title

The simpler quadratic formula | Ep. 1 Lockdown live math

description

Another view on the quadratic formula.
Full playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDP5CVelJJ1bNDouqrAhVPev
Home page: https://www.3blue1brown.com
Brought to you by you: https://3b1b.co/ldm-thanks
Beautiful pictorial summary by @ThuyNganVu:
https://twitter.com/ThuyNganVu/status/1258217451323416577
Po Shen Loh on quadratics:
https://www.youtube.com/watch?v=ZBalWWHYFQc
Welch Labs on imaginary numbers:
https://www.youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJXapKkNaF
Mistakes (there will always be mistakes):
At minute 22, I write "b' / 2" instead of "-b' / 2".
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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
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Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
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3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe
Various social media stuffs:
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detail

{'title': 'The simpler quadratic formula | Ep. 1 Lockdown live math', 'heatmap': [{'end': 1254.312, 'start': 1215.009, 'weight': 0.731}, {'end': 1512.623, 'start': 1470.195, 'weight': 1}, {'end': 2884.522, 'start': 2847.175, 'weight': 0.942}], 'summary': 'Series explores teaching the quadratic formula interactively, emphasizing its connection to other math patterns, aiming for hour-long lectures, and discusses its use in computer graphics, mental arithmetic tricks, finding roots, and real-life applications, along with the systematic approach to finding roots using midpoint and distance.', 'chapters': [{'end': 156.282, 'segs': [{'end': 85.648, 'src': 'embed', 'start': 51.589, 'weight': 0, 'content': [{'end': 53.051, 'text': "But frankly, it's going to be a bit of a pain.", 'start': 51.589, 'duration': 1.462}, {'end': 59.556, 'text': 'What I will talk about is somewhat similar to something that Po Shen Lo put out a couple of months ago.', 'start': 53.992, 'duration': 5.564}, {'end': 62.379, 'text': "For those of you who don't know him, he has a YouTube channel.", 'start': 59.957, 'duration': 2.422}, {'end': 65.041, 'text': "But more than that, he's the coach of the US IMO team.", 'start': 62.419, 'duration': 2.622}, {'end': 68.204, 'text': 'And he has a company called Xpy, does a lot of great math stuff.', 'start': 65.061, 'duration': 3.143}, {'end': 72.065, 'text': 'So he talked about kind of an alternate way that we could teach the quadratic formula.', 'start': 68.884, 'duration': 3.181}, {'end': 77.026, 'text': "The way I'm going to talk about it is a little bit different from how he did, but certainly very similar in spirit.", 'start': 72.965, 'duration': 4.061}, {'end': 85.648, 'text': '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.', 'start': 77.486, 'duration': 8.162}], 'summary': 'Teaching quadratic formula in a different way to improve problem-solving skills.', 'duration': 34.059, 'max_score': 51.589, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY51589.jpg'}], 'start': 0.109, 'title': 'Teaching the quadratic formula', 'summary': 'Introduces an interactive approach to teaching the quadratic formula, emphasizing its connection to other common patterns in math, aiming for hour-long lectures.', 'chapters': [{'end': 156.282, 'start': 0.109, 'title': 'Lockdown math: the quadratic formula', 'summary': 'Introduces a new approach to teaching the quadratic formula, emphasizing its connection to other common patterns in math and problem-solving, aiming for interactive hour-long lectures.', 'duration': 156.173, 'highlights': ['The lecture introduces a new approach to teaching the quadratic formula, emphasizing its connection to other common patterns in math and problem-solving. Emphasizes the importance of connecting the quadratic formula to other common math patterns for better problem-solving.', 'The lecturer aims for hour-long interactive lectures with audience participation. The lectures are expected to be approximately an hour long and interactive, with audience participation.', 'The technology involving audience interaction encounters some technical difficulties during the lecture. Technical issues arise with the technology used for audience interaction during the lecture.']}], 'duration': 156.173, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY109.jpg', 'highlights': ['The lecture introduces a new approach to teaching the quadratic formula, emphasizing its connection to other common patterns in math and problem-solving.', 'The lecturer aims for hour-long interactive lectures with audience participation.', 'Technical issues arise with the technology used for audience interaction during the lecture.']}, {'end': 1417.192, 'segs': [{'end': 402.966, 'src': 'embed', 'start': 377.591, 'weight': 3, 'content': [{'end': 383.174, 'text': "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.", 'start': 377.591, 'duration': 5.583}, {'end': 387.277, 'text': "Because if I asked you hey, let's factor 143, I mean how do you go about it?", 'start': 383.475, 'duration': 3.802}, {'end': 390.079, 'text': 'I think the way a lot of us think about it is.', 'start': 388.157, 'duration': 1.922}, {'end': 394.181, 'text': "we initially think OK, we know that 2 doesn't go into it, because it's not even.", 'start': 390.079, 'duration': 4.102}, {'end': 398.804, 'text': "3, there's a nice divisibility check where you do 1 plus 4 plus 3.", 'start': 394.201, 'duration': 4.603}, {'end': 402.966, 'text': "And that sum, in this case it's 5 plus 3 or 8, is not divisible by 3.", 'start': 398.804, 'duration': 4.162}], 