Common Core Algebra I.Unit 9.Lesson 6.The Quadratic Formula.by eMathIntruction

Hello and welcome to another E math instruction common core algebra one lesson. My name is Kirk weiler, and today we're going to be doing unit 9 lesson number 6 on the quadratic formula. Quadratic formula is one of the most famous formulas in all of mathematics. Anyone who's ever taken algebra one or algebra two has seen it before. And today we're going to be going over it in some depth and detail. I'd like to remind you that if you'd like the worksheet that goes with this lesson, or I follow up homework, just click on the video's description. As well, don't forget at the top of each one of our worksheets there's the QR code that goes along with the video. Scan it with your smartphone or your tablet, then you can watch this video immediately. All right, let's get right into it. The quadratic formula really comes from the method of completing the square, all right? So I thought we would start with that. Little review on how to solve a quadratic using completing the square. So we're going to solve X squared plus 8 X plus three equals zero by completing the square. Remember how we do this as long as a is equal to one, the leading coefficient which it is. What we do is we take the first two terms X squared plus 8 X, we look at our 8, we divide it by two, which gives me four, and then we take that four and we square it. And we add that on. Now, in order to add on absolutely nothing, what we do is we add and subtract 16. Right away. Now what we're going to do is we're literally going to group these three terms together. And we're going to factor them. And if you think about it, right? Factor is X plus four times X plus four, we can now combine these two terms into negative 13. And we'll get X plus four quantity squared -13 equals zero. Now you might say, wow, that was a lot of work for nothing. Except now we can easily solve this equation. Let me do it up here. X plus four quantity squared -13 equals zero. I can now reverse what's been done to X, right? I added four. I squared, I subtracted and got 13. I will undo each of those. In due course. X plus four squared is equal to 13. Undo the squaring by taking the square root. Remember when we do that, we always introduce a plus or minus. Now there's nothing we can do with the square root of 13. All right? 13 is not a perfect square. It's also a prime number, so we can't even simplify that radical. That's as much as we can do. And then we'll subtract four from both sides. That would be completely okay to leave your answer and looking like this. Plus or minus the square root of 13 minus four. That's okay. I'm going to actually write it as follows. Negative four plus or minus the square root of 13. And you'll understand why I did that. Once we work through the next exercise. All right? But simple enough, well, okay, maybe it's not really simple. But it's something that we've covered already in the course. And we've done quite a bit of it. What we're going to do today is actually give you an alternative. Now you're supposed to know how to solve a quadratic by completing the square under common core algebra one. But you're also supposed to know how to solve a quadratic using the quadratic formula. So let's introduce you to that now. I'm going to pause the video just for a second, let you copy down any of this text. All right. And it's gone. Let's move on to the next exercise. Now, not the next exercise. Let's look at the quadratic formula. So the quadratic formula can be used to solve any and this is key. Any quadratic that's written in this form, AX squared plus BX plus C, where a, B and C are just those coefficients, right? They're the numbers that multiply X squared, that number that multiplies X and then the additive constant. Now, the formula itself looks really complicated when you first see it. Take a look at this thing. X equals negative B plus or minus the square root of B squared minus four AC divided by two a this formula has been memorized by students throughout the ages and it's been around for a long time. In fact, I bet if you went online and you Googled the quadratic formula, you could even find people who sing songs to try to try to remember it. Negative B plus or minus, I can't sing. I don't even know why I tried that. The point is, you have to memorize the formula. Now, throughout the worksheet today, I'm going to have that formula on every single screen so that we can basically plug and chug throw numbers in and get answers. All right? So I'm going to clear out that little bit of blue, so it doesn't show up on the next page. That's always embarrassing. And let's start using the quadratic formula. Now, one thing that we have to make sure we can do is always identify a, B and C and there's my quadratic formula. It's kind of fly in every time like that. So in this particular problem, we've already arranged the formula in AX squared plus BX plus C form. And the values of a and B and C are a equals one, B equals 8, and C equals three. Now that's probably the most confusing one because it's not actually written. Okay? And very often a is equal to one in these problems. This is, of course, the