The simpler quadratic formula | Ep. 1 Lockdown live math

hello and welcome to the very first episode of lockdown math so in this first lecture what I'm going to talk with you guys about is the quadratic formula and traditionally I think the quadratic formula is one of the most famously memorized things in high school we almost all kind of have this song that sings in our head related to it and yet actually ask someone how often do you expect you're going to be using this in the real world I mean how many people are going to give an answer that's greater than zero to that question now what I want to do is give you guys a lesson regarding the quadratic formula that's a little bit different from the traditional approach 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 now the tactic that I'm going to talk through it's not the usual completing the squares bit that some people will learn in high school and when you learn that you often think okay in principle I could rederive the quadratic formula but frankly it's going to be a bit of a pain what I will talk about is somewhat similar to something that um post and low put out a couple months ago for those of you who don't know him he has a YouTube channel but more than that he's the coach of the US IMO team and he has a company called xbuy does a lot of great math stuff so he talked about kind of an alternate way that we could teach the quadratic formula the way I'm going to talk about it is a little bit different from how he did 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 Solver and that should be the main takeaway lesson for you know this particular topic now these lectures in general I'm going to be shooting for something roughly on the order of an hour for each one of them and the dynamic that I'm going for is one where there's questions that I'm asking the audience and then the audience interacts with it in a very uh well in a very live way now the thing about live stuff though is sometimes it doesn't go the way that you intended so the general technology that we're going to have for this is going to involve me pulling up some kind of question like for example right here uh the question is asking kind of facetiously about your relationship with the quadratic formula and I even had this going over the intro animation for you guys and then what you see is the statistics associated with what the answers you know other people are entering are um and they'll kind of get overlaid we see the bars at the moment all you see are question marks we don't know what the most common answer was but at some point I can choose to grade that and then we'll see except for the fact that it looks like there's quite a few of you guys and it's giving our servers a little bit of trouble so uh we have a couple people who are going to be working that on that in the the background in the meantime I'll just be asking the questions that are pushing us through the lesson for future streams this will hopefully be a much more Dynamic situation 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 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 so one of the questions that I was going to ask is how many times do you expect to use the quadratic formula and when we were doing this uh a little bit earlier with a couple people for sure the most common answer was zero 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 brought it up as this example of something that we teach every student um and it's unclear why they're not going to use it 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 oh ho 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 something I think he was pitching me on um and the basic idea he he was talking through an image that he created purely with computer Graphics I think this is totally beautiful you would think that this is a photograph of some reflective billiard balls but this is entirely computer graphics and the tactics used behind this involve something called Ray tracing 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 now part of that problem though is to know hey if I'm shooting a ray in some Direction will it hit a given sphere and so what you might do is Define a certain Vector you might imagine an amount of time that the beam of light is moving and then you can come up with a graph for how close is that Ray to the sphere as a function of time and in some cases if it passes through the sphere you're going to have two solutions there and once you work out the math it turns out this is a quadratic equation so what you need is a systematic way to plug in these solutions to a quadratic equation and it needs to be programmatic it needs to be some kind of formula because you're doing this for every single Pixel on the screen you also need to be able to know is it just glancing off of the sphere or does it not go through it at all and again all of that comes down to solving a quadratic and this particular engineer at Pixar he did a loose back of the envelope calculation for me that suggested the movie Coco probably use the quadratic formula over a trillion times just given how many lights there are for every single frame in that shot so in some very real sense the quadratic formula is useful to the point of being used a trillion times to produce a movie nevertheless I don't necessarily think that's representative of most people's future and it could be the case that the reason we teach people this formula is because maybe they happen to be that Pixar employee you never know what's going to be useful but I want to make the case again that we can connect it to other patterns and problem solving bits in math 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 I just want to talk about some mental math tricks okay because I think if you are thinking about arithmetic something very basic 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. you know it's easy to see that 5 goes into it and uh it goes into it seven times each of those is prime great we know how to factor 35. now factoring is kind of an annoying task the bigger the numbers get especially if there's no obvious small factors that go into it because if I ask to hey let's Factor 143 I mean how do you go about it I think the way a lot of us think about it is we initially think okay we know