'summary': 'Factoring large numbers like 143 can be challenging without obvious small factors.', 'duration': 25.375, 'max_score': 377.591, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY377591.jpg'}, {'end': 442.551, 'src': 'embed', 'start': 415.331, 'weight': 4, 'content': [{'end': 421.994, 'text': '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.', 'start': 415.331, 'duration': 6.663}, {'end': 428.342, 'text': 'Actually, 11 does go into it, and it ends up being 11 times 13.', 'start': 421.994, 'duration': 6.348}, {'end': 433.865, 'text': '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,', 'start': 428.342, 'duration': 5.523}, {'end': 434.526, 'text': 'highly related.', 'start': 433.865, 'duration': 0.661}, {'end': 442.551, 'text': "Because what if I asked you 3,599? And this particular example isn't randomly chosen, by the way.", 'start': 435.686, 'duration': 6.865}], 'summary': 'The quadratic formula is related to 11 times 13, not randomly chosen.', 'duration': 27.22, 'max_score': 415.331, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY415331.jpg'}, {'end': 518.058, 'src': 'embed', 'start': 489.478, 'weight': 1, 'content': [{'end': 494.441, 'text': 'And maybe it kind of makes sense that its factors are each kind of close to six.', 'start': 489.478, 'duration': 4.963}, {'end': 500.924, 'text': "You know, five times seven, it's probably not gonna equal six times six, but the fact that they're of similar sizes should be intuitive.", 'start': 495.261, 'duration': 5.663}, {'end': 503.106, 'text': "And as it happens, it's precisely one less.", 'start': 501.285, 'duration': 1.821}, {'end': 507.874, 'text': 'And something similar happens with the other example I chose.', 'start': 504.232, 'duration': 3.642}, {'end': 512.196, 'text': '143 is rather close to a square number, 144.', 'start': 507.894, 'duration': 4.302}, {'end': 513.076, 'text': "In fact, it's one less.", 'start': 512.196, 'duration': 0.88}, {'end': 518.058, 'text': "And it's no coincidence that its two factors are hovering right around the square root of that value.", 'start': 513.957, 'duration': 4.101}], 'summary': 'Factors close to six, 143 close to square of 144, factors near square root.', 'duration': 28.58, 'max_score': 489.478, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY489478.jpg'}, {'end': 609.066, 'src': 'embed', 'start': 576.842, 'weight': 2, 'content': [{'end': 578.323, 'text': 'So let me just pull up an image of a square.', 'start': 576.842, 'duration': 1.481}, {'end': 584.027, 'text': 'and if we want to say can i factor a number which is one less than a square?', 'start': 579.324, 'duration': 4.703}, {'end': 588.25, 'text': 'what i might have us imagine doing is taking the corner off of that square.', 'start': 584.027, 'duration': 4.223}, {'end': 591.953, 'text': "okay, so let's say it's x by x, whatever x is.", 'start': 588.25, 'duration': 3.703}, {'end': 597.66, 'text': "in this case it happens to be six, but you want to think a little bit more abstractly than that, And I'm going to take that bottom right corner,", 'start': 591.953, 'duration': 5.707}, {'end': 598.24, 'text': 'get it out of here.', 'start': 597.66, 'duration': 0.58}, {'end': 598.841, 'text': "We don't want it.", 'start': 598.38, 'duration': 0.461}, {'end': 605.204, 'text': 'The question of factoring is to say, can I rearrange the quantity that remains into some kind of rectangle?', 'start': 599.481, 'duration': 5.723}, {'end': 609.066, 'text': 'And in this case, if I take that bottom rope, sloop it on up to the right.', 'start': 605.844, 'duration': 3.222}], 'summary': 'Explaining factoring using a square image, with an example of x by x, where x=6.', 'duration': 32.224, 'max_score': 576.842, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY576842.jpg'}, {'end': 1254.312, 'src': 'heatmap', 'start': 1215.009, 'weight': 0.731, 'content': [{'end': 1221.37, 'text': "And because we know the sum of the two numbers is negative b prime, that's the same as negative b prime over two,", 'start': 1215.009, 'duration': 6.361}, {'end': 1227.351, 'text': "which you can basically read off of the equation as just negative one half times whatever's sitting right there,", 'start': 1221.37, 'duration': 5.981}, {'end': 1228.712, 'text': 'which in this case will be negative three.', 'start': 1227.351, 'duration': 1.361}, {'end': 1230.612, 'text': 'Awesome, we know what m is.', 'start': 1229.432, 'duration': 1.18}, {'end': 1233.692, 'text': 'But look at the equation we have up here.', 'start': 1231.912, 'duration': 1.78}, {'end': 1241.554, 'text': "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.", 'start': 1234.173, 'duration': 7.381}, {'end': 1243.379, 'text': 'So we could rearrange this.', 'start': 1242.318, 'duration': 1.061}, {'end': 1245.021, 'text': 'So that was saying C prime is that.', 'start': 1243.78, 'duration': 1.241}, {'end': 1254.312, 'text': 'We could say that D squared the square of this kind of standard deviation between our roots is M squared, minus, C prime,', 'start': 1245.442, 'duration': 8.87}], 'summary': 'Equation manipulation leads to finding values for m and d.', 'duration': 39.303, 'max_score': 1215.009, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY1215009.jpg'}, {'end': 1305.919, 'src': 'embed', 'start': 1281.596, 'weight': 0, 'content': [{'end': 1287.563, 'text': 'So look, when we said r and s is some midpoint plus or minus a distance, that midpoint is negative.', 'start': 1281.596, 'duration': 5.967}, {'end': 1289.426, 'text': '3, plus or minus.', 'start': 1287.563, 'duration': 1.863}, {'end': 1294.171, 'text': 'well, if d squared is 2, that means d is the square root of 2..', 