Skitter cool to be proud of these things, although they aren’t particularly impressive and you do sound like a douche. That said, if you don’t just not stutter but also don’t use pause words (like Grant, who practically never says uh/um/like/etc), did you practice it or did it happen naturally? (I want to learn to do this as well)
0:00 - Introduction 2:56 - "How often am I going to use this?" + ray tracing example 5:38 - Mental math tricks (factoring numbers, differences of squares) 13:36 - Properties of quadratic functions 18:40 - Deriving the variant quadratic formula 23:07 - Practice problems (ft. complex numbers!) 34:10 - Deriving the traditional quadratic formula from the variant 41:07 - Conclusion (key takeaways) 43:21 - Fun with joke poll questions
This is brilliant! I didn't know they had this in the 80s :) I especially like the writing on the notepad bits. Excellent sound quality as well. The what? The maths? (yeah, I'm from the UK :-) That, well, went whoosh above my head :-) Still. I shall return :-)
m ± √(m² - p) is such a meaningful way to put it. Leibniz stressed the importance of choosing meaningful symbols to serve as a vehicle for thought. That is exactly what that is! All that a, b, c, D stuff brings no mental image to my mind, which is now officially blown. Having taught maths for so long, even using that factoring trick, but the dots were still not connected until now. Thank you!
@@pieters286 Possibly. I am not that versed in non-germanic languages, but I would suggest writing it as mean +- sqrt(mean^2 - product) and translating those words to the desired language and abbreviate to single letters. The point is that the letters signify some easy-to-understand property of the problem.
@@eskilwadsholt4289, oof, no thank you, people would abuse the hell out of that. You can open Character Map and search for particular Unicode characters by name, too.
Grant, I use the quadratic formula just everyday. It's built in my spreadsheets for computing the steel area needed for reinforced concrete beams. The same thing we (Structural Engineers) use for calculating reinforced concrete slabs. For columns, the coefficients change and the forces applied harden the problem, and the result isn't a parabola, but a revolution elipsoid (yes, I know, fellows, that the exact shape is uncertain and is not an exact elipsoid, but think not of the full problem, with second order effects, but of the basic problem, just first order effect). Quadratic formula is my everyday breakfast, lunch and dinner. Is my drink. I breath it. I bath on it. And yet you have something to teach me! Thanks for the update, Grant. Thanks for all you do.
@@hulmey676 And bending moments diagram for an uniformly loaded beam. And shear forces for an linearly loaded one. The deflections, rigorously, are a 4th grade polynomial for the simplest case. Anyway, quadratics. Quadratics everywhere! 😅
@@vinitagupta4760 It surely depends. It varies from one to university another. But calculus is way beyond this, I assure you. If there is calculus in your course, you will need to grasp this and much more.
55yo Ph.D. physicist here, 25 years post-graduate. I watch 3b1b all the time, and I never tire of it. Grant's descriptions of math are beyond compare. I expected nothing less of his lesson on the quadratic formula. I was not disappointed.
17 year old high school junior, looking into going into computer science. I love the way that he explains things in different ways to help build intuitions
25 yr old vfx artist, so the correlation with ray tracing was kinda interesting to me. Always wondered how renderers algorithmically solve intersections with spheres as opposed to triangles made of three vertexes.
You're ability to make complex mathematical topics easy to understand with excellent visuals is astonishing. I think I speak for the entire Internet when I say how profoundly grateful we are that you're not just some stuffy professor in a university where only a small handful of students get the privilege of learning from you. Instead, you generously share your talents with the world and we have all become your students. I'm tremendously grateful for your videos Grant and they have caused me to fall in love with math again! Also, it's nice to finally see your face.
In class with normal teacher: If a negative is in a square root, then we'll just say the roots don't exist. Don't worry about it. 3b1b: **Keeps going**
Fun fact, there's a re-arrangement of terms of the quadratic formula, known in computer graphics circles as "citardauq" ("quadratic" spelled backwards), which provides better precision on computers with floating-point numbers. Whenever you draw a Bezier curve with the pen tool in Photoshop, root-finding is one of the approaches we use to draw it, and the "citardauq" formula is what's used there.
