Simpler Quadratic Formula — Full Breakdown & Transcript

[00:00] So in this first lecture, what I'm going to talk And traditionally, I think the quadratic formula is We almost all kind of have this song that sings in our head related to it.

[00:17] expect you're going to be using this in the real world? that's greater than zero to that question? formula that's a little bit different from the traditional approach.

[00:33] And the main thing I want to focus on is connecting it to other common patterns in math that are useful for general problem solving. it's not the usual completing the squares bit that some people will learn in high school.

[00:46] I could re-derive the quadratic formula, but frankly, it's going to be a bit of a pain. something that Po Shen-Lo put out a couple months ago.

[00:59] but more than that, he's the coach of the USIMO team, So he talked about kind of an alternate way that we could teach the quadratic formula.

[01:12] but certainly very similar in spirit. Again, the upshot that I want you guys to come away with is the fact that you should connect this to other common patterns in math so as to make yourself a better problem

[01:25] And that should be the main takeaway lesson for, you know, this particular topic. roughly on the order of an hour for each one of them,

[01:37] asking the audience, and then the audience interacts with it in a very, Now the thing about live stuff though is sometimes So the general technology that we're going to have for this is going to involve

[01:54] me pulling up some kind of question, like for example right here, the question is asking kind of facetiously about your relationship with the And I even had this going over the intro animation for you guys.

[02:06] And then what you see is the statistics associated with what the answers, And they'll kind of get overlaid, we see the bars. We don't know what the most common answer was,

[02:19] Except for the fact that it looks like there's quite a few of So we have a couple people who are going to be working on that in the background.

[02:32] In the meantime I'll just be asking the questions that are pushing us through the lesson. where as you answer we're kind of grading it and you see how it's going. For this particular one, who knows, some magic might happen halfway through.

[02:47] But there's a lot of excitement among you guys right now which is awesome and I love that. And let's see if we can translate that into some of the math itself. many times do you expect to use the quadratic formula?

[03:01] And when we were doing this a little bit earlier with a couple people, Zero times is when I think I'll use the quadratic formula outside of school. And a funny story about that, I actually was once giving this talk at Pixar where I

[03:16] brought it up as this example of something that we teach every student and it's unclear And I should have known better than to say this to a bunch of engineers at Pixar because one of them looked at me and he was like

[03:30] oh I'll have you know I actually use the quadratic formula all the time. His name was Tim Babb and he very kindly sent over basically a storyboard for a video, And the basic idea he was talking through an image

[03:44] I think this is totally beautiful. billiard balls but this is entirely computer graphics. And the tactics used behind this involve something called ray tracing.

[03:59] So the basic idea there is you sort of imagine a camera shooting out a bunch of rays through each pixel of your screen and then depending on whether it hits the sphere and how it hits the sphere that tells you how to color each pixel.

[04:12] shooting a ray in some direction will it hit a given sphere? you might imagine an amount of time that the beam of light is moving,

[04:26] and then you can come up with a graph for how close is that ray to the And in some cases if it passes through the sphere And once you work out the math it turns out this is a quadratic equation

[04:40] so what you need is a systematic way to plug in these solutions to a kind of formula because you're doing this for every single pixel on the screen. off of the sphere or does it not go through it at all?

[04:56] And this particular engineer at Pixar he did a loose back of the envelope calculation for me that suggested the movie Coco probably used the quadratic formula over a

[05:08] trillion times just given how many lights there are for every single frame in that shot. the point of being used a trillion times to produce a movie.

[05:20] Nevertheless I don't necessarily think that's representative of most this formula is because maybe they happen to be that Pixar employee. But I want to make the case again that we can connect

[05:35] And the way that I'm going to go about this is to start in a place that's frankly completely unrelated to quadratics and quadratic functions. Because I think if you are thinking about arithmetic, something very basic,

[05:53] something we do in elementary school, and you think about it deeply enough, some of the patterns that you observe become relevant to stuff that you're doing later on. So for example let's say I ask you to factor the number 35.

