Examples: A Different Way to Solve Quadratic Equations
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- Опубликовано: 5 сен 2024
- If you'd like to continue learning in this style, Prof. Loh runs a systematic course series which focuses on teaching people how to invent their own mathematical techniques (live.poshenloh..., as well as regular streams on this RUclips channel.
Want to understand how to solve any quadratic equation, without just memorizing the quadratic formula? In this lesson full of worked examples, Daily Challenge founder, Carnegie Mellon University professor, and U.S. International Math Olympiad coach Po-Shen Loh introduces a simple, different method for solving quadratic equations. Instead of guess-and-check, this method uses students’ existing experience searching for a pair of numbers with a given sum and product to solve quadratics.
This video explains all of the details behind Prof. Loh's initial quadratic method video (ruclips.net/user/playlist?l..., including different examples of how to use the method to solve more and more general quadratic equations.
Prof. Loh came to discover this method because he has been systematically thinking about how to explain school math concepts in more thoughtful and interesting ways, while creating lessons for our Daily Challenge (live.poshenloh....
Thanks to the Templeton World Charity Foundation for their support of this work.
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UNIQUE APPLICATIONS OF THE METHOD
Looking for a specific kind of problem that this method can solve? Check out the timestamps below:
21:24 Coefficients with different signs
25:03 If the coefficient of x is odd
27:28 If x² has a coefficient other than 1
30:41 Deriving a Quadratic Formula
33:14 Deriving the commonly learned Quadratic Formula
HISTORICAL PARTS OF THE SOLUTION
38:18 Viète’s Relations
39:00 Babylonian Mathematics
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ABOUT DAILY CHALLENGE
The Daily Challenge with Po-Shen Loh is a series of online math courses designed to teach people how to invent, taught by an actual inventor. The novel teaching style continually presents students with problems that they barely don't know how to solve, and guides them to invent their own solution methods through hints. It covers a wide range of ability levels, starting from basics which only require Pre-Algebra, and reaching up to university-level topics such as Number Theory. Instead of simply teaching the standard curriculum earlier, it goes deeper, providing additional challenges outside of the classroom. The course series also provides an efficient way to systematically cover math competition topics. To learn more about Daily Challenge with Po-Shen Loh, visit daily.poshenlo....
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Expii is a free math and science education platform that embraces what makes us all individuals: our voices, experiences, and cultures. We encourage choice and experimentation by giving students the power to choose how they learn. Through creative, fun, practical, and personal lessons each person can have their own aha moment.
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This is the type of math teacher I like. The teacher who actually lets their kids understand the math instead of making them pass the exam so he doesn’t have to worry about it again.
ruclips.net/video/L4ImyFn1xLk/видео.html
Right bro
I wish I saw this lecture before JEE exam lol. The graph method makes the questions INCREDIBLY easy
he also has an iq of like 230 so that might have something to do with it
Hmm
I am a math and computer science major and a math teacher (middle school) and, like others have commented, I have never seen this method before. I love how you can easily get non-integer solution sets! Perhaps instead of the "Po-Shen Method" I shall henceforth call this the "Magic Quadratic Po-Shen" (just to add a little wordplay)!
I saw a video in which a college professor did the same thing -- because he couldn't remember how to pronounce his name.
Portion
This is one of the best lectures on quadratic equations I have seen. Thank you Mr Loh for clear and understandable information presented in a concise manner.
Thanks for your feedback!
@@psloh sir, could you please reply me??
ruclips.net/video/L4ImyFn1xLk/видео.html
Why 14 doesn't become -14 as in x^2-14x+24, can someone explain it pl
I salute you man! This is absolutely amazing....have been teaching math for 35 years, you have just made my job easier for whatever remaining number of years I will be able to teach. That's why they say....it doesn't matter you learned something new so late in life, what matters is you got new knowledge (new, I say, for the learner), new wisdom. And last but not the least, what is the primary objective of Problem solving....in any branch of knowledge or education....it is to achieve the objective in as simple and as uncomplicated a manner as possible....so that it can bring a smile on the faces of our students. Thanks a zillion bro!!!
Sir pls do research on vedic maths which will change your life 🇮🇳🇮🇳
That is the easiest method I have ever used. I can't imagine that I did not know this before. Thank you sir
Google PQ
I think the Colombian Method is easier to be honest.
DominicanOps what Is the colombian method?