'start': 1289.426, 'duration': 4.745}, {'end': 1294.571, 'text': 'There you go.', 'start': 1294.171, 'duration': 0.4}, {'end': 1297.373, 'text': 'You could do that for any quadratic that I give you.', 'start': 1295.192, 'duration': 2.181}, {'end': 1299.515, 'text': 'You could just walk through that particular process.', 'start': 1297.393, 'duration': 2.122}, {'end': 1305.919, 'text': '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.', 'start': 1300.195, 'duration': 5.724}], 'summary': 'Midpoint is negative, d squared is 2, and can be applied to any quadratic.', 'duration': 24.323, 'max_score': 1281.596, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY1281596.jpg'}], 'start': 156.663, 'title': 'Understanding quadratic functions', 'summary': "Discusses the anticipation for dynamic future streams, the use of the quadratic formula in computer graphics, mental arithmetic tricks, and finding the roots of a quadratic equation, with a pixar engineer's calculation suggesting its use over a trillion times in the movie 'coco' and the systematic approach to finding the roots using the midpoint and distance between them.", 'chapters': [{'end': 213.425, 'start': 156.663, 'title': 'Engaging math stream with quadratic formula', 'summary': 'Discusses the anticipation for dynamic future streams, the common expectation of not using the quadratic formula, and a humorous anecdote about its practical use at pixar.', 'duration': 56.762, 'highlights': ['The speaker anticipates more dynamic future streams and hopes to engage the audience in grading the answers, fostering excitement among viewers.', 'Most respondents, including the speaker, expect to use the quadratic formula zero times outside of school, reflecting a common perception of its practical irrelevance.', 'An anecdote is shared about an engineer at Pixar who uses the quadratic formula frequently, humorously highlighting its unexpected practical application.']}, {'end': 434.526, 'start': 214.345, 'title': 'Quadratic formula in computer graphics', 'summary': "Discusses the use of the quadratic formula in computer graphics, particularly in ray tracing, with a pixar engineer's loose calculation suggesting its use over a trillion times in the movie 'coco', and connects it to mental math tricks for problem-solving in math.", 'duration': 220.181, 'highlights': ['The use of the quadratic formula in computer graphics, particularly in ray tracing, was illustrated through an example of creating an image using computer graphics, involving the calculation of how rays hit reflective spheres to color each pixel on the screen.', "A Pixar engineer's loose calculation suggested that the quadratic formula was used over a trillion times in the movie 'Coco', due to the numerous lights for every single frame in a shot, highlighting the practical significance of the quadratic formula in movie production.", 'The speaker emphasizes the relevance of mental math tricks, such as factoring numbers, to problem-solving in math, suggesting that deeply understanding arithmetic patterns can be connected to more advanced mathematical concepts, like the quadratic formula.']}, {'end': 761.318, 'start': 435.686, 'title': 'Mental arithmetic tricks', 'summary': 'Explores mental arithmetic tricks, including the factors of specific numbers close to perfect squares and the geometric and algebraic explanations behind them, using the example of factor 3,599 and its relation to perfect squares.', 'duration': 325.632, 'highlights': ['The mental arithmetic trick of factoring numbers close to perfect squares is demonstrated through the examples of the factors of 3,599 and 143, showcasing their proximity to perfect squares and the relation of their factors to the square roots, providing a visual and algebraic understanding of the concept. demonstrated mental arithmetic trick; factors of 3,599 and 143; proximity to perfect squares; relation of factors to square roots; visual and algebraic understanding', 'The visual representation of factoring a number one less than a square by taking the corner off the square and rearranging it into a rectangle, as well as the geometric interpretation of factoring a difference of squares, provides a clear and satisfying understanding of the mental arithmetic trick. visual representation of factoring; rearranging into a rectangle; geometric interpretation of factoring; clear and satisfying understanding', 'The algebraic explanation of factoring a difference of squares using the expression (m - d)(m + d) = m^2 - d^2 and its relation to any two numbers expressed as a midpoint plus or minus a distance, presents a deeper and intriguing insight into the mental arithmetic concept. algebraic explanation of factoring; relation to any two numbers expressed as a midpoint plus or minus a distance; deeper and intriguing insight']}, {'end': 1120.085, 'start': 761.318, 'title': 'Understanding quadratic functions', 'summary': 'Discusses the relationship between products, primes, and difference of squares, and how it relates to understanding quadratic functions, highlighting the importance of rescaling and the key facts needed to solve any quadratic.', 'duration': 358.767, 'highlights': ["The relationship between products, primes, and difference of squares is discussed, emphasizing that only primes can't be broken down into a product of two smaller whole numbers.", 'The method of rescaling quadratic functions by dividing everything by the leading coefficient is highlighted, making it easier to work with and solve random quadratic functions.', 'The key facts needed to solve any quadratic without needing to memorize much are highlighted, including the b value being the negative sum of the roots and the c value being the product of the roots.']