[06:08] You know it's easy to see that five goes into it and goes into it seven times. We know how to factor 35. especially if there's no obvious small factors that go into it.

[06:23] Because if I asked you, hey let's factor 143, I mean how do you go about it? okay, we know that two doesn't go into it because it's not even.

[06:35] Three, there's a nice divisibility check where you do one plus four plus three, and that sum in this case is five plus three or eight is not divisible by three, Seven, okay you have to think about that for a moment,

[06:49] but you might see that seven goes into 140 evenly because that's 20 times seven. And at this point you might be annoyed with whoever was asking you the question, Actually 11 does go into it and it ends up being 11 times 13.

[07:08] this has to do with the quadratic formula, but I assure you, highly related. Because what if I asked you 3,599?

[07:20] I was reading this article about a Russian programmer in an interview that he had, the intensity of that interview environment. And in one of them he just walks in the door, hadn't even met the person,

[07:35] You know the programmer's like, uh, all right. And he actually was able to spit out the factors particularly quickly, all of the different primes, you know, does two go into it?

[07:50] You're going to be sitting there for quite a while. So clearly something different was happening in his head, and the question is, what? I've chosen here is that 35 is awfully close to a square.

[08:06] And maybe it kind of makes sense that its factors are each kind of close to six. but the fact that they're of similar sizes should be intuitive,

[08:21] And something similar happens with the other example I chose. 143 is rather close to a square number, 144. hovering right around the square root of that value.

[08:38] One of them is 12 minus 1, and the other one is 12 plus 1. So it kind of makes sense that the product will be something around 12 times 12. maybe any time that you have some number that's one less than a perfect square,

[08:54] and this, you know, it's suspiciously close to a very round seeming number, 3600. We recognize the 36 as being six squared and 100 is 10 squared, so that's 60 times 60.

[09:09] And maybe you think, okay, the guess, if it's going to follow the same pattern as what we've seen above, is that this would be one less than 60 times one more than 60. Let's see if there's a nice reason that it's a true.

[09:23] You can work it out algebraically, which we will, but as you guys know I love animation, so let's just see if there's a nice visual way to understand this particular property. place to turn is to think about squares.

[09:36] So let me just pull up an image of a square, and if we want to say, can I factor a number which is one less than a square, what I might have us imagine doing is taking the corner off of that square, okay?

[09:49] but you want to think a little bit more abstractly than that. And I'm going to take that bottom right corner, get it out of here, we don't want it. the quantity that remains into some kind of rectangle?

[10:05] And in this case if I take that bottom row, sloop it on up to the right, and another with a side length x minus one. So this general property holds quite well, and I think that's really cool, right?

[10:20] arithmetic trick where any time you have some sort of square number minus one, you can factor that as x plus one times x minus one.

[10:32] It doesn't have to be one sitting there. If we go back to our diagram, what you might instead do is say, What if I wasn't taking x squared minus one, I was taking x squared minus y squared?

[10:49] Well, I can rearrange what remains into a rectangle where it's x plus y on one side and then x minus y on the other side. which I think gives a certain satisfaction.

[11:02] And when we view it in terms of numbers, it gives us a reminder of how powerful it is. And yet somehow when we do this algebraically, the magic is a little bit lost. It's definitely easier to see, and that demonstrates the power of algebra.

[11:16] you know what, let me change the variable names there. I don't like x and y because those don't necessarily have the same meaning. What I might do is try to picture, you know, let's picture the original

[11:30] two numbers that we had, something that's like 59 and 61 on a number line. And I want to think in terms of the midpoint m and then the distance between m and each of the other numbers.

[11:44] In this case, d is just one, but in principle, it might be something more. you know, at this point, it's pretty straightforward algebra.

[11:58] Next, we have negative d times m. Next, we have m times plus d, m plus d.

