to the 6. And distribute the division to the four. I do not distribute the division to this too. 6 divided by two is three. And of course, four divided by two is two. And there we have the answer. And it was the same when we got before. Which is good. Otherwise something weird would have happened. All right, so take a look. Take a moment, pause the video and write down anything you need to. Okay, I'm going to clear this out. Let's keep going. All right. Exercise three. Which of the following represents the solutions to this equation? Well, pause the video now and take a shot at plugging all the numbers into the quadratic formula and simplify them. Okay, let's go through it. Again, first things first, let's identify a, B, and C, a is again one in this problem. B is again negative, and it's negative ten. And C is 20. All right, so let's do it. Let's do negative B so negative negative ten. Plus or minus the square root of negative ten squared minus four times a times C, all divided by two times a again, being careful. Negative ten squared is positive 100. Minus four times one times 20, which would be 80. That's going to be ten plus or minus square root of 20. All over two, now we have to be careful. Many of these answers have been kind of put there to try to catch us on common algebraic mistakes. So for instance, it would be tempting to divide ten by two and get 5 and divide 20 by two and get ten and think the choice is choice one. But it's not. Never divide a number outside of a radical. Into a number inside of a radical, okay? Just don't do it. It doesn't work. You've got to get the numbers outside of the radical. So I'm going to simplify root 20 as root four root 5. Divided by two, that's going to be ten plus or minus two root 5. Divided by two, and now I can distribute that to both the ten and the two. And that's going to be ten divided by two, which is 5, two divided by two, which is one. So I'm not going to write it. And I just get 5 plus or minus root 5. Kind of a cool answer. All right. Pause the video now, and we'll move on to the next sheet. All right. Backs out of the sheath. Let's take a look. Okay. So one thing that has happened so far as we've done these problems is we keep getting these answers that are irrational numbers. Irrational one type of irrational number is when we're left with a non perfect square underneath the square root. That is an irrational number. All right, but we could certainly have rational answers to these solutions. So I don't want you to get the feeling that you can't. So what I'd like to do is take a look at solving a quadratic equation in two different ways. One way I'd like to solve it is by the use of factoring. And the other way with the use of the quadratic formula, in other words, the zero product law. So what I'd like you to do is try to solve this problem in both ways, both by factoring, and by using the quadratic formula, and then we'll go through it. Pause the video now. All righty. Let's take a look. Well, the zero product law says that we've got to be able to factor this. And this one's a little bit more challenging to factor because that leading coefficient is not one. But it's going to be two X and X and remember our intelligent factoring, the 6 could be one times 6. It could also be two times three. Whoops, how about two times three? But either way, you can't put another even with the two because then there'd be a GCF. So let me try three and two. Well, that work. I'll get a four and a three that'll be 7. That's not going to work. Okay, so I'm going to get rid of that. Get rid of that. All right, so let me try the one and the 6th, the one would have to be here, the 6 would have to be here. That'll give me a 12 and a one. Yeah, I can make that into an 11. By doing this, right? 12 X minus one X is 11 X so then we'll get two X minus one equals zero. I'm sure you can solve that in your head X equals one half. And then we'll have X plus 6 equals zero. And we'll get X equals negative 6. All right. So let's use the quadratic formula and make sure we get one half and negative 6 as well. All right, let's take a look. In this case, a is no longer one, a is two. B is 11. And C is negative 6. So let's do it. Negative B plus or minus square root here, it's not as dangerous because B is positive. But there's another little piece that's going to show up for the first time in this problem. Which is this double negative. One there. And one there. Now again, if you put all of this in your calculator, it's no big deal, you know? You might not even notice the double negative. But you've got to put this all in your calculator correctly. And you could do it all at once. There's no question about that. And if you do, all that stuff under the radical becomes one 69. Now, unlike the other problems that we've done so far, one 69 is a perfect square. So we can actually take its square root entirely, just ends up being 13. Now, what's cool about this is we get two answers, right? We get negative 11 plus 13 divided by four. And we get negative 11 -13 divided by four. So let's do it negative 11 plus 13 is two. Divided by four, and that gives me one half, and then negative 11 -13 is negative 24 divided by four. And that gives me negative 6. And we get the same answers. So the quadratic formula is wonderful. Doesn't mean you can like stop knowing how to factor or zero product law or completing the square. But it's a really nice formula that allows us to evaluate and find the solutions the zeros to any quadratic equation. All right. Pause the video now. And write down anything you need to. All right. Let's clear out the text. Keep going. Okay, so the last four problems that we have. Are just all practice problems. That's all they are. And it says solve each of the following quadratic equations by using the quadratic formula. Some answers will be purely rational numbers, meaning no square roots left. And some will involve irrational numbers. Place all answers in simplest form. Okay, so there's four of these