}, {'end': 1417.192, 'start': 1120.765, 'title': 'Quadratic equation and roots', 'summary': 'Explains a systematic approach to finding the roots of a quadratic equation using the midpoint and distance between the roots, with the key formula being m plus or minus the square root of m squared minus p.', 'duration': 296.427, 'highlights': ['The formula m plus or minus the square root of m squared minus p can be used to find the roots of any quadratic equation, making it simpler than the traditional quadratic formula. The formula m plus or minus the square root of m squared minus p can be used to find the roots of any quadratic equation, simplifying the process compared to the traditional quadratic formula.', 'The midpoint for any quadratic equation is a rescaled version of b divided by 2, while the distance between the roots is the square root of the midpoint squared minus the product of the roots. The midpoint for any quadratic equation is a rescaled version of b divided by 2, while the distance between the roots is the square root of the midpoint squared minus the product of the roots.', 'The product of the roots can be expressed as m squared minus D squared, where m is the mean of the roots and D is the distance between the roots. The product of the roots can be expressed as m squared minus D squared, where m is the mean of the roots and D is the distance between the roots.']}], 'duration': 1260.529, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY156663.jpg', 'highlights': ["The quadratic formula was used over a trillion times in the movie 'Coco', highlighting its practical significance in movie production.", 'The formula m plus or minus the square root of m squared minus p can be used to find the roots of any quadratic equation, simplifying the process compared to the traditional quadratic formula.', 'The midpoint for any quadratic equation is a rescaled version of b divided by 2, while the distance between the roots is the square root of the midpoint squared minus the product of the roots.', 'The mental arithmetic trick of factoring numbers close to perfect squares provides a visual and algebraic understanding of the concept.', "The relationship between products, primes, and difference of squares is discussed, emphasizing that only primes can't be broken down into a product of two smaller whole numbers."]}, {'end': 2105.381, 'segs': [{'end': 1512.623, 'src': 'heatmap', 'start': 1470.195, 'weight': 1, 'content': [{'end': 1478.44, 'text': 'I could always just go through this little rigmarole again and say okay, if I systematically wanted it to be a quadratic with roots r and s,', 'start': 1470.195, 'duration': 8.245}, {'end': 1479.381, 'text': 'this is what it would look like.', 'start': 1478.44, 'duration': 0.941}, {'end': 1480.782, 'text': 'This is how it would expand.', 'start': 1479.681, 'duration': 1.101}, {'end': 1482.263, 'text': 'So you can re-derive it on the fly.', 'start': 1481.042, 'duration': 1.221}, {'end': 1483.724, 'text': "There's not too much memorization needed.", 'start': 1482.303, 'duration': 1.421}, {'end': 1488.127, 'text': 'In this context, that works out to be negative 5.', 'start': 1483.744, 'duration': 4.383}, {'end': 1491.309, 'text': "And, by the way, if I do ever make any mistakes, which I'm quite positive,", 'start': 1488.127, 'duration': 3.182}, {'end': 1496.232, 'text': "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.", 'start': 1491.309, 'duration': 4.923}, {'end': 1497.953, 'text': 'So we know the midpoint.', 'start': 1497.193, 'duration': 0.76}, {'end': 1502.176, 'text': "And then we just ask ourselves what's the square of the distance?", 'start': 1498.754, 'duration': 3.422}, {'end': 1512.623, 'text': "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,", 'start': 1503.277, 'duration': 9.346}], 'summary': 'Teaching quadratic expansion and midpoint calculation with minimal memorization, correcting mistakes on the fly.', 'duration': 42.428, 'max_score': 1470.195, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY1470195.jpg'}, {'end': 1574.846, 'src': 'embed', 'start': 1548.127, 'weight': 6, 'content': [{'end': 1553.631, 'text': "If you're watching this in the future, by the way, as with any video, I highly encourage you to pause and ponder.", 'start': 1548.127, 'duration': 5.504}, {'end': 1557.273, 'text': "I really think that's the best way to learn math if you're looking at some kind of lecture.", 'start': 1553.991, 'duration': 3.282}, {'end': 1563.638, 'text': "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.", 'start': 1557.734, 'duration': 5.904}, {'end': 1565.679, 'text': "I guarantee you'll learn more effectively that way.", 'start': 1563.918, 'duration': 1.761}, {'end': 1571.043, 'text': "I don't know, what should we do? Let's do maybe 3x squared.", 'start': 1567.04, 'duration': 4.003}, {'end': 1574.846, 'text': "I think I wrote one down earlier, didn't I? Just as kind of an offhanded thing.", 'start': 1571.444, 'duration': 3.402}], 'summary': 'Pausing during math lectures increases learning effectiveness.', 'duration': 26.719, 'max_score': 1548.127, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY1548127.jpg'}, {'end': 1617.905, 'src': 'embed', 'start': 1588.718, 'weight': 0, 'content': [{'end': 1592.28, 'text': "All right, so in this case, step one, we've got to rescale things.", 'start': 1588.718, 'duration': 3.562}, {'end': 1595.403, 'text': "That's what gives the coefficients a nice readable meaning.", 'start': 1593.381, 'duration': 2.022}, {'end': 1602.372, 'text': 'minus four thirds x plus five thirds.', 