It's actually hard to find by googling the name. Try looking up "quadratic formula for floating points". There's a good stackexchange post talking about where the quadratic formula is lossy for large floating numbers.
This represents the difficulty in education: does one define such that the formulas are simple or does one write the formulas in such a way to make the relationships they represent be obvious?
Teachers need to spend more time on one question. It would be cool to do just one day of analyzing the quadratic formula. When you go deep into a question, the beauty of math comes out
@FriedIcecreamIsAReality It is, at the end of the day, always a time constraint, ultimately that imposed by death, that is the difficulty in anything. Heck, some economists even define value that way. Everything in math is a definition; in some sense that's the entire point. If one takes the time to give 3 definitions where one suffices to provide a prerequisite you just made all of math 3 times longer. So instead of being able to contribute to mathematical research at, say, 20, now one has to learn until 60 to write their first published result. This is how we are able to teach what was the cutting-edge 180 years ago to undergraduates today: we've filtered out the redundancies.
@FriedIcecreamIsAReality The question that matters is what teaching strategy leads to the student's greatest understanding in the most useful (read: smallest) amount of time. Teaching both doesn't *seem* to suit that requirement, although I suppose it might under certain circumstances. In general, my observation has been that you teach with simpler concepts and more involved formulas to begin with, and then gradually simplify the formulas and ramp up the conceptual requirements. Perfect understanding of math would have extremely simple formulas contain entire universes of depth to them, like the Maxwell equations, for example.
Same problem here - I'm procrastinating with my maths homework to watch maths. The issue is that this particular homework is tedious, boring, and uses a website that is not at all user friendly. And they've set me ~300 tasks to do on it, with 10-20 questions each :c
Great ideas from a great teacher: - The best way to expand your understanding about any subject is when you connect it with real life problems. - Try to see the concept by different visions since that lets it sink deep in your mind. As an example: look to the quadratic formula using geometry by changing the variable X by a rectangular and the rest by the suitibale geometrical shapes. - The ultimate way to learn is to practice what you learn immediately. - Do not stop if you find yourself go out of the topic. Math likes making connections. - To go from A to C, look for another point B which will decrease the difficulty.
It’s been bugging me for a while as a math teacher that the formula for finding the x-coordinate of the vertex and the descriminant are taught separately and then students are taught the quadratic formula as though they are separate ideas. The quadratic formula is this beautiful combination of the two that you explain in a unique way, here. I love how this explanation focuses on that idea by making “m” your “-b/2a” (in a slightly different order). Then you attach meaning to how the desriminant finishes finding your solutions, or zeros. Conceptually this is beautiful on several levels.
Grant, I had asked my 14-yr old son to watch this live yesterday (he loved it!), and I came back to enjoy it myself today...truly wonderful. Thank you! So great to see you adding such mathematical beauty into this world, and sharing it with all of us. Your channel is a favorite of mine, and now even more so. By the way, my number would have been 137 for the fine structure constant. Please keep doing these. I will contribute to your Patreon in my appreciation for what you add to this world
It is crazy how something as notoriously difficult as quadratics can be expressed in a way an 8th or 9th grader can understand. A testament to his skill in explaining mathematical concepts.
@@ryansajeev123 yesterday when he was doing a secret test-stream for a small audience, people answered 69 too, but he just laughed at it saying "of course it would be one of the answers" or something. This shows that he expected that and prepared for today's stream lol. Next time let's do 420
I know its been a couple of years, but these live streams were something I looked forward to during lockdown. It was like having someone come over to my house and say, 'hey - want to learn something interesting?'. Thank you for that. It was great and it still is.
I have switched careers and no longer a techie. However I was always condescending about the "put the formula get the answer" approach as lots of questions went unanswered. I wish my generation was taught like this. Keep these videos coming. Subscriber from a long time now. My curiosities are finally getting answered. Thank you so much!
People wonder why teachers don't get paid like superstars... well, if they were all like Grant, they would. This guy is a prodigy of math communication. The programming skills, the verbal fluency, the pacing, the warm-hearted nature... a true gem. Thank you for the vids, Grant.