that two doesn't go into it because it's not even three there's a nice divisibility check where you do one plus four plus three uh and that sum in this case is five plus three or eight is not divisible by three so that's out five we can see doesn't go into it seven okay you have to think about that for a moment but you might see that seven goes into 140 evenly because that's 20 times 7 okay so that's right out and at this point you might be annoyed with whoever was asking you the question but if you go one step further you might see okay 11. actually 11 does go into it um and it ends up being 11 times 13. I just love the fact that right now people probably have no idea what any of this has to do with the quadratic formula but I assure you highly related because what if I asked you 3599. and this particular example isn't randomly chosen by the way I was reading this article about a Russian programmer and an interview that he had I think it was at Goldman Sachs and he was describing kind of the intensity of that interview environment and in one of them he just walks in the door hadn't even met the person and the person just shouts at him Factor 3599. you know the program is like uh all right and he actually was able to spit out the factors particularly quickly which is impressive given the fact that if we imagine going through all of the different primes you know does two go into it no does three go into it no five no 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 well one thing you might notice about the numbers that I've chosen here is that 35 is awfully close to a square it's actually just one less than 6 squared and maybe it kind of makes sense that its factors are each kind of close to Six you know five times seven it's probably not going to equal six times six but the fact that there are of similar sizes should be intuitive and as it happens it's precisely one less and something similar happens with the other example I chose 143 is rather close to a square number 144. in fact it's one less and it's no coincidence that its two factors are hovering right around the square root of that value one of them is 12 minus one and the other one is 12 plus 1. so it kind of makes sense that the product will be something around 12 times 12. and so from this you might make a guess that hey maybe any time that you have some number that's one less than a perfect square and this you know it's suspiciously close to a very round seeming number 3600. well what is 3600 we recognize the 36 as being 6 squared and 100 is 10 squared so that's 60 times 60. 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 1 less than 60 times 1 more than 60. but of course you don't just want a pattern match let's see if there's a nice reason that it's a true 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 and if we're going to think about this the natural place to turn is to think about squares 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 so let's say it's X by X whatever X is in this case it happens to be 6 but you want to think a little bit more abstractly than that I'm going to take that bottom right corner get it out of here we don't want it the question of factoring is to say can I rearrange the quantity that remains into some kind of rectangle and in this case if I take that bottom rope shloop it on up to the right what you can see is it forms a rectangle one with a side length x plus one and another with a side length x minus 1. so this general property holds quite well and I think that's pretty cool right you see this visually you see that it's this neat mental arithmetic trick where anytime you have some sort of square number minus one you can factor that as X plus one times x minus 1. but of course we can go one step further than that if we really want to it doesn't have to be one sitting there if we go back to our diagram what you might instead do is say hey what if I chopped off a bigger corner of that square what if I wasn't taking x squared minus 1 I was taking x squared minus y squared 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 now that's the geometric way that you can view it which I think gives a certain satisfaction 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 here to see and that demonstrates the power of algebra but if I just said X Plus y times x minus y actually 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 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 so plus and minus d in this case D is just one but in principle it might be something more so if I ask you multiply M minus D times M plus d you know at this point it's pretty straightforward algebra the first term is M times M we get M squared next we have negative D times M so that's minus d m next we have M times plus d m times plus d and then negative D times D so minus D Squared and indeed what comes out is M squared minus D Squared and if you're anything like me the first time you see this in an algebra class it's just one of a whole bunch of exercises that you're doing you're like yeah okay can I expand this one way I guess some terms cancel out nicely but the more you think about this the more intriguing it is because any two numbers we can express you know if I just take any numbers R and S 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 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 and I want you to systematically tell me can each one of them be broken down as a product of two smaller whole numbers well we know that what that's asking is what are the primes because only the primes can't be broken down like that and yet evidently that's a very similar question to saying when can I walk through those numbers and say hey am I able to express you as a difference of squares could I add a square number to you to get another Square that still feels hard but that feels like a very different question to me so I think if you were doing some arithmetic and you had this in the back of your mind that's a good pattern and that's a pattern that's going to come up later in life for example let's say later in life you find yourself wanting to understand quadratic functions all right so here we're going to talk about the the simpler version of the quadratic formula which is really just refactoring the original thing but