'start': 1596.127, 'duration': 6.245}, {'end': 1605.014, 'text': 'great. and now same process.', 'start': 1602.372, 'duration': 2.642}, {'end': 1609.398, 'text': "i'm sort of thinking in my head of this particular image where i want to know the midpoint and the distance.", 'start': 1605.014, 'duration': 4.384}, {'end': 1617.905, 'text': 'so i say that midpoint is negative of this second coefficient divided by two, so that will become positive four thirds,', 'start': 1609.398, 'duration': 8.507}], 'summary': 'Rescale coefficients for readable meaning, finding midpoint and distance.', 'duration': 29.187, 'max_score': 1588.718, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY1588718.jpg'}, {'end': 1672.984, 'src': 'embed', 'start': 1639.639, 'weight': 1, 'content': [{'end': 1640.94, 'text': 'So this one gets a little messier.', 'start': 1639.639, 'duration': 1.301}, {'end': 1643.542, 'text': "We've got to work out our fractions, but that's not too bad.", 'start': 1640.96, 'duration': 2.582}, {'end': 1645.963, 'text': '2 thirds squared is going to be 4 ninths.', 'start': 1643.562, 'duration': 2.401}, {'end': 1648.865, 'text': "I'm off screen a little.", 'start': 1648.124, 'duration': 0.741}, {'end': 1657.838, 'text': "And then what is five thirds in terms of ninths? That's gonna be 15 ninths, if I'm not wrong.", 'start': 1652.83, 'duration': 5.008}, {'end': 1661.539, 'text': 'So we end up getting a negative number, which is always fun.', 'start': 1659.438, 'duration': 2.101}, {'end': 1664.82, 'text': 'So here we have negative 11 ninths.', 'start': 1662.079, 'duration': 2.741}, {'end': 1667.001, 'text': 'So what that means?', 'start': 1666.261, 'duration': 0.74}, {'end': 1672.984, 'text': 'is that our final answer the roots of this polynomial r and s, the values that will make the polynomial 0,', 'start': 1667.001, 'duration': 5.983}], 'summary': 'Working with fractions, 2/3 squared equals 4/9, 5/3 in ninths is 15/9, resulting in a final answer of -11/9.', 'duration': 33.345, 'max_score': 1639.639, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY1639639.jpg'}, {'end': 2065.71, 'src': 'embed', 'start': 2041.975, 'weight': 3, 'content': [{'end': 2048.9, 'text': 'The only thing that looks remotely like memorization is if you want to jumpstart to the end and just say m plus or minus m squared minus p.', 'start': 2041.975, 'duration': 6.925}, {'end': 2051.54, 'text': 'Now, to finish things off,', 'start': 2050.379, 'duration': 1.161}, {'end': 2057.143, 'text': 'I think it would be very satisfying to remind ourselves that this is actually equivalent to the traditional quadratic formula,', 'start': 2051.54, 'duration': 5.603}, {'end': 2058.185, 'text': 'something that looks much bigger.', 'start': 2057.143, 'duration': 1.042}, {'end': 2061.547, 'text': "So let's go ahead and actually do that exercise.", 'start': 2059.045, 'duration': 2.502}, {'end': 2065.71, 'text': 'And again, if you can, pause and just work it out for yourself right now.', 'start': 2062.507, 'duration': 3.203}], 'summary': 'Equivalent to traditional quadratic formula, m plus or minus m squared minus p.', 'duration': 23.735, 'max_score': 2041.975, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY2041975.jpg'}], 'start': 1418.277, 'title': 'Quadratic equations and roots', 'summary': 'Covers the process of finding roots of quadratic equations by completing the square, solving two quadratic equations, and deriving their roots. it also discusses finding complex roots of quadratic polynomials and delves into the geometric and algebraic insights of quadratic equations with examples, ultimately leading to the roots 2/3 + sqrt(-11)/9 and 2/3 - sqrt(-11)/9, and also 3 +/- i.', 'chapters': [{'end': 1563.638, 'start': 1418.277, 'title': 'Quadratic equations and roots', 'summary': 'Demonstrates the process of finding roots of quadratic equations by completing the square, providing a detailed explanation and encouraging active participation, ultimately solving two quadratic equations and deriving their roots.', 'duration': 145.361, 'highlights': ['The process of finding roots of quadratic equations by completing the square is demonstrated, emphasizing the importance of understanding the underlying concepts rather than memorization, promoting active learning and participation.', 'Detailed steps and calculations are provided for solving two quadratic equations, showcasing the derivation of their roots and the application of the difference of squares method for obtaining the solutions.', 'Encouragement is given for active participation and engagement in the learning process, with the instructor emphasizing the benefits of pausing to ponder and attempting the exercises independently for effective learning.']}, {'end': 1839.929, 'start': 1563.918, 'title': 'Complex roots of quadratic polynomials', 'summary': 'Discusses finding complex roots of quadratic polynomials, demonstrating the process with examples using midpoint, distance, and complex plane, ultimately leading to the roots 2/3 + sqrt(-11)/9 and 2/3 - sqrt(-11)/9, and also 3 +/- i.', 'duration': 276.011, 'highlights': ['The midpoint and distance method is used to find the roots of the polynomial 3x squared - 4x + 5, resulting in the roots 2/3 + sqrt(-11)/9 and 2/3 - sqrt(-11)/9.', 'A simpler example with the polynomial x squared - 6x + 10 is also demonstrated, resulting in the roots 3 + i and 3 - i.', 'The concept of complex roots is related to the complex plane, where the roots 3 + i and 3 - i are situated, expanding beyond the real number line.', 'The discussion aims to connect patterns in math that have practical applications beyond the specific class being studied.']