[12:10] So minus d squared. And indeed, what comes out is m squared minus d squared. it's just one of a whole bunch of exercises that you're doing.

[12:24] I guess some terms cancel out nicely. because any two numbers we can express, you know, if I just take any numbers r and s,

[12:36] I could write that down as some midpoint plus or minus a distance. And what this is telling us is to think about products is always the same as thinking about a difference of squares, which is weird, because products can be very chaotic.

[12:49] If I just walk through the number line, and I don't know, we I say look at 101, 102, 103, and I just say, take a look at all the numbers on the number line. them be broken down as a product of two smaller whole numbers?

[13:06] Well, we know that what that's asking is, what are the primes? And yet, evidently, that's a very similar question to saying, am I able to express you as a difference of squares?

[13:21] That still feels hard. So I think if you were doing some arithmetic, and you had this in the back of your mind, And that's a pattern that's going to come up later in life.

[13:36] yourself wanting to understand quadratic functions. which is really just refactoring the original thing.

[13:49] quadratic that's thrown at you if you're going by um if you're going by the method that I want to show you right here. Anytime you have some function that looks like ax squared plus bx plus c,

[14:07] So maybe that's something like 3x squared minus 4x plus 5. And you want to know when that equals zero, what are the two roots?

[14:20] Now, this is actually equivalent to rescaling everything. where I'm going to divide everything by a. Because that's way easier to work with.

[14:33] So I'm going to call that x squared plus b prime x plus c prime, where b prime and c prime are just the rescaled versions there. It is a different quadratic function, but the roots are going to be the same.

[14:46] So let's go over and pull up our best friend of Desmos, right? roots is going to be two and the other is four.

[14:58] And then g of x here is just the scaled version of the expanded version of that. But let's say I wanted to scale that up and down, right? So I might take this and then I will multiply it by some kind of constant.

[15:13] If I had the question framework working right now, And we would see the statistics come up with what people had. And as I change that, the function changes.

[15:27] It is a different function, but the roots are still two and four. So solving for this rescaled version of the quadratic is the same. So it doesn't matter how much we scale it.

[15:42] the reason that I think it's much nicer to make sure that that leading coefficient is a one is because if we're thinking of our quadratic in terms of its roots,

[15:54] and I say, okay, you know, it's going to intersect the x-axis at r and s, or maybe it does, maybe it doesn't, but let's just write one that does like this. Another way that I could express that particular

[16:07] quadratic is as x minus r times x minus s. Because if I plug in r, it's clear that we're going to get zero. So if I expand this, whatever r and s are, they're the unknowns,

[16:23] And this is generally useful. This is not just for the quadratic formula, but a very good relationship to have with polynomials is to know how the roots correspond to the coefficients,

[16:35] and the whole polynomial has been kind of normalized in that way. And in this case, if you just expand it out, what we'll get is x squared minus

[16:48] r plus s times x, because we have this minus r times x and then x times minus s. And then the constant term will be negative r times negative s, Okay, so what does this tell us?

[17:04] This tells us the first two of the three key facts that are needed to be able to solve any quadratic without really needing to memorize all that much. So the first key fact is that this b value, I'll call it b prime,

[17:17] just to remind ourselves that it's after you've scaled things down, is the same as the negative sum of the two roots. Okay, and then similarly, c prime is going to be the product of those two roots,

[17:33] which is kind of cool, because what we have right here is, you know, it's a system that feels like it should have a solution. not obvious how we would go about solving it.

[17:46] And every now and then, I think classes will have a unit in factoring quadratics, So in some very fortunate circumstances, if you have something like,

[17:59] you know, x squared, let's see, minus 7x plus 12. different numbers that add up to be 7 and that multiply to be 12.

[18:12] okay, can I find any two numbers that add to 7 and multiply to 12? Well factoring 12, we get 3 and 4, and 3 and 4 do add to 7. And you could write this as x minus 3 and x minus 4, because those are the two roots.