I I hope you see it's much easier to actually solve a random 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 so what's the task anytime you have some function that looks like ax squared plus BX plus C okay for any numbers a b and c 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 now this is actually equivalent to rescaling everything and I think it's a good common pattern to rescaling where I'm going to divide everything by a so I just want that first coefficient to be a one because that's way easier to work with 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 and this is a different function it is a different quadratic function but the roots are going to be the same and one way that you might see this is if you just graph them so let's go over and pull up our best friend of Desmos right so let's say I have some kind of Parabola in this case I've written it so that one of the roots is going to be 2 and the other is 4 and then G of X here is just the scaled version of the expanded version of that it's exactly the same graph 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 what do you guys want to call that constant see this is the thing if I had the question framework working right now I could just ask you what do you want the constant to be and we would see the statistics come up with what people had but instead maybe I'll throw in a slider and as I change that the function changes it is a different function but the roots are still two and four okay so solving for this rescaled version of the quadratic is the same and that should kind of make sense because s times 0 is always going to be 0. so it doesn't matter how much we scale it so going back to uh to what we're working with here 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 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 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 0 and if I plug in s again this term will also cancel out to 0. so if I expand this whatever R and S are they're the unknowns I should get the same thing as what we have up here 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 especially in a circumstance where that leading coefficient is one and the whole polynomial has been kind of normalized in that way in this case if you just expand it out what we'll get is x squared minus 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 so that becomes a positive R times s okay so what does this tell us 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 uh B value I'll call it B Prime 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 which is kind of cool because what we have right here is um you know it's an it's an it's a system that feels like it should have a solution we have two equations and two unknowns but it's not obvious how we would go about solving it and every now and then uh I think classes will have a unit in factoring quadratics where they basically tell you to just guess and check so in some very fortunate circumstances if you have something like you know x squared uh let's see minus seven X Plus 12. they basically say see if you can guess and find two different numbers that add up to be seven and that multiply to be 12. and if you just kind of sit back there and think you're like okay can I find any two numbers that add to seven and multiply to 12 well factoring 12 we get three and four and three and four do add to seven so yeah in that case you're just totally luck out and you could write this as x minus 3 and x minus 4 because those are the two roots and then you typically move on from that chapter and it's like well in most cases you won't be able to just guess and check and in any case what you're going to look for is a general formula anyway so hope you had fun with that but it turns out there's actually a systematic way to take this puzzle of finding 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 of them and you can see where this is going this is why we talked about difference of squares because the third key factor to Come Away with were 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 and you see where I'm going right that means that the product that we know looks like M squared minus D Squared so uh just to give an example here it's often much more helpful to have numbers let's say that you were given a quadratic like x squared uh I don't know let's do six even numbers will make this easier for us and then seven okay and you were tasked with knowing when does this equal zero I haven't told you how to solve it yet but these three key facts are going to be enough to just basically walk yourself into the answer well what is what is M right because we're going to ultimately Express Our Roots R and S as M plus or minus D for some kind of midpoint well that midpoint is the sum of the two numbers over two that's how we Define averages 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 uh one-half times whatever sitting right there which in this case will be negative three awesome we know what m is but look at the equation we don't have up here 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 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 but we know both of those values we could just write that down over here 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 9 minus C Prime which is that last coefficient or seven well seven handwriting is terrible but I think you guys can work with me you see why I usually animate stuff so that means that's two 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 that means D is the square root of 2. there you go you could do that for any quadratic that I give you you could just walk through that particular process and let me just show you what it looks like in general 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 version of B if the leading coefficient wasn't already one divided by two and then that distance well what did we just do we took the square root of the midpoint squared minus the product of the two and when I'm sort of thinking into my head I've been saying like N squared minus P just because P has a readable meaning I think of it as the midpoint squared minus the product but of course p in this context is just whatever the last coefficient of our quadratic was 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 