}, {'end': 2105.381, 'start': 1841.889, 'title': 'Quadratic equation insights', 'summary': 'Delves into the geometric and algebraic insights of quadratic equations, touching upon the pythagorean theorem, complex numbers, and the significance of roots, highlighting the connection to various mathematical concepts, including prime regularities and factoring problems.', 'duration': 263.492, 'highlights': ['The product of the magnitudes of the two separate roots of a quadratic equation needs to be 10, leading to a magnitude of the square root of 10, reflecting a connection to complex numbers and geometric concepts. The product of the magnitudes of the two separate roots of a quadratic equation needs to be 10, resulting in a magnitude of the square root of 10. This demonstrates the connection to complex numbers and geometric concepts.', 'The connection between expressing sums of squares, the Pythagorean theorem, and factoring, indicating the broader applicability of quadratic formula insights in diverse mathematical contexts. The connection between expressing sums of squares, the Pythagorean theorem, and factoring highlights the broader applicability of quadratic formula insights in diverse mathematical contexts.', 'The significance of understanding key facts related to quadratics to rediscover the quadratic formula on the fly, emphasizing the value of knowing the coefficients and the product of the roots. The significance of understanding key facts related to quadratics to rediscover the quadratic formula on the fly emphasizes the value of knowing the coefficients and the product of the roots.']}], 'duration': 687.104, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY1418277.jpg', 'highlights': ['The process of finding roots of quadratic equations by completing the square is demonstrated, emphasizing the importance of understanding the underlying concepts rather than memorization, promoting active learning and participation.', 'Detailed steps and calculations are provided for solving two quadratic equations, showcasing the derivation of their roots and the application of the difference of squares method for obtaining the solutions.', 'The midpoint and distance method is used to find the roots of the polynomial 3x squared - 4x + 5, resulting in the roots 2/3 + sqrt(-11)/9 and 2/3 - sqrt(-11)/9.', 'A simpler example with the polynomial x squared - 6x + 10 is also demonstrated, resulting in the roots 3 + i and 3 - i.', 'The product of the magnitudes of the two separate roots of a quadratic equation needs to be 10, leading to a magnitude of the square root of 10, reflecting a connection to complex numbers and geometric concepts.', 'The connection between expressing sums of squares, the Pythagorean theorem, and factoring, indicating the broader applicability of quadratic formula insights in diverse mathematical contexts.', 'The significance of understanding key facts related to quadratics to rediscover the quadratic formula on the fly, emphasizing the value of knowing the coefficients and the product of the roots.']}, {'end': 3129.9, 'segs': [{'end': 2156.382, 'src': 'embed', 'start': 2106.821, 'weight': 5, 'content': [{'end': 2109.602, 'text': "And of course, what is that math? That math is exactly what we're doing right now.", 'start': 2106.821, 'duration': 2.781}, {'end': 2111.303, 'text': "So let's work it out again.", 'start': 2110.382, 'duration': 0.921}, {'end': 2114.504, 'text': 'ax squared plus bx plus c equals 0.', 'start': 2111.323, 'duration': 3.181}, {'end': 2117.805, 'text': "This time, we're just going to write everything in terms of a, b, and c.", 'start': 2114.504, 'duration': 3.301}, {'end': 2118.945, 'text': 'No new variables coming up.', 'start': 2117.805, 'duration': 1.14}, {'end': 2130.809, 'text': '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.', 'start': 2119.305, 'duration': 11.504}, {'end': 2131.869, 'text': "We don't give it any new names.", 'start': 2130.809, 'duration': 1.06}, {'end': 2134.455, 'text': 'Now remember how our trick works.', 'start': 2133.055, 'duration': 1.4}, {'end': 2139.057, 'text': 'We sort of picture in our head, hey, imagine this quadratic has some roots.', 'start': 2135.396, 'duration': 3.661}, {'end': 2142.158, 'text': "Let's call them R and S.", 'start': 2139.717, 'duration': 2.441}, {'end': 2146.339, 'text': "And we're trying to find the midpoint and the standard deviation.", 'start': 2142.158, 'duration': 4.181}, {'end': 2151.881, 'text': 'And we can read off that that midpoint is the negative of the second term divided by two.', 'start': 2146.899, 'duration': 4.982}, {'end': 2156.382, 'text': "So in this case, that's gonna be negative B over two A.", 'start': 2151.901, 'duration': 4.481}], 'summary': 'Rescaling the quadratic equation using a, b, and c without introducing new variables, finding the midpoint as -b/2a.', 'duration': 49.561, 'max_score': 2106.821, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY2106821.jpg'}, {'end': 2371.698, 'src': 'embed', 'start': 2340.169, 'weight': 4, 'content': [{'end': 2341.832, 'text': "And if this feels tedious, that's kind of the point.", 'start': 2340.169, 'duration': 1.663}, {'end': 2344.596, 'text': 'The whole quadratic formula is more complicated than it needs to be.', 'start': 2341.872, 'duration': 2.724}, {'end': 2348.121, 'text': 'Because we were just solving any quadratic that was thrown at us without having to do this.', 'start': 2344.796, 'duration': 3.325}, {'end': 2351.946, 'text': "But this is what happens if you don't introduce any new variable names on your way.", 'start': 2348.501, 'duration': 3.445}, {'end': 2354.49, 'text': "It's like code that