I am sorry, and sure that you just wanted to point out your appreciation for Ben, but people don't wonder why Teacher don't get paid like Superstars, they wonder why they don't get paid like they impact the future of our children. And yeah Grant is great, but he probably couldn't do what he does having 5 classes with 30 people in them.
Sorry, who is Ben? I am a high school teacher myself with 4 preps and 5 full periods (2 Algebra II sections and 3 different programming courses), so I concur with what you said. The job is hard! In my district, we get paid fairly well. I'm in my 6th year in public schools and I get about 78k, which I feel like is not too bad especially considering we get full benefits.
I actually also wrote a ray-tracing graphics engine and the first formula I used was the quadratic formula. I also used the formula for finding the intersection between a line and a plane. So, lots of these algebraic formulas are found in geometry, as well. It was my most favourite activity in maths - solving complex geometric relationships by simplifying them down to simple algebraic formulas. Great lesson. Thanks for the video.
Honestly with such an in depth explanation, it's so easy to recognize where the pieces for the original quadratic formula come from that I find it easier to remember the form with a, b and C now.
Wow I never thought about it this way. This phrase unlocked in me so many new possibilities. Possibilities that I've learned for code, but never thought to apply to math. A huge "thank you" for that!
Solving a formula : representing data in a more useful way for the problem at hand. Second "mind blown" moment. I knew that I'd learn something by watching this video, even if I thought I was already quite familiar with the topic.
And then, you learn that π isn't the true one, τ is. Then, that power notation is a mess. Later, that sets are not the most fundamentals objects, mult-fuzzysets are way more powerful. And finally, you discover that infinity might be a lie. Good luck! ^^
@@MindcraftMax Honestly pi vs tau is such meaningless and pointless discussion, and power notation is only a problem for high schoolers. Most people get through math fine with sets as the most fundamental object and what do you even mean by 'infinity might be a lie'? I feel this is the problem when people get too much math exposure on YT without really doing math.
I spent two years having memorized the vertex formula, the weird guess and check factoring, and the messy quadradic formula, now I not only know a much neater formula, but I know why things work! THANK YOU SO MUCH!!!!!
I've been using the difference of squares trick to multiply numbers quickly (e.g. 17 * 22 is close to 18 * 22 = 20^2 - 2^2, so subtract another copy of 22 to get 374). The trick has been ingrained in my head ever since I understood the algebraic reasoning behind it. But somehow, I had never bothered to think of the geometric interpretation of (a-b)(a+b). I love learning things in different ways.
Here in Austria, this modified quadratic formula is the only one we learn ("small quadratic formula"), the original ("big quadratic formula") is mostly just listed in formula books. x^2 + px + q = 0 ==> x1,2 = .-p/2 +- sqrt((p/2)^2-q) ("p-q-formula")
I don’t get why one would use this instead of the simpler one you get if you scale to x^2/2 + bx + c = 0 which gets rid of those half factors that are pretty ugly in the p-q one.
Can't imagine how good my math would become if I were to watch this several years before! Thank you soooo much for reigniting my enthusiasm for math when I am about to lose confidence during recent study! God bless you❤
I have always wanted to know how the quadratic formula was derived and for some reason, I never learnt it. I'm so glad that I realized it today. That is such a beautiful explanation. Thank you so much, Grant!
"How many times do you expect to use the quadratic formula?" I just used it yesterday when I was extrapolating the familiar category 1-5 hurricane scale beyond the familiar categories of 1-5 by fitting a quadratic to the minimum windspeeds, plus those for the "proposed" category 6 and 7. The answer I got was a category 17 hurricane. (Sometime afterwards, I realized an exponential would be better, then we'd have a log scale like with earthquakes.) I also remember a^2 - 1 = (a+1)*(a-1) extraordinarily well from high school.