two on this page two on the next. What I'd like you to do is pause the video now, work on these two. When you're either done, or you feel like you can't get any further, unpause the video, we'll go through those two, and then we'll do the same on the next two. Pause the video now. Okay, let's go through it. Now again, up front, I don't know. I'm right. I ended up having completely rational answers. I could have some irrational answers. All right, whoops. Let's make B into 5. That's a little bit better. Here we go. X equals negative B plus or minus the square root of B squared minus four times a times C, all divided by two a great, so far so good. Negative 5 plus or minus. Again, if you can do all this at once, that's great if you need to break it up a little bit at a time. That's fine too. Sorry, I know I wrote those a little bit close together. It can be confusing. Oh, look at this. Underneath that radical, we're just going to have a one 25 -24. But once a perfect square, right? So it's square root as itself. So we're going to have two nice rational answers. One is going to be negative 5 minus one divided by 6. And one is going to be negative one plus one divided by 6. That's going to be negative 6 divided by 6. And that's going to give me one answer is negative one. And this one's going to be negative four divided by 6, so that's going to give me an answer. Of negative two thirds. So nice rational answers. Okay. Who knows? Maybe let her be will have rational answers. Maybe it'll have irrational answers. Let's dive into it. Okay, we're back to a equaling one. B is negative 8. And C is 13. So X will equal negative B or sorry negative B plus or minus the square root of B squared minus four times a times C, all divided by. Two times a that's going to give us 8 plus or minus square root of 64. Minus 52 all divided by two. 8 plus or minus the square root of 12 divided by two. Let me just take a moment. To say stop or store, apparently that's supposed to be a P beautiful. Stop at this point for a moment. All right? This is kind of moment of truth. This is when you look at this number and you say, is that a perfect square or not? If it is a perfect square, like it was in letter a, kind of all is good with the world. On the other hand, if it's not a perfect square, and it isn't here, that's when you have to start doing the kind of annoying simplification process, right? You have to say, well, radical 12 is radical four times radical three. Then radical four is two. And now I can distribute. 8 divided by two is four. Two divided by two is one so those are my final simplified answers. So four plus or minus the square root of three. Simple enough. Pause the video now, write down anything you need to. Okay. Let's clear out the text. Let's do a couple more exercises. Again, what I'd like you to do is pause the video now, make an attempt at both of these, and then we'll go through them. All right. Let's do it. What do we have here? A is two, B is negative two. And C is negative 5. All right? So X is negative B notice how I'm just immediately making that a positive. Plus or minus B squared, which is a little fly in here, which is four. Minus four times a times C. All divided by two a so simplifying the number underneath the radical. Again, watch out. Double negatives there. All right, end up giving me a 44 under the radical. And then a four down here. Again, this is that point where I always encourage you to stop and look at that 44. Is it a perfect square? And the answer is no, which is a little unfortunate. All right. But we can break it up as the square root of four times the square root of 11. All divided by four, which will be two plus or minus two root 11. Divided by four. Now, how do we go about simplifying this at this point? And that's a really good question. I would encourage you to rewrite it like this. All right, that really ensures you distribute that for. Now, again, two fourths, well, that's one half. And plus or minus one half root 11. I think this is a good way to write it. Sometimes you'll see it as one plus or minus root 11 all divided by two. So we'll just common denominator, put the two fractions together. And that's it. All right, let's look at D here we have a is 5. B is 8. And C is negative four. So X will be negative B, plus or minus B squared, just 64. Minus four times a almost looks like a day. Times C, all divided by twice a negative 8 plus or minus all of that simplifies to one 44. All divided by ten. Now again, here's where you sort of stop, you pause, and you say, well, that one 44 is that a perfect square or not, and the answer is it is. So we're going to get negative 8 plus or minus the square root of one 44, but that's just 12. All right. So great. We have negative 8 -12 divided by ten. And negative 8 plus 12 divided by ten. What do those equal? Negative 8 -12 is negative 20. Divided by ten. Oh, that turns out really nice, negative two. And then negative 8 plus 12 divided by ten. Well, that's going to be four divided by ten. And that's two fifths. So nice rational answers. There we go. All right. So some practice with the quadratic formula, very basic, nothing all that applied, or all that interesting, just solving these equations, making sure we can substitute the numbers in, evaluate, simplify, et cetera. Okay, pause the video now, write down anything you need to, and then we'll finish up the lesson. All right, let's clear out the text. And finish up. So this was a very basic review lesson. What does the quadratic formula look like? What is its purpose? What can it do for us? What are some of the pitfalls and how do we simplify it? How do we simplify how do we evaluate it? Most of this material, you should have seen before. And we're going to work with the quadratic formula a little bit more in the next lesson just to make sure you definitely have it. All right. For now, I want to thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler, and until next time, keep thinking. Thank you for solving problems.