@
ruclips.net/video/jcRW2R42azE/видео.html
ruclips.net/video/tRblwTsX6hQ/видео.html
RUclipsrs discovered it years ago
My teacher had me explain this to her and now she's recommending your technique and referencing to you! Thanks from Sweden!
not your but as a tip
Of course other countries know this while USA still wants kids to learn the route method
ruclips.net/video/L4ImyFn1xLk/видео.html
@@muhammadazeem9775 Just a tip, based on my personal preference. I almost never click on a video link that does not include a brief statement of what the content/summary of that video is...you'll attract more clicks with a brief statement. Nice rose picture.
Sir please keep posting such methods so that people start analyzing mathematics and taste the true essence of it, Thank you so much for sharing this🙏.
:)
This is exactly how we learned in my country East Africa. God how did I forget this, and our professor has zero clue and got me confused and spent days to really figure this out. I wish I found you early. I would have saved a lot of time. But thank you so much! You did an amazing job. keep them coming.
Every highschool teacher should learn this method
Yes
True talk
ruclips.net/video/L4ImyFn1xLk/видео.html
X²+6x-27=0
No plssss. This is complicated 😭😭 like why can't we just use the quadractic formula. It isn't that difficult to remember 😔
I have been teaching this method to my high school students back in 1999 and yes, this is another way. Students will definitely learn if a teacher will expose them to different ways of solving the problem. Nice presentation...
I realise that this is basically not much different than completing the square. Why was this method not widespread before? It would have saved many students a whole lot of suffering...
@@geraldillo The one thing I don't like about this method relative to completing the square is that it doesn't teach the "add and subtract" trick, which is useful elsewhere in math. To clarify what I mean, I think of completing the square in this kind of format: x^2+2x+2=x^2+2x+1-1+2=(x+1)^2+1. The "I can just put a +1 there as long as I take it away later" concept is kind of magical to an algebra student, but is widely used elsewhere in math. Unfortunately I sometimes see students learning this in terms of adding the desired quantity to both sides of an *equation* which is really the less common way to think about it in more advanced math.
I wish we could have learnt all subjects this way in school. So systematically and brilliantly explained
What a marvellous lecture in every way! Not just a genius new method for solving quadratic equations, but also clear and precisely presented.
I have never seen this method. Beautiful. Thank you. Retired math teacher.
Thank you for your career of teaching!
ruclips.net/video/L4ImyFn1xLk/видео.html
Haven't you taught your students one of the methods in finding for the roots of a quadratic equations is by completing the perfect square trinomial? This method is used how we arrived at the so-called quadratic formula😀
@@jecelaguilar9319 At first, I didn't get what he was doing. So, I Completed The Square on a couple of problems, using an area model. I'm a really visual and conceptual math learner, so things like area models really help me. After I did some problems, I realized that his method is basically a shortcut for Completing The Square. I had the idea because he said his method always works. To those of us in the U.S., this approach seems different because it's not the exact way we are taught to Complete The Square in school.
This is so elegant, intuitive, and forces the solver to think about what's happening and how things relate to each other. I wish I had RUclips back when I was in school.
If there were RUclips when I was in school, I would never have gone to school. Attendance and tuition would have been a waste of resources.
VIO stand your life.
Damn I've never been able to solve quadratic equations with irrational solutions purely in my head but now I can.
Do your work and find some podophiles not quadratic lol
@@aaditrangnekar lmao
I’m reviewing all of my precalculus math that I had 50 years (!) ago. I wish I’d had you for a math teacher back then. Thank you so much for the teaching an old dog a new trick! I’m planning to learn a lot more new tricks. I’ll be watching more of your videos in the future! Thanks so much!!
Dear Prof Loh, I am now retired at 61, and I have worked in IT for most of my working life. I have, however, always dabbled in mathematics. The thrill of discovering this only recently, is one the reasons for my continued love of the subject. If you have not done so, adding the graphical interpretation would help many other students. Thank you very much.
Could we try it on Desmos?
@@southernkatrina8161 Desmos?
Desmos is a free download app that takes a function and graphs it. It's good fun and helps learning.
I had to find videos that explained it, graphically, in order to fully understand what was going on.