This video kind of underscores how I always felt about math - that it is an interconnected language that makes sense if you see the right patterns. My problem is I don't naturally perceive those patterns. I like that this video has given me a small glimpse.
this is exactly why people don't find math useful or are confused by it. there is no relationship whatsoever between the language that is typically used to describe wildly different mental images. for some people with a) better educators, b) better access to knowledge, c) better arrangements of thought patterns, it's a completely different experience to analyze a symbol-ridden mental gymnastics from another world, only to conjure an admittedly beautiful solution, but the one that a 7-year-old could imagine just as vividly, given enough patience and encouragement to break through the language barrier. I'm 40 years old, and have been programming for 30 years, and solving progressively hard-ish-er math problems all the time, but was never formally educated at a level I'm readily operating with, and I'm always hardcore struggling with math papers; sometimes it truly feels the same as staring at the ancient egyptian hieroglyphs. like a child standing against the adults' way of passively forbidding me some hidden knowledge employing conspiratory ciphers for my "own good". but because I'm not a child, and things won't ever improve on that front, instead a voice inside me screams of gratuitous elitism, of an artificial language barrier and a consensual unwillingness to communicate in plain symbols. of course, I rationally understand that's only the half-truth. in the meantime, I am extremely grateful that Grant and people like Grant exist, because I have significantly pushed myself beyond the comfort zone in the last decade thanks to them. whatever my previous schooling did, had exactly the opposite effect on my self-esteem when it comes to mastering the math that I can apply to real problem solving.
Hi!, I am a former math teacher and I love this video. One way I used to teach Quadratics was starting with the open-ended questions, "If you have 50 meters of fencing what is the largest fenced-in dog pen you could construct?" This question leads us into the whole discussion about rectangles, squares, lines, area, etc. After a week or so about discussing this question, we get to the fact that a quadratic is simply the product of two sides of a rectangle or even more simply the product of two lines, i.e (x+1)(x+2). Anyways again, thank you for the great content.
This video is both amazing and so calming. I really wish I'd a) been taught quadratic equations like this (so I could actually see what they're doing!), and b) had you to lead my lessons because you've just got such a calm and affirming persona!
I would like to start off by saying, Thank you grant! For taking the time to share such an interesting take on such a well known topic! As soon as you wrote down the 3 key takeaways, I saw that there was a connection to the the original quadratic formula and I actually tried proving it! I succeeded and was elated that I could do it. I wanted to write down the proof in the comments, when you explained it your self a few minutes later....
Ah.. I saw this presumably after Grant fixed this and coincidentally now if we hit the time stamp it takes to classic 3B1B animation.. now I get it that Grant starts talking from there..😅
I do find it interesting, I live in Germany and we did learn that "simpler" quadratic formula, even through almost that exact derivation (tho that last fact probably changes as teachers do). In Germany it's called the "p-q-formula", as the polynomial is then usually described as *x² + px + q = 0* and the result is then *x = -(p/2) ± √((p/2)²-q)* I always thought it funny that many people I know from other countries found this version of the formula more pleasing and easy to work with, but seamingly only a few countries (obviously including Germany ^^) teach it that way. Not to bash on, or praise any country. Just some observation I had over the years
Yes, actually previously I've seen Grant's method described by Po Shen Lo but it's always the traditional formula that's derived from completing the square that's conventionally used in most countries. I'm very happy that more of the sum/product of the roots/midpoint version of the quadratic formula method and rhetoric is being spread out. NOT ONLY is the proof good but short BUT ALGORHYTHMICALLY it is a lot simpler to do the computation.
In for beginners it´s just derived differently than in the viedeo: ax^2 + bx + c = 0 | divide by a (b/a = p ; c/a = q) x^2 + px + q = 0 | -q x^2 + px = -q | adding (p/2)^2 to complete the binomial formula x^2 + px + (p/2)^2 = (p/2)^2 - q | rewrite with the binomial formula (x+(p/2))^2 = (p/2)^2 -q | take the root x + (p/2) = ± sqrt((p/2)^2 - q) | - (p/2) x = - (p/2) ± sqrt((p/2)^2 - q)
Ahhh...die gute alte Mitternachtsformel :) Context for everyone who isn't german: A wide spread nickname for the pq-formula is "Mitternachtsformel" = "midnight-formula" because when your maths teacher wakes you up at midnight they'd expect you to have it memorized.