one from exercise one. Now what we're going to do is we're going to substitute a, B, and C into this formula, and find out what we get. And I'm going to do it right here to give myself plenty of room. X equals negative 8 plus or minus negative B, the square root, always a good idea whether it's positive or negative to put that B in parentheses before you square it. Minus four times a times C, all divided by two times a now, this quantity underneath the square root. Very important. So important it's given a name, it's called the discriminant, but you don't really need to know that for our course. That number you've got to calculate correctly. All right, whether you do it in stages or whether you do it all at once in your calculator, is kind of up to you. I tend to take my calculator out. I tend to actually throw this in all at once. I'll literally put this in my calculator. And I'll write it exactly like this. I'll write it exactly like that. What my calculator will tell me is that that number is equal to 52. So that's the number under the square root. All right, notice I have not taken the square root of it yet. Now that is a perfectly good answer. All right. But this problem asks us to simplify our expression. And that means a couple different things. One, it means that we've got to deal with that square root of 52. We've got to simplify it. Simplest radical form. You remember that? So I need to find a perfect square that goes into 52 and a perfect square that goes in that is four. So I'll break up the square root of 52 as the square root of four times the square root of 13, four times 13 is 52. So that becomes two times the square root of 13. All right, now resist the urge. It would be very tempting to try to combine that 8 in that two right now, but we can't. It would be like combining negative 8 and two X, we can't do that. So one more thing, we're going to distribute the division by the two. All right, and what we find is that negative 8 divided by two is negative four. Two divided by two is one, and we get exactly the same answer. We got from exercise one. Now, after all, of this, you might be convinced that the quadratic formula isn't the way to go. Maybe completing the square is. Maybe factoring is. What we need to know how to do each one of these methods, and each one of them has its advantages. The biggest advantage of the quadratic formula, as beastly as it is, as scary as it looks, is that it works. Every time. I don't need to worry about whether a is one or two or three. I don't have to worry about whether I can factor this thing. It will work every time to solve a quadratic equation. Which is why it's called the quadratic formula. All right. This is a very important screen, all right? So pause the video for as long as you need to. Think about it and write everything down. Okay, here it goes. All right, we can also use the quadratic formula when we can factor. So I'd like to compare and contrast those two methods. We'll bring up the quadratic formula when it's time. But letter ray says, find the solutions to this equation by factoring, all right? It's also known as the zero product law. The zero product law, when we solve a quadratic by factoring. So remember how that works, right? I want to take this trinomial, and I want to figure out how to factor it. All right? The two binomials that will give me that as a trinomial. Now look, the center term is negative, and this thing is positive. So that means both of these have to be negative. And I could guess and check and play around with this for a while. But that's going to be the right answer. Remember, here's the quick check, negative one X, negative 8 X, combined to give me negative 9 X so that checks. Now remember, once we have it factored, then all we have to do is set two X minus one equals zero. Set X minus four equal to zero. A little more work on this one. Add one to both sides. We'll get two X equals one, divide both sides by two. And we'll get X equals one half. Here much simpler at four to both sides. And we'll get X equals four. Now what we're going to do is we're going to use our friend the quadratic formula to find the exact same numbers. What's remarkable is that we're going to get one half and four by using this formula, and it's pretty hard to see why at the beginning. Let's first identify the values of a, B, and C a is the number that multiplies X squared, that's two. B is the number that multiplies X that's negative 9, very important that you have the negative. And C is positive for. So let's just crank through the formula. A equals negative B but that's negative negative 9. Eventually, we'd like to just write down a 9 there, but I'll write it that way at the beginning because we've just started working with this. All right, now I've got that negative 9 squared minus four times a times C, all divided by two times a now again, this is probably where I go somewhere on my calculator and I type this in. Negative 9 squared minus four times two times four. Notice these parentheses. If you don't have the parentheses around the negative 9, you're going to get the wrong answer. The wrong answer. But if I do all that, what I find out is it's all equal