[18:30] well, in most cases, you won't be able to just guess and check. so hope you had fun with that. But it turns out there's actually a systematic way to take this puzzle of finding

[18:45] two numbers that have a known sum and a known product and figure out what they are. And the key comes down to thinking in terms of not the two numbers themselves, but the mean of those two numbers, and then the distance between that mean and each one

[19:01] This is why we talked about difference of squares, because the third key fact to come away with, or to come into it with, I should maybe say, is that we could re-express that product as m minus d times m plus d.

[19:20] That means that the product that we know looks like m squared minus d squared. So just to give an example here, it's often much more helpful to have numbers.

[19:35] Let's say that you were given a quadratic like x squared, I don't know, let's do six, even numbers will make this easier for us, and then seven.

[19:47] So I haven't told you how to solve it yet, but these three key facts Well, what is what is m, right? Because we're

[20:00] As m plus or minus d for some kind of midpoint. Well, that midpoint is the sum of the two numbers over two.

[20:12] And because we know the sum of the two numbers is negative b prime, that's the same as negative b prime over two, which you can basically read off of the equation as just negative one half times whatever's sitting right there,

[20:26] Awesome, we know what m is. We have an expression for c, the last coefficient, in terms of m, which we now know, and d, which is the only thing we don't know left.

[20:42] So we could rearrange this, and so that was saying c prime is that, we could say that d squared, the square of this kind of standard deviation between our roots, is m squared minus c prime, the product of the two roots.

[20:55] Maybe I'll change colors again, be a little flamboyant. d squared is equal to m squared, which in our example turns out to be nine,

[21:07] minus c prime, which is that last coefficient, or seven. Handwriting is terrible, but I think you guys can work with me. So that means that's two.

[21:22] So look, when we said r and s is some midpoint plus or minus a distance, that midpoint is negative three, plus or minus, well if d squared is two, There you go.

[21:35] You could just walk through that particular process. so that you can maybe remember it as a formula if you wanted to. In general, for any quadratic, that midpoint is just the rescaled

[21:52] version of b if the leading coefficient wasn't already one, divided by two. We took the square root of the midpoint squared minus the product of the two.

[22:10] And when I'm sort of thinking into my head, I've been saying like m squared minus p, I think of it as the midpoint squared minus the product. the last coefficient of our quadratic was.

[22:22] So over in this example, the product of the two coefficients was seven. So for me, what I think the simpler quadratic formula is, if you're going to memorize anything, is to come away and say it's m plus or minus the

[22:36] All you have to do is first find m, which is, you know, just a factor times one of the coefficients you're looking at, and then find p, which is also, if not already, a factor, a coefficient that you're looking at,

[22:51] So this to me is way simpler than the traditional quadratic formula. You're just sitting there like m plus or minus square root of m squared minus p.

[23:04] So let's do a couple practice problems, because I do think practice will make it easier. And for future streams, again, it'll ultimately be the case that I'm giving you these questions, and then you'll be able to go to 3b1b.co.live and answer them.

[23:20] This is the equivalent of trying to run a class where you have, you know, And then there's just people banging at the doors and trying to cram themselves in,

[23:32] You can't have this the way that you hoped for. But it's cool that there's so many of you here enthusiastic to learn about math. That'll kind of highlight what this process looks like.

[23:49] Let's say we had x squared plus 10x plus three. That's always lovely. whether or not the two roots turn out to be real or even positive.

[24:04] We're looking for where the two roots are. products is to think about a difference of squares. that in terms of their mean and the kind of standard deviation.

[24:19] So I just write down for myself, what does that mean? And if I forget that fact, if I forget that that's what the sum of the two roots is, I could always just go through this little rigmarole again and say, okay,

[24:34] this is what it would look like. There's not too much memorization needed. And by the way, if I do ever make any mistakes,

[24:50] chat and those will be forwarded to me and I'll be able to correct myself there. And then we just ask ourselves, what's the square of the distance?