Thank you from my 19 person math class who enjoyed learning this method
This is why I love mathematics, 😀😀
ruclips.net/video/L4ImyFn1xLk/видео.html
Mind Blown! Genious! Not just in the solution but in the expert EXPLAINATION, very clear and understandable, just amazing! Thank you for taking the time to produce this video. I will go over it several times to make sure I understand its simplicity. I am now a follower and will look over your other videos. Wow, just amazing how ancient secrets from hundreds and thousands of years ago are lost over time but come back to life. I wonder how many other things are like this?
You are simply great.
I have never seen such a pleasant mathematics teacher during my learning period. I am now 72. Today you taught me a novel technique. I salute you
I watched this entire video without getting bored or wanting to click off. This proves that this video is godly
I am from Czech republic and I know this since elementary school. It's one of the taught methods to solve cuadratic ecuations.
Same in Kenya
What a joy that this method fits naturally in each number domain! The intuition of "finding the average" leads so smoothly into creating a complex conjugate pair. It's as if we knew more than we knew we knew, all along. Thanks for sharing this so well!
As a 4th grade teacher, I can tell you that it would be incredible to have this kind of talent exposed to the kids right from 1st grade on. So much of the math problems we have is because we lack our natural curiosity and stop having fun. Math gets to be all about 'testing' pretty quick once they leave 6th grade. Thanks for posting, great lesson.
You could teach them!
The method at 00:46 is just the quadratic formula. - B/2A +- u, where u^2 = B^2/4 - AC. The x-coordinate of the vertex of a parabola is -B/2A. It is also the line of symmetry. If you extend the line of symmetry through the x-axis, you will see that one root is a distance SQRT(B^2 - 4AC) to the right and the other is a distance -SQRT(B^2 - 4AC) to the left. The method is a bit different than just plugging numbers into the quadratic formula, but there is really no new math here, just a different way to apply the quadratic formula.
No new maths but simpler maths 😊
The Indian mathematician Brahmagupta (597-668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD,[24] but written in words instead of symbols.[25] His solution of the quadratic equation ax2 + bx = c was as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."
This was amazing! I actually had to pause the video and derive the quadratic formula because I was so excited about noticing how it relates to this method. As soon as I finished I unpaused the video and Professor Loh started talking about how to use the method to derive the quadratic formula. The logical pathway is so satisfying in this video and I am grateful to finally understand the quadratic formula!
Hi....
This man is a LEGEND!
The Indian mathematician Brahmagupta (597-668 AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD,[24] but written in words instead of symbols.[25] His solution of the quadratic equation ax2 + bx = c was as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."
@@creativeclub2023 stfu
I actually watched the 40:05 minute video without skipping!😂 Just like learning golf, when you can connect with the instructor through the way the knowledge is properly conveyed, then you find gold at the end of the rainbow!🎉😊
What have I been learning all my life🙆🏾♀️🙆🏾♀️
This is beautiful 🔥🔥🔥
Showed this to my 7 year old math obsessed daughter and she immediately understood. She loves learning advanced math and we love your explanations. Thank you!
You're such a friendly person! It's a pleasure to follow your explanations, Mr. Loh.
This is the first thing our teacher taught us to use and never waste time!
I was taught this, since class 8 in India.
Po-Shen Loh's enthusiasm is the best of the video
I love the way he teaches, how he anticipates future content, let’s viewers know there is a reason for doing something, his enthusiasm.
Thank you for sharing your thoughts! Math is about reasoning, not about memorizing. Everything in math can be taught in a way which illuminates the reasons. :)
Po-Shen Loh yes! It’s no good to just memorize. That will be quickly lost. But the ability to explain your way through a concept gives access to so much more.
@@belmer73 :)
This is actually the pq-Formula, but i never understood how it came to live. Thank you for this illuminating explaination.
thanks Po-Shen Loh, I never studied this method in school or in University but I'm so thankful for sharing your knowledge and understanding of different methods to solve quadratic equations. I prefer learning this method for all students to know is the best choice than the normal way. Glad how you ended up connecting or even linking to the Quadratic Formula itself.
Great method! It's so rare you see an improvement over such a basic thing as quadratic equations. I just love your enthusiasm. Science Alert brought me here.
Thank you for your fantastic teaching skills. I'm very happy with your lessons, because it's so easy, but I don't know why they don't teach us in College like this beautiful way. Thanks a lot.
Teachers like you and my best maths teachers(Chandan Sir,Azizul Sir)really inspire the students to discover and love the maths.Much respect to every good and ideal teachers.