square root of M squared minus p 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 a rescaling of one of them so this to me is way simpler than the traditional quadratic formula and if you were to try to sing a song to yourself for the song is almost too short it's no fun you're just sitting there like M plus or minus square root of M squared minus P that's it no song No song to be had so let's do a couple practice problems because I do think practice will make it easier and for future streams again um it'll ultimately be the case that I'm giving you these questions and then you'll be able to go to 3d1b.co live and answer them but because there's too many of you we can't have any fun today this is the equivalent of trying to run a class where you have you know I don't know 20 seats sitting out and you're going to do a normal lecture and then there's just people banging at the doors and trying to cram themselves in and the fire marshal comes in and they're like ah you got to cut out the class 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 so let's just do some examples okay that'll kind of highlight what this process looks like let's say we had x squared plus 10x Plus 3. so in this case it's nicely already rescaled for us that's always lovely so I often I just like to draw the picture for myself whether or not the two Roots turn out to be real or even positive it's just nice to remind myself what we're looking for we're looking for where the two roots are and I know that a different way to think about products is to think about a difference of squares so I just kind of think in my head okay I'll be thinking of that in terms of their mean and a kind of standard deviation so I just write down for myself what does that mean well it's negative B Prime over two and if I forget that fact if I forget that that's what the the sum of the two Roots is I could always just go through this little rhythm rule again and say okay if I systematically wanted it to be a quadratic with roots R and S this is what it would look like this is how it would expand so you can rederive it on the Fly there's not too much memorization needed in this context that works out to be negative five and by the way if I do ever make any mistakes which I'm quite positive I will go ahead and throw them in the chat and those will be forwarded to me and I'll be able to correct myself there so we know the midpoint and then we just ask ourselves what's the square of the distance and based on difference of squares that'll be that midpoint squared minus the product which in this context is negative 5 squared or 25 minus the product which is that last coefficient minus 3. so what are the roots r and s well it's negative 5. plus or minus the square root of 25 minus 3 or 22. there you go another quadratic soft ain't no thing let's try another 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 if you can just kind of race ahead and do the same process before I do it that's awesome that is the best kind of learning experience if you're watching this in the future by the way as with any video I highly encourage you to pause and Ponder I really think that's the best way to learn math if you're looking at some kind of lecture in those crucial moments where there's a question asked pause see if you can do it yourself and then see what the answer turns out to be I guarantee you'll learn more effectively that way uh I don't know what should we do let's do maybe 3x squared I think I wrote one down earlier didn't I this is kind of a offhanded thing what did I write that wasn't there oh yeah it was at the top let's do that one 3x squared minus 4X plus 5. why not 3x squared minus 4X Plus 5. all right so in this case step one we've got to rescale things that's what gives the coefficients a nice readable meaning minus four-thirds X Plus Five Thirds great and now same process I'm sort of thinking in my head of this particular image where I want to know the midpoint and the distance so I say that midpoint is uh negative of this second coefficient divided by two so that would become positive four thirds but then that four divided by 2 gives us a 2. so that midpoint will be two-thirds and then the distance squared is M squared minus the product which in this case is Five Thirds okay and M squared in this case is let's see two-thirds squared minus five-thirds all right so this one you know it gets a little Messier we've got a work out our fractions but that's not too bad two-thirds squared is going to be 4 9. I'm off screen a little and then uh what is Five Thirds in terms of ninths that's going to be fifteen ninths if I'm not wrong so we end up getting a negative number which is always fun so here we have negative 11 ninth okay so what that means is that our final answer the roots of this polynomial are an S the values that will make the polynomial zero are going to be two-thirds plus or minus the square root of negative 11 over D Okay negative 11 over d d what am I saying negative 11 over nine okay so that's fun in this case the square root is a negative so that means we have complex roots and there's actually a very fun way to think about the way that complex Roots play into this difference of squares perspective on the quadratic formula um let me actually try it with a simpler example but it ends up relating to the Pythagorean theorem which I think is cool because the whole purpose of this is to try to connect various patterns in math that come up a lot things that might be useful outside of this particular class that you're doing um so I actually have written down for myself an example that will work out with nice numbers here I mean nice ish x squared minus 6X Plus 10. okay so we go through the same Rhythm rule we say m the midpoint is going to be negative B Prime over two so in that case it works out to just be 3 because we take the negative of this term and divide by two D Squared is going to be M squared so 3 squared is nine minus the product which in this case is 10. so nicely that's exactly negative one which means that our two Roots I'm down down to the wire on the page that I've been writing with here our two Roots our three plus or minus I okay so what that means for us is that the actual uh Parabola here it doesn't look like something that