[25:03] And based on difference of squares, that'll be that midpoint squared minus the product, which in this context is negative five squared or 25 minus the product, So what are the roots r and s?

[25:20] Well, it's negative five plus or minus the square root of 25 minus three or 22. Ain't no thing.

[25:32] Just because I do think it's kind of nice to get a little bit of practice here. And as I'm going, if you have a piece of paper and pencil, please follow along. That is the best kind of learning experience.

[25:47] as with any video, I highly encourage you to pause and ponder. If you're looking at some kind of lecture in those crucial see if you can do it yourself and then see what the answer turns out to be.

[26:03] I don't know. Let's do maybe three x squared. Just as kind of a offhanded thing.

[26:15] That wasn't there. Three x squared minus four x plus five. Three x squared minus four x plus five.

[26:29] So in this case, step one, we've got to rescale things. Minus four thirds x plus five thirds.

[26:42] I'm sort of thinking in my head of this particular So I say that midpoint is negative of this second coefficient divided by two.

[26:55] But then that four divided by two gives us a two. And then the distance squared is m squared minus the product,

[27:07] which in this case is five thirds. And m squared in this case is, let's see, two thirds squared minus five thirds.

[27:19] We've got to work out our fractions, but that's not too bad. Two thirds squared is going to be four ninths. Oh, I'm off screen a little.

[27:32] That's going to be fifteen ninths if I'm not wrong. So here we have negative eleven ninths.

[27:45] So what that means is that our final answer, the roots of this polynomial are in s. The values that will make the polynomial zero are going to be two thirds plus or minus the square root of negative eleven over d.

[28:02] Negative eleven over nine. So that means we have complex roots. And there's actually a very fun way to think about the way that complex roots

[28:16] play into this difference of squares perspective on the quadratic formula. but it ends up relating to the Pythagorean theorem, connect various patterns in math that come up a lot.

[28:30] Things that might be useful outside of this particular class that you're doing. example that will work out with nice numbers here. X squared minus six x plus ten.

[28:46] We say m, the midpoint, is going to be negative b prime over two. we take the negative of this term and divide by two. d squared is going to be m squared.

[29:02] So three squared is nine minus the product, which in this case is ten. So nicely that's exactly negative one, which means that our two roots, and I'm down to the y or on the page that I've been writing with here,

[29:16] our two roots are three plus or minus i. So what that means for us is that the actual parabola here, it doesn't look like something that crosses the x-axis.

[29:28] But it does have imaginary roots. So if we were to look at the input space, not just in terms of the real number line, Black will be our complex plane color for this moment.

[29:46] We'll call this our imaginary axis, where numbers like i and negative i, the square root of one, or maybe I should say square roots of one, it's got two of them. And then we've got just the real numbers.

[29:59] One, two, three, four. So in this case, our two roots live at three plus i and three minus i. So this, I should be very clear, this is not an x-axis and a y-axis.

[30:13] This entire plane now is where the input lives, where x lives. If you were going to graph it, you'd get some kind of graph that's outside of the paper. you absolutely should, might be bringing to mind right now an absolutely awesome

[30:27] like artificial reality effect he does, where he sort of pulls out that graph. But what's interesting about this is if we look at the magnitude of the roots, okay?

[30:41] Because I could ask you, what is the magnitude of that root? And based on the Pythagorean theorem, it'll be the square root of one of the lengths squared, which is in this case three squared,

[30:56] plus the other length squared, which is one squared. So square root of three squared plus one squared, and that ends up being root 10. It is not a coincidence at all that the magnitude of our roots,

[31:10] I guess sort of no pun intended, the magnitude of the roots of our quadratic equation here are the square root of that constant coefficient. telling us what is the product of the two roots.