hasn't been refactored properly.", 'start': 2352.467, 'duration': 2.023}, {'end': 2356.392, 'text': 'Okay, so what can we do here?', 'start': 2355.311, 'duration': 1.081}, {'end': 2364.875, 'text': "We can factor out the 1 over 4a squared, and because that's in a radical, its square root will also be 1 over 2a,", 'start': 2356.432, 'duration': 8.443}, {'end': 2371.698, 'text': 'and then what sits inside is what remains the well-familiar negative b squared, minus 4a.', 'start': 2364.875, 'duration': 6.823}], 'summary': 'Quadratic formula simplification by factoring out 1 over 4a squared.', 'duration': 31.529, 'max_score': 2340.169, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY2340169.jpg'}, {'end': 2543.275, 'src': 'embed', 'start': 2496.328, 'weight': 0, 'content': [{'end': 2501.772, 'text': "And the whole lesson here, if we think about what's going on, it's about representing the same information in different ways.", 'start': 2496.328, 'duration': 5.444}, {'end': 2514.462, 'text': "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??", 'start': 2501.792, 'duration': 12.67}, {'end': 2522.468, 'text': "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.", 'start': 2515.603, 'duration': 6.865}, {'end': 2529.224, 'text': "And really because that a can get scaled out, it doesn't add any information to the system.", 'start': 2523.959, 'duration': 5.265}, {'end': 2531.786, 'text': "There's really as much information as two different constants in there.", 'start': 2529.244, 'duration': 2.542}, {'end': 2536.851, 'text': 'So one of these directions is easy and one of these directions is hard.', 'start': 2532.387, 'duration': 4.464}, {'end': 2543.275, 'text': 'And this idea of information that can be expressed in separate ways and translation one direction being easy,', 'start': 2538.533, 'duration': 4.742}], 'summary': 'Quadratic formula represents coefficients as roots, with one direction easy and one hard.', 'duration': 46.947, 'max_score': 2496.328, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY2496328.jpg'}, {'end': 2830.307, 'src': 'embed', 'start': 2806.614, 'weight': 1, 'content': [{'end': 2813.501, 'text': "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.", 'start': 2806.614, 'duration': 6.887}, {'end': 2816.583, 'text': "So this one, actually, now there isn't.", 'start': 2815.023, 'duration': 1.56}, {'end': 2818.304, 'text': 'Where do I type? Help.', 'start': 2817.584, 'duration': 0.72}, {'end': 2819.904, 'text': 'Bars are attacking me.', 'start': 2819.104, 'duration': 0.8}, {'end': 2825.646, 'text': 'OK There is an objectively correct answer, because there is going to be some number that most people enter.', 'start': 2819.924, 'duration': 5.722}, {'end': 2830.307, 'text': "And it looks like 919 of you think that it'll be one particular thing.", 'start': 2826.186, 'duration': 4.121}], 'summary': "About 919 people think there's an objectively correct answer.", 'duration': 23.693, 'max_score': 2806.614, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY2806614.jpg'}, {'end': 2884.522, 'src': 'heatmap', 'start': 2847.175, 'weight': 0.942, 'content': [{'end': 2855.265, 'text': 'And in a weird way, like, the plurality of you are definitionally correct that 7 was the most commonly entered expression.', 'start': 2847.175, 'duration': 8.09}, {'end': 2857.988, 'text': '69 being the second most common.', 'start': 2855.285, 'duration': 2.703}, {'end': 2860.189, 'text': 'I can make a guess for why that might be the case.', 'start': 2858.208, 'duration': 1.981}, {'end': 2865.692, 'text': 'Did you know that 69 is the first number where, if you square it and then you cube it,', 'start': 2860.209, 'duration': 5.483}, {'end': 2871.595, 'text': 'those two values between them encounter the numbers or the digits zero through nine once and only once?', 'start': 2865.692, 'duration': 5.903}, {'end': 2877.618, 'text': "Yeah, it's the first number with that property, which I assume is why that was the second most popular answer.", 'start': 2872.175, 'duration': 5.443}, {'end': 2884.522, 'text': 'But the very end, which is actually apropos at this point, we can pull up another question.', 'start': 2879.159, 'duration': 5.363}], 'summary': '7 was the most commonly entered expression, followed by 69. 69 has a unique mathematical property, hence its popularity.', 'duration': 37.347, 'max_score': 2847.175, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY2847175.jpg'}], 'start': 2106.821, 'title': 'Understanding quadratic formula', 'summary': 'Covers the derivation and understanding of the quadratic formula, emphasizing the importance of comprehension, multiple pathways for problem-solving, and engaging students with fun math activities, including real-life applications.', 'chapters': [{'end': 2421.944, 'start': 2106.821, 'title': 'Quadratic formula derivation', 'summary': 'Discusses the derivation of the quadratic formula using a, b, and c as variables, and emphasizes the importance of understanding the meaning behind the formula to minimize errors and improve comprehension.', 'duration': 315.123, 'highlights': ['The step-by-step process of deriving the quadratic formula using a, b, and c as variables. The chapter explains the step-by-step process of deriving the quadratic formula using a, b, and c as variables, emphasizing the rescaling and visualization techniques to find the midpoint and standard deviation of the roots.', 'Emphasizing the importance of understanding the meaning behind the formula to minimize errors and improve comprehension. The chapter stresses the significance of understanding the meanings behind the formulas to reduce errors and improve comprehension, particularly for individuals who may find the arithmetic process error-prone.']