crosses the x-axis instead it would be something that maybe sits above it 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 but let me draw it out let me go back to black black will be our complex plain color for this moment 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 you interchange them it's fine and then we've got just the real numbers one two three four so in this case our two Roots uh 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 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 and any of you have watched the channel Welsh Labs which if you haven't you absolutely should might be bringing to mine right now an absolutely awesome like artificial reality effect he does where he sort of pulls out that graph highly worth watching super great moment but what's interesting about this is if we look at the magnitude of the roots okay because I could ask you what is the magnitude of that root and it's a right triangle that we're looking at we've switched from algebra to Geometry now and based on the Pythagorean theorem it'll be the square root of one of the lengths squared which is in this case 3 squared plus the other length squared which is 1 squared so square root of 3 squared plus 1 squared and that ends up being root 10. it is not a coincidence at all that the magnitude of Our Roots 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 because remember that constant coefficient is telling us what is the product of the two roots and if you know something about complex numbers 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 you know it's going to be 3 plus or minus some imaginary number each of them is going to have the same magnitude the product of their magnitudes needs to be 10 we know that offhand so you kind of know ahead of time that it should be magnitude of square root of 10. and the the reason that this is happening is basically because when you do difference of squares something like M plus d times M minus d but that distance is an imaginary value I what you get is M squared minus I times D Squared but because I squared is by definition negative 1 you get a sum of squares so even sums of squares which come up in Geometry 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 one of my favorite videos that I've made actually is um oh what did I title it I think Pi hiding in Prime regularities where you're counting lattice points inside a circle and the key Insight associated with doing that is to realize that asking when you can express something as the sum of two squares is a sort of factoring problem but it's factoring not where you're dealing with prime numbers on the real number line but primes in complex numbers these things called gaussian primes so even simple I shouldn't say simple stuff but even stuff that comes up in high school like the quadratic formula I think if you're learning it the right way it ends up connecting to all sorts of other delightful things and remembering these patterns comes up like I said so just to re-emphasize how did we get here what three key facts do you need to be aware of with quadratics to be able to kind of ReDiscover a kind of quadratic formula on the Fly the first one is how to read the con 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 you can read that first coefficient as the negative sum of the roots if you don't know that already you can ReDiscover it on the fly but that is actually worth coming away with similar with how we can read the other coefficient it's the product of the roots then the only other thing you need to know is that we could express that product of roots as a difference of squares with respect to the mean and the kind of standard deviation of those roots and you can just solve any quadratic that comes to you on the Fly the only thing that looks remotely like memorization is if you want to jump start to the end and just say M plus or minus M squared minus p now to finish things off I think it would be very satisfying to remind ourselves that this is actually equivalent to the traditional quadratic formula something that looks much bigger so let's go ahead and let's go ahead and actually do that exercise and again if you can pause and just work it out for yourself right now so let's remind ourselves of how that goes so we've got uh what are we solving any kind of quadratic ax squared plus BX plus c we want to say no matter what a B and C are give me a systematic way to find these roots again maybe you're doing something beautiful like Ray tracing let me pull up that image again that was a really nice image I just can't believe that this is something that a computer generated I guess I should believe it because like Pixar movies are amazing but um evidently if you know the kind of math it can lead you to create an image like this one that's the kind of thing that can get you a job as an engineer at Pixar and of course what is that math that math is exactly what we're doing right now so let's work it out again ax squared plus BX plus C equals zero this time we're just going to write everything in terms of a b and c no new variables coming up so when 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 we don't give it any new names now remember how our trick works we sort of picture in our head hey imagine this quadratic has some Roots let's call them r and s 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 2. so in this case that's going to be negative B over 2A great and then that standard deviation is going to be M squared minus the product of the roots 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 and remember if you if you don't remember this idea that oh that's what the distance is just rederive it for yourself on the Fly just go and say okay I remember that the product p is uh just the product of my two Roots which can be expressed as M minus D and plus d oh okay that's something I can write as M squared minus D Squared oh okay that's what gives me an expression for D Squared in terms of M squared and P don't feel like you have to just come in and uh know it off the top of your head but after you do it a couple times it it sticks in your memory so that's M that's D and the quadratic formula is just telling us that the roots are M plus and minus that's standard deviation which in this case looks