[31:24] you might know that if I have two complex numbers and I multiply them together, the magnitude of the product is the same as the product of the magnitudes. So in this case, given that I'm going to have two separate roots who are symmetric,

[31:39] you know, it's going to be three plus or minus some imaginary number, The product of their magnitudes needs to be 10. So you kind of know ahead of time that it should be magnitude of square root of 10.

[31:54] And the reason that this is happening is basically because when you do difference of squares, something like m plus d times m minus d,

[32:07] but that distance is an imaginary value, i, what you get is m squared minus But because i squared is by definition negative one, you get a sum of squares.

[32:24] Pythagorean theorem stuff, all of that, can be expressed as a kind of difference of squares, which itself gives a kind of factoring. And this, this shows up in a lot of very, very beautiful math later down the road.

[32:37] One of my favorite videos that I've made actually is, oh, what did I title it? you're counting lattice points inside a circle. you can express something as the sum of two squares is a sort of factoring problem,

[32:54] but it's factoring not where you're dealing with prime numbers on the real number line, So even simple, I shouldn't say simple stuff, but even stuff that comes up

[33:06] in high school, like the quadratic formula, I think if you're learning it the right way, And remembering these patterns comes up, like I said. What three key facts do you need to be aware of with quadratics to

[33:22] be able to kind of rediscover a kind of quadratic formula on the fly? The first one is how to read the coefficient sitting in front of x. And if we have a quadratic that looks like x squared plus b prime x plus c prime,

[33:36] you can read that first coefficient as the negative sum of the roots. but that is actually worth coming away with. Then the only other thing you need to know is that we could

[33:51] respect to the mean and the kind of standard deviation of those roots. The only thing that looks remotely like memorization is if you want

[34:05] to jumpstart to the end and just say m plus or minus m squared minus p. to remind ourselves that this is actually equivalent to the So let's go ahead and actually do that exercise.

[34:22] And again, if you can, pause and just work it out for yourself right now. So we've got, what are we solving? Ax squared plus bx plus c.

[34:38] give me a systematic way to find these roots. That was a really nice image. I just can't believe that this is something that a computer generated.

[34:53] But evidently, if you know the kind of math that can lead you to create an image like this one, that's the kind of thing that can get your job as an engineer at Pixar.

[35:06] That math is exactly what we're doing right now. Ax squared plus bx plus c equals zero. No new variables coming up.

[35:19] So when we do the first step of rescaling, we say x squared is equal to b divided by a times x plus c divided by a. Now remember how our trick works.

[35:35] We sort of picture in our head, hey, imagine this quadratic has some roots. And we're trying to find the midpoint and the standard deviation. And we can read off that that midpoint is the negative of the second term divided by two.

[35:52] So in this case, that's going to be negative b over 2a. And then that standard deviation is going to be m squared minus the product of the roots,

[36:05] which in this case looks like negative b over 2a squared minus, and the product here is what that last term is, c divided by a, c over a.

[36:18] that's what the distance is, just re-derive it for yourself on the fly. Just go and say, okay, I remember that the product p is just the product of my two roots,

[36:30] which can be expressed as m minus d, m plus d. Oh, okay, that's what gives me an expression for d squared in terms of m squared and p.

[36:45] Don't feel like you have to just come in and know it off the top of your head. So that's m, that's d, and the quadratic formula is just telling us that the

[36:57] roots are m plus and minus, that's standard deviation, which in this case looks like, maybe I'll write it out on two lines because this will be a lot, negative b, actually no, I'll write it on one line for this one, negative b plus or minus,

[37:13] I'm jumping to the original quadratic formula, a little bit too hard ingrained, divided by 2a plus or minus the square root, oh, I wrote this incorrectly, oh,

[37:28] This is what d squared is equal to, d squared is this whole thing, I'm sure lots of people were shouting that in the live chat, I don't have it pulled up now, but to those of you who did, much appreciated.

[37:42] Well, it's going to be the square of negative b over 2a minus c over a, minus c over a.

[37:55] Okay, now we just got to expand this thing, which is frankly not super fun, but it'll connect it to the original quadratic formula to for us.