to 49. I'm going to go back to the blue. Negative negative 9 is positive 9. Plus or minus the square root of 49, all divided by two times two, which is four. Now in the previous problem, we had an ugly number under the square root. That number of the square was called the erratic hand. We had 52. But 49 is a nice number, right? The square root of 49, that's a rational number. That's 7. So we're going to get two answers. We're going to get 9 plus 7 divided by four. And we're going to get 9 -7 divided by four. 9 plus 7, of course, 16. And then we divide that by four, we just get four. Look at that, X equals four. When we do 9 -7, we get two. Now be careful. Two divided by four is not two, it's one half. And there they are. There are two answers. X equals four and X equals one half. All right. So the quadratic formula can be used instead of factoring. Students really like that. I find even an upper level classes that I teach. I could give that same quadratic two X squared -9 X plus four, and a lot of students will sort of ought to use the quadratic formula to solve it, set equal to zero rather than use factory. Okay, I'm going to clear out the text to get a very important screen. Please write down anything you need to. All right, here it goes. Next page. All right. So we really want to get some factor or some practice using this formula. All right? A lot of times on tests, especially if your student in New York State. They will ask you to use the quadratic formula and express your answers in simplest radical form. That's actually what we did in the first problem actually, not the first problem, the second problem. When we simplified that square root of 52. All right? So I'm going to put a little line here just to separate the two problems or what I thought was going to be a line. And let's work through one. All right, here we have a equals one. B equals 6, C equals negative 9. Let's work through this one together and then have you work on letter B on your own. All right, X equals negative B plus or minus the square root. Of B squared minus four times a times C, all divided by two times a again, somewhere on your calculator. You plug this exact expression in and you find out it's equal to 72. All right, resist the urge of taking the two and dividing it into the 72. Never let a number that's outside of a square root divide into a number inside of a square root. They're not. They're not the same kind of beasts. So try to resist. Now, of course, we want to simplify the square root of 72. So we want to look for the biggest perfect square that goes into 72. Certainly 9 goes into 72, but it's not the largest perfect square. The biggest one that goes into 72 is actually 36. So we can break up route 72s route 36 times root two, so that's going to be 6 root two. Going back to here, that means I'll have negative 6 plus or -6 times the square root of two, divided by two, now I'll distribute that division. So I'll have negative 6 divided by two. Plus or -6 root two divided by two. And that will be negative three plus or minus three times the square root of two. That's a long problem, isn't it? Especially for algebra one. All right, what I'd like you to do is pause the video now and work on letter B on your own. Try to get it in simplest radical form. All right. Let's go through it. In this particular problem, a is equal to three. B is equal to four. C is equal to negative one. Here we go. X equals negative B plus or minus the square root of B squared minus four times a times C I actually think saying the formula out loud is quite helpful. Sorry if it's annoying you. But it really kind of drills it into your head. Negative B plus or minus the square root of. Okay, again, using your calculator being very careful in using the exact parentheses I use. You're going to substitute that in. And you're going to get negative four plus or minus, figure out that, what is that end up being 16 plus 12? 28. Temporary loss my solutions. All right, we have to simplify that route 28. Okay. Biggest perfect square that goes into 28 is actually quite small. It's just four. So we can simplify square root of 28 as two times the square root of 7. So I'll get negative four plus or minus two square root of 7 divided by 6. Now there's kind of two ways that this could be simplified. Either the division could be distributed the way we've been doing it. In which case, this fraction would reduce to negative two thirds, and this would reduce to square root of 7 divided by three. Or so that's a great way to leave the answer. Or sometimes what they'll do is they'll say, look at all the numbers outside of the radicals. Is there a common divisor? You should know that term, a common divisor, and there is. It's two, right? Two goes into all of them. Therefore, we can divide a two out. Now, two divided by two is one, but we tend to not write that. So we could also see the answers expressed this way. It's a little unfortunate that there are so many different ways the answers can be written in these problems. But that's kind of what we have to deal with. Equivalent answers, right? Or equivalency. You can kind of check your answers by looking at the decimals on your calculator, but