like maybe I'll write it down 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 I'm jumping to the original quadratic formula a little bit too hard ingrained we'll go back to two lines negative B divided by 2A divided by 2A plus or minus the square root oh I wrote this incorrectly oh someone should have corrected me this is what D Squared is equal to D Squared is this whole thing so D itself is the root of those I'm sure lots of people were shouting that in the live chat I don't have it pulled up now but for those of you who did much appreciated um so what do we have here well it's going to be the square of negative B over 2A minus C over a minus C over a okay now we just gotta expand this thing which is frankly not not super fun but it'll connect it to the original quadratic formula to for us so I can pull out this 2A squared and I'm just going to write that as 1 over 4A times 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 the last term also looks like 1 over 4A times something um and that's something to make it equal to C over a would have to cancel out the four it would have to cancel out an extra a and then C sorry because this is really 1 over 4A squared let's see have I done this right yeah because I pulled out the 2A so that should be 4 times a squared so over here I have 4 times a squared I want that to cancel to become C over a which it looks like it does so that's awesome so we can go over here negative B over 2A plus or minus and if this feels tedious that's kind of the point the whole quadratic formula is more complicated than it needs to be because we were just solving any quadratic that was thrown at us without having to do this but this is what happens if you don't introduce any new variable names on your way it's like code that hasn't been refactored properly okay so what can we do here we can factor out the 1 over 4A squared because that's in a radical its square root will also be 1 over 2 a and then what sits inside is what Remains the well-familiar negative B squared minus 4 a okay and this this is starting to look like the traditional form because if we pull onto the numerator negative B plus or minus square root of why am I writing negative B squared yeah sorry this is a negative B squared same as positive B squared what should sit on the inside here is B squared minus 4AC for AC you can see how scatterbrained I am when I'm just doing some arithmetic at times it's okay we all forget a variable or two here and there but I get 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 if I'm just trying to work through with a bunch of symbols that I don't 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 right if I basically refactor it and I'm saying okay does the midpoint equal negative B over 2. I can kind of fact check yeah that that makes sense especially if you know a little calculus those of you will be able to look at that quadratic formula and understand how to find the maximal or minimal point 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 I can ask myself if it makes sense that the distance looks like this again that that has a readable meaning so when I'm looking at the simpler version of the quadratic formula we have M plus or minus a kind of square root of the variance a kind of standard deviation so it's a refactoring now I titled this thing this is the simpler version of the quadratic formula and I think one could rightfully complain like like truly simpler because you know you've got a lot of steps you've introduced new variables into it now in addition to thinking of the three coefficients a b and c you're telling me I also have to think about like a new term M and a distance between them but for me math is very much about trying to draw connections to other patterns and I think things are much better remembered if you have that web of Connections in your head rather than just isolated cases and the whole list whole lesson here if we think about what's going on it's about representing the same information in different ways right because what the quadratic formula is doing for us it's saying can I go for 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 because we can expand something like x minus r and x minus s and really because that a can get Scaled out it doesn't add any information to the system there's really as much information as two different constants in there so one of these directions is easy in one of these directions is hard and this idea of information that can be expressed in separate ways and translation One Direction being easy One Direction hard that comes up all the time in math that's a very useful thing to think about and another useful thing to think about is how sometimes going in that 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 um just as they are doing your M plus or minus D trick 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 and that's all this is It's just talking about different information flow paths that can help go between various ways to just Express two numbers whether those two numbers are the coefficients of your quadratic are the roots of the quadratic or the mean and standard deviation so if the lesson You Come Away with is one of thinking oh wow sometimes there's a lot of different ways that I can represent my data 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 really want to thank everyone who showed up for this uh definitely a ton of fun next time we're gonna have the live quizzing Dynamic where we're gonna get stats up on the screen so um it's gonna be pretty cool I think this is a very cool thing that two former co-workers of mine put together from Khan Academy but obviously this is the first time that we're trying it it's a little rough around the edges so in the end what it's going to look like is These Bars that you're seeing live oh interesting these are bigger numbers than they were before okay clear release they're updating they're updating oh this is exciting do I have actual access here 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 are you ready for this guys okay I'm grading the answers oh oh wonderful oh we can see what people answered 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 so it looks like the most common answer what's our question here what best describes