[38:07] So I can pull out this 2a squared, and I'm just going to write that as 1 over 4a times negative b squared, negative b, yeah, negative b squared, and then I want to also pull out 1 over 4a, I want to be able to say that

[38:23] the last term also looks like 1 over 4a times something, and that something to make it equal to c over a would have to cancel out the 4, it would have to cancel out an extra a, and then c, sorry,

[38:38] because this is really 1 over 4a squared. Yeah, because I pulled out the 2a, so that should be 4 times a squared, I want that to cancel to become c over a, which it looks like it does, so that's awesome.

[38:55] and if this feels tedious, that's kind of the point. because we were just solving any quadratic that was thrown at us without having

[39:08] on your way. Okay, so what can we do here? We can factor out the 1 over 4a squared, and because that's in a radical,

[39:22] its square root will also be 1 over 2a, and then what sits inside is what remains, Okay, and this this is starting to look like the traditional form,

[39:35] because if we pull onto the numerator, negative b plus or minus square root of... Yeah, so this is a negative b squared, same as positive b squared.

[39:47] What should sit on the inside here is b squared minus 4ac. You can see how scatterbrained I am when I'm just doing some arithmetic at times.

[39:59] We all forget a variable or two here and there, but maybe that's why I actually care so much about formulas having nice readable meanings, because I think this is a very error-prone process for someone like me.

[40:12] necessarily know how to read them, I get to the end result, and it's hard for me to say like, oh yeah, of course that's what the answer is. Whereas if I look at something like the simpler variant of the quadratic formula,

[40:27] does the midpoint equal negative b over 2? especially if you know a little calculus. and understand how to find the maximal or minimal point.

[40:42] So that's a thing that gets reinforced with a better pattern later on in your mathematical life, always a good sign that you're learning things well. Again, that has a readable meaning.

[40:55] So when I'm looking at the simpler version of the quadratic formula, a kind of standard deviation. Now I titled this thing, this is the simpler version of the quadratic formula,

[41:10] Like, is this actually simpler? Because, you know, you've got a lot of steps, you've introduced new variables into it. you're telling me I also have to think about like a new term m and a distance

[41:25] But for me, math is very much about trying to draw connections to other patterns, connections in your head rather than just isolated cases. it's about representing the same information in different ways, right?

[41:44] Because what the quadratic formula is doing for us, it's saying, can I go from my coefficients a, b, and c, and can I get to the roots r and s? And we know that there's a very easy way to go the other way around,

[41:59] because we can expand something like x minus r and x minus s. it doesn't add any information to the system, there's really as much information So one of these directions is easy, and one of these directions is hard.

[42:18] one direction being easy, one direction hard, that comes up all the time in math. And another useful thing to think about is how sometimes going in that

[42:30] harder direction will be better if you go through some intermediate step. In this case, rather than thinking of your pair of numbers on the number line just as they are, doing your m plus or minus d trick,

[42:42] because we know that that changes how you think about products. If you go through the intermediate step of expressing that same information with a mean and a standard deviation, that can get you to the roots.

[42:54] And that's all this is, it's just talking about different information flow paths whether those two numbers are the coefficients of your quadratic, So if the lesson you come away with is one of thinking, oh wow,

[43:10] and some of them lend themselves to certain kind of problem solving better than others, well then that is the proper lesson to have with the quadratic equation. Okay, so I think with that I'm going to call it an end to lesson number one.

[43:25] Really want to thank everyone who showed up for this, definitely a ton of fun. going to get stats up on the screen, so it's going to be pretty cool. put together from Khan Academy, but obviously this is the first time

[43:42] so in the end what it's going to look like is these bars that you're seeing live. Okay, clearly they're updating, they're updating.

[43:55] Oh it would be so fun to do this properly before we end. Oh my god, oh this is so great, I think we're, I think we can do it. Okay, I'm grading the answers.