even that is sort of a job. All right, although I do encourage you to check. So important screen, pause the video if you need to. All right, here goes the text. Okay. Last problem. Come on, quadratic formula. There it is. A projectile is fired vertically from the top of a 60 foot tall building. Its height and feet above the ground after T seconds is given by this formula. H equals negative 16 T squared plus 20 T plus 60. All right, first thing it tells me it says using your calculator sketch a graph of the projectile's height, H using the indicated window. At what time T does the ball hit the ground. Solve by using the quadratic formula to the nearest tenth of a second. All right. Well, first things first, let's get a graph of the height of this object as a function of time. To do that, I'm going to open up the TI 84 plus right now. Hello, calculator. All right, I'm going to go into Y equals. So let's do that. Now, I realize that this is H equals and there's a T but what I'm going to do is I'm going to be putting into my calculator. As Y one equals negative 16 X X squared plus 20 X plus 60. All right? So let me put that in. Negative 16 X squared plus 20 X plus 60. As always, once you've got that equation in there, make sure to look at it carefully. Make sure it's correct. All right. Now, in terms of the window, what's neat is that the problem is given me the window. Right? Because what I know is I know that the X axis, I know here it's the T axis, just goes from zero to three. So my minimum X value is zero. And my maximum X value is three. On the other hand, the Y axis, which I realize here is H it's smallest value, really, is the ground height. It's zero. And it's largest value is 70. So let's set zero for Y min and 70 for Y max. Now we have everything in there, we need to check the equation, check the window, and we're ready to graph. Let's hit the graph button. All right. Let me just sketch this curve. All right. Starts right about there. And nice parabolic path. We're used to seeing these at this point and then hits the ground. All right? The question then becomes after we've got the sketch at what time does it hit the ground? Right? Now how do we solve that? Well, the ground by definition is where H is equal to zero. So I'm going to take this equation negative 16 T squared plus 20 T plus 60, and I'm going to set that equal to zero. And I'm going to use the quadratic formula with a equaling negative 16 B equaling 20 and C equaling 60. Now the numbers are going to be big here. They're going to be quite big, but that's okay. That's why you have your calculator. So let's do it. We get X equals. Oh, not X, that was silly. There's no X in this problem. But what there is is a T T equals negative B plus or minus the square root of B squared minus four times a times C, all divided by two times a, just like in the previous problems on my calculator. I'm going to go somewhere. I'm going to enter this calculation. I'm going to figure out what it is. And it's large. It ends up being. What is it? 4000 240. Now, be careful, it's negative 32 in the denominator, not positive 32. In previous problems, we took that square root of whatever, and we simplified it. Thank goodness we don't have to do that this time. Thank goodness. In fact, I need to evaluate two things now. Negative 20 plus the square root of 4240. Divided by negative 32, and then minus. Let's do them one at a time. Now to do this calculation on your calculator, here's what I advise. I advise doing this calculation first, all right? Getting an answer and then dividing it by the negative 32. All right? Do it in two steps. And when you do that, you get an answer that's something like negative 1.4, et cetera. It's quite ugly. Let's do the other one. The other one is T equals negative 20. Minus. Square root of 4240. All divided by negative 32. Again, and you should do this on your own. Do this calculation first. Then divide by negative 32 second. Why don't you go ahead and try that on your own, okay? All right, let's go through it. When you do that calculation, you work it all out. It's kind of messy again. It's 2.659, et cetera, but that's the one we want, right? We knew our answer was between two and three seconds. They ask us to round to the nearest tenth of a second. So 2.7 seconds. That negative 1.4, that should be rejected as a non viable solution. All right. Nice applied problem much bigger numbers. I get it. But a good problem to work out. Okay, pause the video now if you need to. All right, here it goes. All right, let's finish up. So the quadratic formula, one of the most famous formulas in all of mathematics because it allows us to solve equations that have X squareds in them as their highest power. Without having to do factoring or completing the square or any other real method. All right? So it is very, very convenient, although those other methods you're still supposed to know. We're going to get more work on it in the next lesson as well as the other techniques. For now though, let me remind you, this has been another E math instruction. Common core algebra one lesson. My name is Kirk weiler, and until next time, keep taking. Thank you solving problems.