your relationship with the quadratic formula and the most common answer is C I'm a big fan you might say I'm rooting for it um so it looks like uh 1724 of you five short of ramanujan's constant um are addicted to puns 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 but I actually think this is the perfect way to end the whole lesson is to just kind of wind down with some of what were meant to be introductory jokes oh it's working I love this so much 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 what would it be okay well it looks like around 800 of you I think it will be something but by the way I um I def I'm being told right now that if there's too many of you who access it we're for sure gonna 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 so uh I'm being told not to say this but please go to 3v1b.co live and enter questions to this um and then you know when everything's break that would be a perfect time to end the stream because I just think that's hilarious okay so oh again 19 1791 oh I guess we blew past ramanu John's constant maybe I can I can try to see if I can grade this at a point right where we're going to lock in answers where the majority is uh 1729 I think that would be fun okay so uh it looks like a majority of people when with C if the quadratic formula had a Patronus it would be an old man hunched over a chessboard um which is the correct answer actually you know this this question was structured as a poll where there's no correct or incorrect grading but I don't think that's right I think it should have been structured where C is the objectively correct JK Rowling would agree style answer all right let's do the the warm-up question number three here oh this is a fun one okay uh what does it say what integer will most people enter into this box okay oh lots of answers coming in um again I really want my friends to like struggle bars all over my face um you know this actually seems apropos given that the whole title of this is lockdown math that it starts to look like I'm slowly being imprisoned by everyone's answers and just getting locked down further and further into the quarantine situation um so this one actually now there is where do I type 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 that it'll be one particular thing but we've got a wide spread this is actually pretty fun um so again if you want to partake in this head on over to 3b1b.com live I actually think the best way that you could do this um oh what is the seven wow I would not have guessed that most people entered seven and in a weird way like the plurality of you are definitionally correct that seven was the most commonly entered expression um 69 being the second most common I can I could have guessed for why that might be the case did you know that 69 is the first number where if you square it and then you Cube it those two values between them encounter the numbers or the digits zero through nine once and only once uh yeah it's the first number with that property which I 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 the whole lesson with so you can kind of see how the plan went here how many times do you expect to use the quadratic formula in your real life outside of school um and in this case luckily I'm getting a little bit less you know locked down by the bars uh 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 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 2 a 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 can when are you going to use the principles of expressing your data in terms of a mean and a standard deviation can help you solve problems that I think would give you wildly different answers but at the moment 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 um 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 we can all guess what it might be but let's see if you can go to 3b1b.co 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 phone to uh 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 um okay so I'm just going to wait until we get that top bar up to be ramanu John's constant of 1729. for those of you who don't know the famous story behind that number is that I think it was the mathematician Hardy was visiting ramanujan when he was in the hospital and he said I found this taxi cab that I was following on my way here and the number on it was 1729 and I thought to myself what a wholly unremarkable integer there's nothing special about it you know it's not a square or prime or cube or anything like that but we're getting close so let me not let me not mess this up oh God I over blew it okay um you know and then so he tells this to ramanu John he says 1729 uh has no remarkable properties and ramanujan says oh but can't you see 1729 is the first number that can be expressed as the sum of two cubes in two separate ways uh he says I don't understand how you can't see it Hardy because 1729 is on the one hand 12 cubed plus one cubed but on the other hand it's 10 cubed plus 9 cubed and it is the first natural number that has this property that you can express it as the sum of two cubes in two separate ways and that's just such a delightfully random fact that I think it became immortalized in the history of math so it looks like 1769 of you which is appropriate given the second quest uh second most common entry here um said that you thought you would use a quadratic formula zero times and honestly I agree with you I don't think you're ever going to use the quadratic formula unless you're a Pixar engineer in its actual form but I think you will use some of the other lessons that I tried to get across here so with that I think that is the perfect time to call things an end here really appreciate you guys joining on this first stream the next lesson is probably but don't hold me to it going to be about how to not memorize uh trig formulas some of the things like cosine of 2 Theta sine of 3 Theta all of those things and it'll be on Tuesday at the same time unless something changes and you can check the banner of the channel about that and then for those of you watching in the future hopefully this is just a long playlist of high school lectures that you can join and appreciate so thank you so much for joining I hope you to see you next time and keep loving math