I am now in MSc in Engineering and even though I had most of your subjects already I still find your channel amusing. I wish I had your depth of understanding and imagination. And, like you, I wish we had people teaching us with passion, not just commitment. Greetings from UK
19:32 "its often much more helpful to have numbers", this guy truly knows how to teach maths. Some people cant just visualize the general rule without first given an example first
I think this method of simplifying formulae to contain terms that can be thought of in concrete terms (and are not like the a,b,c,d in the quadratic formula which don't form any mental image) would be extremely helpful for *Coordinate Geometry!* Equations of a circle, parabola, and other curves all have terms like a,h,b,g,f but they don't project any mental image. It's hard to think how the curve's shape will change if we tweak one of those variables. It would be really great if you consider this as an idea for a future episode
You are doing crucial work for our community. As a student, I find these videos very important for further studing, bacause of the way you explain all the topics. Please, keep doing that. Dont be afraid to include even more challenging topics. Thank you in advance.
43:00 actually introducing the mean and deviation into the problem, making the "hard" way of solving more easy, is exactly analogous to involving a catalyst in a chemical reaction.
Thank you, my passion with math wane in highschool with poor marks and seemebly so much to memorize is such a short amounts of times, math educators like you are what make remember what I love about this subject:)
Blimey, I just realized that this is the pq formula! It's the standard formula taught in Sweden for solving quadratic equations. Brilliant explanation, as always! I hope I'll catch your next stream.
Seems like I'm in a minority here but I find the standard formula quick and easy to use. If a != 1 and you have to normalize the quadratic, I think my (former) students would be more likely to make an error.
Huge, huge thank you for this series of videos. Whilst I’m no longer in school the wonder of these is in the community, spirit of adventure and warmth. Gosh I wish all maths was presented with these elements.
I dont know why but every time I see your videos, it reminds me of Professor Gilbert Strang. Maybe, I can already see you turning into one of the best math teachers as you grow old. The way you said Keep loving math, shows the love that you have for the subject and the passion with which you teach it.
That's pretty crazy. In Sweden, this is actually the formula we get learned from the beginning when solving quadratic equations but here in Sweden we call it the PQ-formula where "p" is b'/2 and is basically your "m" and your "p" is our "q". Never thought about it in such a way that was presented in this video tho so this was definitely still a learning experience for me :) Thank you for doing this Grant!
Hey Grant! Great lecture as always. I’d like to suggest you to do a video on books you enjoyed reading. I know it is a bit off topic, but that amount of books behind you just activated my curiosity
In Italy, I was taught in school that (x+a)(x+b)=x^2+sx+p, where s and p stand for "sum" and "product" of a and b. We were also assigned "x^2+sx+p" exercises to solve as "(x+a)(x+b)". By extension, also that (x+a)(x-a)=x^2-a^2, but I'm not sure which one is taught first. But you explain it so nicely that it's a pleasure to listen to things I already know...
Its basically the same thing as the main general formula. Since x= (-b±(√b2-4ac))/2a Since a=1 It becomes x=(-b±(√b2-4c))/2 On division it becomes- x=( -b/2)±√((b2/4)-c) Taking -b/2 as (m) We get m±√(m2-c)
Please consider doing a Linear Algebra in this speed and this kind of format. It really helps us to follow your thought and give us time to think about the problem. Thanks for sharing your knowledge !
34:05 What you wrote down here is basically what I know as the "pq formula". We only learned that one in school (in Austria) and I'm surprised people all around the world learn the general one
The real name is Viète's formulas (for the case of quadratics, it can also be expanded to higher order equations). In the Czech Republic it's taught as well, along with the sing-songy general formula.
Degree in math, engineering, over a decade numerical modeling ... I thought I knew the formula. But if I don't use it for a while, I still have to double-check I haven't made a mistake. Now I *understand* the method. No formula required. Thank you kindly!
@3Blue1Brown - I just wanted to let you know this: I'm currently 33 and studying a Physics and Nanotechnology degree. I went back to university because of people like you sparking my interest in physics and math. I have found out since, that I am a very visual learner. I don't look at an equation and see properties in the language. I see an equation as objects. Thus, I see things like ke + pe = te as a block being lifted a certain height as something that increases the block total energy. This is equating the force in the visualised version in my mind to work expended. Visually seeing this also intuitively helped me connect the idea of Forces in Newtonian physics to the ideas of relativity which are extremely unintuitive. In line with this, your relation of perfect squares minus one being factorable by two numbers made so much sense to me, after having paused the video and attempting to explain it through factoring the numbers separately without visuals. I tried everything and I couldn't figure it out. Then I unpaused, watched the visual, and it made sense instantly. Please do not stop making these videos. I guarantee, you will inspire an entire generation of students in a way that schools currently can't. Thank you.