I can't believe that im finished watching this video, its amazing how Mr. PO makes math looks very interesting. Respect ❤️
It uses the implication of transitive relation 'if' this exists 'and' the next condition Exists 'then' the third condition must exists to satisfy the relation. Thanks a lot sir for this method it's more solid compared to guessing factors
You have made me cry
I have just one word for this "AWESOME"
Many thanks for this video! It gave me (and my high school students) new insight into understanding the cubic equation formula which uses the same beautiful ideas explained so well in this video. It never crossed my mind to use these ideas to solve the simpler quadratic equation. So, thank you! Best regards, Anna/Helsinki/Finland
I'm a little surprised that this is considered a novel thing in the US. I learned this as the "standard" method for solving quadratic equations in junior high in China. But Prof. Loh still kept me engaged. Great professor!
we did this thing called the quadratic formula that enabled us to solve any quadratic equation in Hong Kong.
if i remember correctly, it was (-b+/-(b^2-4ac)^1/2)/2a
if u do manage to remember it, solving quadratic equations is quite simple as well
What clarity, elegance & humility. Thank you.
Love this clever use of dimension reduction. A = x + y -> A = (A/2 + u) + (A/2 - u). I also found a fun little generalization: If A = Sum ( x_i ) for i = 1, 2, ... , N then let x_i = (1/N)A + iu for i < N and x_N = (1 / N)A - ((N - 1)N / 2)u. One number is the counterweight and the rest are various multiples of u.
Are you suggesting that this can be used to factorise cubics (N=3) and higher order polynomials? If so, then how does your method work? If not, then what is this a "generalisation" of?
Thank you so much! I'm studying for the GMAT right now and this method is going to help me work through quant problems involving quadratic equations much more quickly and efficiently!!
He's actually "completing the square"! Throw the constant on the other side as a negative, take half of b, square that and add it to the number on the other side. Then you take plus or minus the square root, then add the number from the left side.
I was actually thinking along those lines a few weeks ago, but didn't have time to follow it through. Hats off to you!
Thank you sir. It's a brilliant way of solving quadratic equations rather than memorising the quadratic formula. Hoping for you to come up with many other tricks to simplify Math
Very interesting and amazingly explained. never imagined i should have seen this 40 min continously without stopping. Great. Love to learn more.
Thank you very much sir for this wonderful method. I m an indian JEE ASPIRANT and even i m very thankful for this method.
I really going to share it with my friends.
But Isn't It Just Completing The Square?
@@anshumanagrawal346 it will come in handy for tricky quartic quadratic equations
@@ayushmansharma4362 oh ok thanks
Even student should have a teacher like him.
Can't wait to find this in text books in a few years or so.
shoutout to people like this on youtube for carrying us through school single-handedly. THANKYOU
Brilliant! Very well explained. I wasn’t aware of this technique before...something different.
All indians must have studied it.. It's called Shree Dhara Acharya formula. But this is the first time i have understood the essence of the formula. That's why it's imp to understand the proof of a formula or how a formula came..this may make students buildup an interest in Mathematics. Loved it👏
I am quite surprised that this has not been more widely known.
In Germany it is called the pq-Formula (because of x^2+px+q) and it has probably been the most common way it is being taught.
@JJPhenom I'm American. When I was in school, we were just expected to memorize the Quadratic Formula. It was never explained to us. I had never heard of the PQ Formula -- until I heard of the Po-Shen Loh method and began doing research and watching a lot of videos. I think the way the Po-Shen Loh method is presented (especially when done with a graph) makes it easier to understand & makes the formulas make sense.
@@chocolateangel8743 I absolutely agree. I was only surprised that this is not necessarily the way it has been taught.
@@jjphenom2831 I think it's because of how generations of math educators have been taught (at least in America). I've talked to a lot, and the idea of learning math on a deeper level, from a more conceptual perspective, is pretty new for us. Back-in-the-day, being good at math just meant that you were good at memorizing algorithms (even if you didn't understand them). Plus, the standards that teachers had to meet in order to teach math were pretty low.
When they first raised them, many teachers that had been teaching for years, couldn't pass them. It was a big deal because they also couldn't replace all these teachers. The kids basically got screwed. These teachers were talking about how hard the tests were to pass. So, I asked some professors that had taught at the high school and college levels. They said they weren't that had -- if you understood math. If just tried to get through everything using memorization, you were in trouble.
@@chocolateangel8743 Oh I see. That sounds like an interesting shift in approaches. I will be following up on how successful this turns out to be.