[44:09] This is great, this is how I wanted to end the stream. While I was talking just in the background some magic was being done, this is wonderful. What best describes your relationship with the quadratic formula?

[44:25] I'm a big fan, you might say I'm rooting for it. So it looks like 1724 of you, five short of Ramanujan's constant, are addicted to puns.

[44:39] Let's do a couple more, this is going to be pretty fun. These were just like the joke warm-up questions before we got into the real lesson, is to just kind of wind down with some of what were meant to be introductory jokes.

[44:57] Oh and there's so many of you answering, this makes me so happy. All right so what's our question here? If the quadratic formula had a patronus, Okay well it looks like around 800 of you think it will be something.

[45:18] By the way, I'm being told right now that if there's too many of you who access it, we're for sure going to break the system, and I'm purposefully ignoring that because I'm having fun with this and if it breaks that kind of tickles me.

[45:30] So I'm being told not to say this, but please go to 3b1b.co live and enter questions to this, and then you know whenever things break that would be a perfect time to end the stream because I just think that's hilarious.

[45:43] Okay so oh again 1791, oh I guess we blew past Ramanujan's constant. going to lock in answers where the majority is 1729, I think that would be fun.

[45:57] Okay so it looks like a majority of people went with C. If the quadratic formula had a patronus, it would be an old man hunched over a chessboard, which is the correct answer actually.

[46:10] no correct or incorrect rating, but I don't think that's right. objectively correct JK Rowling would agree style answer. All right let's do the the warm-up question number three here.

[46:27] What integer will most people enter into this box? Again I really want my friends to like struggle, I have bars all over my face.

[46:42] You know this actually seems apropos given that the whole title of this is locked answers and just getting locked down further and further into the quarantine situation.

[46:54] So this one actually now there is a, where do I talk? Help! Bars are attacking me, okay there's an objectively correct answer because there is going to be some number that most people enter. And it looks like 919 of you think

[47:07] that it'll be one particular thing, but we've got a widespread, So again if you want to partake in this head on over to 3b1.co.live.

[47:19] Oh what is the seven? Wow I would not have guessed that most people entered seven correct that seven was the most commonly entered expression.

[47:36] I can make a guess for why that might be the case. Did you know that 69 is the first number where if you square the numbers or the digits zero through nine once and only once.

[47:52] assume is why that was the second most popular answer. But the very end which is actually apropos at this point we can pull up another question which is going to be what I was going to open

[48:06] the whole lesson with so you can kind of see how the plan went here. formula in your real life outside of school?

[48:18] And in this case luckily I'm getting a little bit less you know locked down by the bars trapping me in here because it seems like there's a little bit more consensus around how many times people think they will need to use the quadratic formula in the real life.

[48:32] What I could do is a plot twist on this and say you know interpret this question in light of the lesson rather than when will you literally use negative b plus or minus square root of I always forget it square root of b squared minus 4ac over

[48:45] 2a 2a that whole thing to when are you going to use the principles of recognizing that a product of numbers expressed as a difference of squares can help you solve problems or we're going when are you going to use the principles of expressing

[48:59] your data in terms of a mean and a standard deviation can help you solve problems. we've got a lot of you on the system and it's not breaking and I'm so happy right now I just can't tell you how much this tickles me.

[49:15] So it looks like we've got a wide forming consensus you know for for my sake can we can we just like keep going on this I would love to see if we can get that top bar up to 1729 whatever it might be at the moment we can all guess what it might be but

[49:29] let's see if you can go to 3b1b.co slash live wherever you're watching this I think honestly the best dynamic that I could imagine is if you just pull up your phone and you're watching this on a screen with the one hand and then you're using your

[49:42] phone to to answer questions a lot of you are already watching it on your phone so that wouldn't necessarily work but that is the dynamic I would I would most expect. Okay so I'm just going to wait until we get that

[51:03] top bar up to be Ramanujan's constant of 1729.