Channel: 3Blue1Brown

Views: 1,201,515

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Id: MHXO86wKeDY

Length: 52min 11sec (3131 seconds)

Published: Fri Apr 17 2020

Reddit Comments

[u/KnowsAboutMath]

There's an alternative quadratic formula I like that I learned from the book Numerical Recipes. You start with the usual formula:

ax² + bx + c = 0

yields

x = [-b +- sqrt(b² - 4ac)] / (2a)

Suppose you're evaluating this numerically, e.g. as part of some program that is simulating something. What if b >0 and we're taking the + root:

x = [sqrt(b² - 4ac) - b] / (2a) (Eq. 1)

Now suppose that b² is very very large compared to 4ac. Then you have a precision problem because you're taking the difference of two large numbers that differ only by a little bit. In finite precision, your result will be garbage.

The trick is to rationalize the numerator in Eq. 1 by multiplying the top and bottom by sqrt(b² - 4ac) + b. Then you are left with

x = -2c / [sqrt(b² - 4ac) + b]

and your precision problem has disappeared!

[u/seamanroses]

The video will be live 9 hours from this comment.

[u/thereligiousatheists]

I actually self-discovered this trick to make calculations easier; whenever b is even, you just keep dividing out by 2 until you get ac = (some integer) / 4 or until what's left as the coefficient of x is an odd number. This helps because all those factors of 2 would have cancelled out or come out common anyway, but doing it beforehand with the equation is less prone to silly mistakes as compared to a complicated expression with square roots and fractions.

[u/[deleted]]

Gosh Grant is handsome

[u/theCJoe]

This seems to be very near the pq-Formula. In germany we (the Math teachers) can usually decide if we teach pq or abc formula, but I guess most decide on the easier pq formula for weaker classes. To solve: ax² +bx+c=0 divide by a, b/a = p, c/a = q, you get x² +px+q=0 Then you get x_1 and x_2= -p/2+-sqrt((p/2)² - q) Edit: To clarify. In my experience pq is taught as the standard unless a class is very advanced in their math skills or the teacher/school has another preference/curriculum. Sorry, didn‘t want to diss Sweden or anyone for that matter...

[u/PM_ME_FUNNY_ANECDOTE]

This form was taught by one of my undergrad professors in an offhand way. and I appreciated it!

[u/lolburgerdog]

Alternative method using complex number and no completing the square.

Say you want to solve az² + bz + c = 0

divide by a and make the substitution p = b/a and q = c/a to get the equation z² + pz + q = 0.

If p = 0 then z = ±Sqrt(-q) = ±Sqrt(-c/a) and you are done.

In the case p != 0 assume the answer is the complex number x + iy and plug it in for z.

(x+ iy)² + p(x+iy) + q = 0

Expand everything

x² + 2xyi - y² + px + pyi + q = 0

group by real and imaginary numbers

(x² - y² + px + q) + i(2xy + py) = 0

Note we have two things equal to zero

x² - y² + px + q = 0 and 2xy + py = 0

Use the second equation to solve for x

x = -p/2

and substituting this into the first equation gives

(-p/2)² - y² + p(-p/2) + q = 0

and solving for y gives

y = ± (1/2 sqrt(4 q - p²⁾⁾

And since z = x + iy

z = -p/2 ± i sqrt( 4 q - p² ) /2

and pulling i inside the square root

z = -p/2 ± sqrt(p² - 4q) /2

and switching back to the original constants gives the quadratic formula

z = -b/2a ± sqrt(b² - 4c) /2a

[u/sassyassasyn]

The best method, if you can't remember the formula, is Bhaskara's algorithm - multiply the whole equation by 4a and add b² to complete the square and the equation becomes

(2ax+b)² = b² - 4ac

Easy peesy!

[u/policalcs]

Something to look forward to when it goes live later today (4 hours from now). I wonder if he'll be exploring Po-Shen Loh's alternative method for solving quadratic equations.