Here the formula is often applied quite algorithmically and without going very deep into its derivision. I do not know the reason why this is the predominant method of solving quadratic equatiins though. Would be interesting to see a worldwide distribition
Dr. Loh: The technique of completing the square and its formal form, the quadratic formula, are beautifully re-interpreted to present an easy to handle method. Will always love to tell my students this method, giving your reference (Dr Poh-Shen Loh) and encouraging them to watch on the RUclips.
Seeing this.. i feel like im recovering from my illness... So unreal.. thank you professor.
This method is so much simpler than what I was taught. Bravo Prof. Loh!
Finally I can put factoring aside it's lot more harder guessing than not having to deal with it.
This method actually combines factoring with an understanding of palabras (the midpoint, in particular). Once you understand it, I do think it's easier. However, it works better on some problems than others. If there's a leading coefficient other than one and you have to deal with lots of fractions, things get can get messy.
I like your teaching style. you actually explained how you arrived at a number which a lot of people don't do. Bravo!!!
This is a great way to approach quadratic equations...great job!
This is amazing, I am watching your videos to see if they'd be a good resource for my son who's 14, but you just taught a 43 year old engineer something new! Thank you!
Thanks Mr. Loh! You're a great help to my math career and I can't wait to see more challenge videos! :)
Pleasure to share this one with everyone!
Yes..... He is a genius...
Thanks Dr.
I love Mathematics my favourite
Sir your content is highly discussed by Triangular Kamal which is really easy for all kinds of people.
ruclips.net/video/CMNDwY7q7AU/видео.html
Just to make it clear, the method presented in this video is not new in any way. It is just a relabeled version of the well known PQ-formula being taught to high school students all around the world. The algorithm in the beginning is just a common derivation of it. However, there is nothing wrong with the video it self. It is one of the most pedagogical videos on quadratic equations I have seen, good job on that.
Unbelievably easy and effective!!! Wish I knew this when I was a lot younger though he he. Thanks heaps. Cheers.
Awesome method! I am physics student and even though I am very familiar with quadratics I always find the quadratic equation and factoring very tedious!
Your method is much faster.
Normal methods require quadratic formulas or completing the square.
This method combines the two which makes it easier
Splitting the middle term is also another normal method to solve Q.E
This is very helpful. Thank you Dr. Po Shen Loh for expounding our knowledge of maths in a precise way of Reasoning without guessing.
A brilliant method brilliantly explained. Much appreciated.
He is the BEST teacher ever! I loved how he teaching way. Thank you!!
Amazing! Will start showing students this too! Thank you!
Nice to connect with a fellow educator!
Po-Shen Loh yes! I’m so happy when I saw your video. Really inspiring! Thank you!
I am from India and these type of mathematics are taught to a eight standard students but I liked the lesson great job
This is groundbreaking material, truly eyeopening
This is really cool. You can actually use this to derive the more commonly known quadratic formula. Just keep the variables, start simplifying and rearranging, and eventually you end up with something in the exact same form as the quadratic formula. Would really love to see you show the 2c form of the quadratic formula.
It can be easily derived with simple algebra by taking a quadratic equal to 0, subtracting `c` on both sides, and dividing by `ax + b` to get `-c/(ax + b) = x`. Then, substituting x for the quadratic formula. After simplifying, you get a new quadratic formula (which can even solve lines!) which is equivalent to the original but flipped (reciprocal) and the 2a becomes 2c.
Additionally, the `-c/(ax + b) = x` form you might note sorta looks recursive, and you can use it to solve a whole set of simple continued fractions (e.g. the one for phi aka the golden ration, the square root of two, etc). It basically shows that quadratic equations are continued fractions too! It's very cool.
I discovered both in my senior year of highschool.
So in short, it's still the quadratic formula, but force A=1, then simply rewrite sqrt((B^2-4C))/2 as sqrt( (B/2)^2 - C )
I agree that this makes it simpler, and I think the key is that you're dealing with B/2 on both sides of the plus/minus.
Nice meeting again!
I think this vdo clears the doubt we were having with that method....
This is Shridhar Acharya formula which is already there in class 9 textbooks.
Your method should be in all high school math books.
I’m going to teach this to all my peers at my community college.