23min: The "song" is now a jingle. Say, Grant. You can also think of m and d as center and radius (of a 1-dimensional ball). On that "Ramanujan constant" - I saw a "kicker" to this incident on some other math YT channel I can't recall now (Mathologer? Numberphile?). The story was that Ramanujan was familiar with this property of that number, because of his trials at solving Fermat's Last Problem. It's a near solution (missing equality by a single unit!) of x³ + y³ = z³ with x = 9, y = 10, z = 12 Overall, I think this video/lesson/fun excursion had a great flow, as always. Fred
My cram school has introduced us to a similar method of doing quadratic formulas using the sum and product of its roots, however we were never taught any proper way to find those products without being given the roots. That rescaled formula looked familiar and your video was very enlightening as to how I could use it when solving problems, much appreciated Grant.
You're an amazing teacher, man. It's great to see your face and see how much enthusiasm that just leaps out of the screen. I wish you were around when I went through high school.
I was going to write about a character that has truly been elusive for years, but just did a search on google to verify my claim. Boom! CGP Grey??? With a face??? Nooooooooooooo! (Joking, but his face has been a myth for so many years.)
Equation at 17:57 is following x^2 - 7x + 12. By using this method I get 3.5 ± √{0.25}, but the answer should be -3.5? What am I doing wrong? x^2 - 7x + 12 m = - (-7) / 2 = 3.5 d^2 = 12.25 - 12 = 0.25 3.5 ± √(0.25)
When writing out the solutions to a polynomial, you can express them in two different way. You can write the solutions explicitly: x1 = a, x2 = b, or you can write the factors of the polynomial: (x-a) * (x-b) = 0. Both have the same meaning, as the factored form asks "for what x does the polynomial equal zero?" And the answer to that remains a and b. The factored form is what caused confusion, you got the correct answer. Remember that in the factored form, the numbers are what's needed to subtract from (or add to) x to get zero, not the roots themselves.
I'm 29 years old and i'm just rediscovering my love for math, and all i can think about watching your videos, is why didn't i have a teacher like you when i was in school
As soon as you pulled up the graphing of showing the scaling of a quadratic function relative to the roots, this whole method clicked for me. Brilliant. Sure beats what i was taught at school all those years ago.
Hey Grant! Wonderful video, as always. I'll be showing this version of the formula to my students as well! Just one thing: My "pausing and pondering" caused me to fall far enough behind the live stream to miss out on the polls at the end! I could have just skipped ahead, but I enjoy the comparison of problem solving methods too much. Oh well, next time!
If you fall behind on the stream (either from pausing or from buffering) you can increase the playback rate, which will let you catch up without missing anything
Excellent point. Hopefully in future streams, when we have the live question dynamic, I'll give more downtime for everyone to "pause and ponder" that way.
I remember in my Physics exam there was a question about projectile motion and it can be solved by the quadratic formula even though I know how to derive it but the time was tight. But now I leaned a more efficient way for such situations thanks Grant!
8:33 This is something i used to think abt while raking leaves at work and really hope theres some explanation and relevance but i havent watched the full video yet
this must have taken far less time than his standard videos but I feel like I'm learning far more directly. Because of the familiarity of the setting I am taking in knowledge more easily. It's nice to make eye contact with the person teaching you something
As a middle-aged lover of math, this was fabulous. In high school i never could remember the quadratic formula, so i always had to derive it. (Including on the Calculus BC exam, which I hope the grader enjoyed) I got pretty fast at it, and it was sort of meditative, but I really like your method.
The m and p form was how I derived the formula the first time I derived it for myself, perhaps in grade 9. _m_ is a single distinguished point, and _d_ is a single distinguished distance. It was obvious to me that this had to work out directly. Unfortunately, easily accessible distinguished points doesn't crack the cube root, though there is a formula.