}, {'end': 2739.98, 'start': 2422.725, 'title': 'Understanding the quadratic formula', 'summary': 'Discusses the quadratic formula, its simpler variant, and the connections between different ways of representing information, emphasizing the importance of understanding multiple pathways for problem-solving in mathematics.', 'duration': 317.255, 'highlights': ['The quadratic formula and its simpler variant are discussed, with a focus on understanding the connections between different ways of representing information. The simpler variant of the quadratic formula is explored, emphasizing the connections and patterns between different ways of representing information.', 'The importance of drawing connections to other patterns in mathematics to reinforce learning is emphasized. The chapter stresses the significance of drawing connections to other mathematical patterns for better retention and understanding.', 'The concept of representing the same information in different ways and the ease or difficulty of translating between them is highlighted. The chapter discusses the concept of representing information in different ways and the varying ease of translating between these representations, providing insights into problem-solving in mathematics.']}, {'end': 3129.9, 'start': 2739.98, 'title': 'Lockdown math: fun with numbers and quadratic formula', 'summary': 'Discusses fun ways to engage students in math, including a poll on the most commonly entered integer, a famous story about the number 1729, and the expected use of the quadratic formula in real life.', 'duration': 389.92, 'highlights': ['The majority of people entered 7 as the most commonly entered integer, followed by 69, with a wide spread of responses, making it a fun and engaging activity. 919 people entered 7 as the most common integer, while 69 was the second most common. This activity engaged a wide audience and provided a fun learning experience.', 'The story of the number 1729 being the first number expressible as the sum of two cubes in two separate ways, engaging 1769 participants. 1769 participants thought they would use the quadratic formula zero times, which aligns with the likelihood of using it in its actual form. The story added an engaging and memorable aspect to the discussion.', 'Announcement of the next lesson about not memorizing trigonometric formulas, scheduled for Tuesday, with an invitation for future participation in the high school lectures. The next lesson about not memorizing trigonometric formulas was announced for Tuesday, with an invitation for future participation in the high school lectures, promoting ongoing engagement and learning.']}], 'duration': 1023.079, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/MHXO86wKeDY/pics/MHXO86wKeDY2106821.jpg', 'highlights': ['The step-by-step process of deriving the quadratic formula using a, b, and c as variables, emphasizing rescaling and visualization techniques.', 'Emphasizing the importance of understanding the meaning behind the formula to minimize errors and improve comprehension.', 'The simpler variant of the quadratic formula is explored, emphasizing the connections and patterns between different ways of representing information.', 'The importance of drawing connections to other patterns in mathematics for better retention and understanding.', 'The concept of representing information in different ways and the varying ease of translating between them is highlighted, providing insights into problem-solving in mathematics.', 'The majority of people entered 7 as the most commonly entered integer, followed by 69, making it a fun and engaging activity.', 'The story of the number 1729 being the first number expressible as the sum of two cubes in two separate ways, engaging 1769 participants.', 'Announcement of the next lesson about not memorizing trigonometric formulas, scheduled for Tuesday, with an invitation for future participation in the high school lectures.']}], 'highlights': ["The quadratic formula was used over a trillion times in the movie 'Coco', highlighting its practical significance in movie production.", 'The lecture introduces a new approach to teaching the quadratic formula, emphasizing its connection to other common patterns in math and problem-solving.', 'The lecturer aims for hour-long interactive lectures with audience participation.', 'The process of finding roots of quadratic equations by completing the square is demonstrated, emphasizing the importance of understanding the underlying concepts rather than memorization, promoting active learning and participation.', 'The step-by-step process of deriving the quadratic formula using a, b, and c as variables, emphasizing rescaling and visualization techniques.', 'The midpoint for any quadratic equation is a rescaled version of b divided by 2, while the distance between the roots is the square root of the midpoint squared minus the product of the roots.', 'The mental arithmetic trick of factoring numbers close to perfect squares provides a visual and algebraic understanding of the concept.', 'The connection between expressing sums of squares, the Pythagorean theorem, and factoring, indicating the broader applicability of quadratic formula insights in diverse mathematical contexts.', 'The significance of understanding key facts related to quadratics to rediscover the quadratic formula on the fly, emphasizing the value of knowing the coefficients and the product of the roots.', 'The importance of drawing connections to other patterns in mathematics for better retention and understanding.']}