Excellent Méthode Monsieur Shen Loh : une solution rapide et efficace. Merci
I think this is quite remarkable to be put into words. A lot of my family members struggle with quadratic equations and I think this will be much easier for me to teach them. To be honest, this is already kinda what I use to brute force my way through quadratic equations with very large coefficient's to impress whoever I'm tutoring at the time. This definitely takes it a step further though.
The first time I had my own way started to studying because of you
This is amazing. This is" MIRACULOUS " that was very kind of you for sharing this. THANKS A LOT
I really like they way you're emphasising IF -> THEN logic in this video.
I believe the key to learning this math in general lies in understanding the relationship between the numbers and lines and the points of a line-being able to better see the graph with your minds eye so to speak. Often, too little emphasis is placed on concepts while too much emphasis is placed on finding the answer to problems, i.e. what is the answer to this one, x^2 + x + 1 = 0?, now what about this one, x^2 + 2x + 3 = 0?, now here are ten more problems for your homework.
The advantage of Dr. Loh’s discovery is that it removes the need to recall the quadratic equation, used by high school students studying algebra. The discovery employs principles of geometry, such as that every segment has a midpoint, which frees the student to focus more on basic principles and concepts and the relationship between numbers and lines and the points of a line, rather than the quadratic formula itself.
Lets look at some of the underlying assumptions. As for the example that every segment has a midpoint, that every segment can be bisected was Euclid’s Proposition 10. However, note that Euclid made a tacit assumption in his proof of Proposition 10 that caused Wikipedia to wrongly state that Euclidean and non-Euclidean geometry share as many as 28 of Euclid’s elementary geometry propositions, when the number is fewer than 10. See the Facebook Note, Wikipedia Contradicted by Euclid's Proposition 10, Youngsters with Ruler and Compass facebook.com/notes/reid-barnes/wikipedia-contradicted-by-euclids-proposition-10-youngsters-with-ruler-and-compa/577085739010671/.
Also assumed is the basics of the coordinate system. Along these lines, the following is from the Yahoo article by Caroline Delbert about Dr. Loh's discovery: "Since a line crosses just once through any particular latitude or longitude, its solution is just one value."
This statement depends on Hilbert's Axiom I. 2, that two such lines cannot share the same pair of points.
When David Hilbert added a coordinate line, the line with the features to comprise a number line, to Euclid’s geometry, the very earliest axioms required subtle modifications. From Euclid's to draw a line from one point to any other, and extend it in a straight line, Hilbert first produced, two points determine a line and added, they determine it completely. But this eventually became every pair of points is in some line (Axiom I. 1) and two different lines cannot contain the same pair of points (Axiom I. 2, paraphrased). This 'line' is what became a coordinate line.
The term "line" in Axiom I. 2 is an elementary term, which means it has no definition that is used in a proof. Non-Euclidean geometry depends on the stipulation that its term for "line" is an elementary term and therefore has no definition that is used in a proof. So this opens the door to interpreting the meaning of what is meant by the elementary terms, "line" or "plane," and then applying the logic of the geometry axioms.
One type of non-Euclidean geometry says, there are no parallel lines. Well, if the “lines” are the great circles on the surface of a sphere, and the surface is their “plane,” then there are no parallel “lines” because great circles on the same sphere always intersect. (Parallel lines are defined as “lines” in the same “plane” that do not intersect.) Euclidean geometry says, through a point not on a “line” there is only one parallel to the line. When you interpret the “line” as a straight line, this seems right.
So given an undefined line, the Euclidean geometry and non-Euclidean were seen as both logically consistent (just not logically consistent with each other). But what has been forgotten is that the non-Euclidean geometry with no parallels (called Riemannian geometry) is not logically compatible with the elementary axioms necessary for including coordinates in the geometry, such as Hilbert's Axiom I. 2. Given this inclusion, the non-Euclidean geometry then becomes self-contradicting because you can prove there are parallel lines, which contradicts the assumption that there are no parallel lines. This is described in a brief Facebook Note: Self-Contradicting Non-Euclidean Geometry facebook.com/notes/reid-barnes/self-contradicting-non-euclidean-geometry/766736476712262/
Reid Barnes The relationship between Euclidean and Non-Euclidean is similar to Physics and Quantum Physics. The latter one requires a better understanding and approach.
Reid Barnes, what amazing insight you have. Who would have thought that one has to invoke Hilbert's axioms and non-euclidean geometry to complete the square to solve an algebra problem?
more people need this in their lives
Well explained, Sir. Actually this formula is known as Sridharacharya's formula. Sridharacharya was an Indian mathematician, Sanskrit pandit and philosopher. All the students of high schools are taught this. So I have also learnt this in my school days. Thank you, Sir, for making it more familiar to others. I'm feeling proud to be an Indian. Jai Hind. 🙏🙏🙏🙏
en.wikipedia.org/wiki/Sridhara
You are right.
Thank you sir for this information. Mr Loh has explained very beautifully also, and I was not aware that Sridharacharya had already discovered it.
I think he had derived the quadratic formula we generally use, and this simplified version was discovered by the Babylonians.
Brilliant way of solving Quads in a more generalistic manner , for helping our natural brain go straight without much of guessing !
You have illustrated so nicely , with great patience in the interests of diverse learners and practitioners , dear friend in Maths !
Bless you ! 👍❤😍✨
Thanks professor for very meaningful video, really love the way you express the logic and guide how to produce the result, actually in real use it will be very quick by just writing down neccesary part to calculate. And in fact i learnt 1 new trick of solving sys of equations consists of a product & a sum (the average plus/minus). Also i would like to contribute another approach for this as well: x^2 + bx + c = [x^2 + (2b/2)x + 1/4] + c - 1/4 = (x + 1/2)^2 + c - 1/4 . It's look a bit complicated but if understand how to express (a + b)^2 should be very easy to imagine. But then, just another trick, and the important is still how we understand things and utilize them the right way.
Genius! Why did we not learn this method at school seeing it has been known for hundreds of years. If we had more of these techniques in math classes there would be a greater interest in mathematics and more people will pursue mathematics as a career.
Thankyou for giving us, such a nice method to solve the quadratic equations
The brilliance of this method is only rivalled by the excellence of the presentation! WONDERFUL!!
Much Thanks Prof. Loh ... Outstanding!
Its just the quadratic formula
Po-Shen Loh, I know you are deservingly happy discovering that we can find quadratic roots much simpler by combining these two ancient insights, BUT have you taken the time yet to notice the other side of what you have discovered? A new insight. Yes, the roots do exist but you have more importantly shown that the average of the roots also ALWAYS EXISTS! And we now see a new and strange average of the roots for an equation like x^2 -2x+3 for example yet see similarities of the root averages of X^2 +1 and horizontal translation of it like x^2 -2x +2. This is the new Loh insight of complex root averages and/or complex averages in general.
This is the most helpful video i've ever seen
What a great explanation to the how they came up with the quadratic equation. That was the best I've seen in quite a while. Thanks.
May I complement Po-Shen-Loh on a brilliant exposition of a bit of basic, fundamental mathematics. A good example of plain honesty, simple truth and easy understanding.
I first watched the video a day or so ago and it was only a day later that the penny dropped.
As per Leonhard Euler's 'Elements of Algebra' (x-a)(x-b) = x^2 - (a-b)x + ab where as we know a & b are the roots of the quadratic.
Taking (a+b)^2 and (a-b)^2 [ i.e. props. 4 & 7 from book 2 of Euclid's 'Elements' ] then expanding and subtracting we get the answer 4ab hence we have (a+b)^2 - (a-b)^2 = 4ab. This is a theorem, prop.8 of book 2 of the 'Elements' and for some unknown reason demoted to a RULE alias 'The Quarter Squares Rule'. After a bit of simplification we end with [(a+b)/2]^2 - [(a-b)/2]^2 = ab.
The algorithm given in the video then amounts to [(a+b)/2]^2 - ab = [(a-b)/2]^2 which taking the square root leaves (a-b)/2.
So (a/2+b/2+a/2-b/2)=a & (a/2+b/2 -a/2 -[-b/2])=b.
It is still highly commendable that the 'QSR' has been derived by another route and has been admirably utilised for the factoring of quadratic equations. What I find a bit astounding and some what sad is that together with the hits on the 3blue1brown and MindYourDecisions videos on the same topic a combined total of around 1,453,000 views no one else seems to have spotted the connection.
Finally if we change a & b to x^m & x^n then the answer (ab) becomes x^(m+n) hence all integers raised to a power above the second are the difference of two squares . Further more the bigger the power the more DoS solutions there are for any one integer raised to that power! What does this mean for Fermat's Last Theorem.
3blue1brown U link
MindYourDecisions U link
Professor Po-Shen Loh found and original insight. You are amazing!