This is also worth checking out: "New" Way To Solve Quadratic Equations That Everyone Is Talking About : ruclips.net/video/oWFRU-ula-A/видео.html Professor Po-Shen Loh devised a different way to solve quadratic equations that could be helpful to many students.
The "couple of steps" you skipped in the second half of the video is what I came here to understand. It's been a long time since I studied math, and even my arithmetic is rusty when it comes to subtraction of fractions that include variables. Can you please point me to where I can learn why (-c/a)+(b²/4a²) = (b²-4ac)/(4a²), step by step? I remember it has to do with cross-multiplication, but I'm not getting it. :-(
Everyone made fun of me using completing the square in 12th grade, whilst they were using the quadratic formula - they said I was wasting my time. I believed it was a lot more elegant. So I decided to prove to them that it was the same thing during one of my classes near the end of the year... took me forever - but I eventually got it. IF ONLY I knew this. SUCH an elegant proof - great video explaining it!
@@sunjidasultana4648 Well I mean, most good teachers will show you that all three methods are best depending on the numbers you are working with. Basically you go through and see if it can easily solved with factorization/completing the square. If the answer to both is they can't (too tedious/"not nice" numbers), you just go straight to the quadratic formula.
+Benny Doe I agree, I do hope students share this with one another. I taught high school math years ago, I never made this as clear as the video does in less than ten minutes! Bravo.
+Bill Stokes I thought I'd just tell my maths teacher, so that when he teaches gcse he can show this to his students so they can hopefully understand it better and not think 'its magic' or similar things.
This is how math should be taught. I see too many math teachers who just present formulas out of nowhere without actually deriving them and explaining them, and this is pretty much like giving an instruction booklet to a beginner and ask him/her to use that booklet for everything. Yes, there are situations when certain topics might be useful and important but also too complicated to actually derive at that point, but if something *can* be derived without too much trouble then it *should* be derived. One rule of thumb is that students should be able to reprove those formulas to themselves with absolutely no extra help or reminders - everything should come from logical step-by-step reasonings.
+alejandro cartes you can also tell your teacher that this is the same ancient method that Al-khwarismi used in around 820 ad in his compilation of algebra techniques. this geometric method has been around for millennia.
5:10 you could've explained this as when square rooting both sides you will go from (x+1)²=16, to x+1 = [+/-]4 because when square rooting, you get a negative and postive value. I just thought giving the length a negative value is quite confusing as that's quite impossible..
Definitely true. Ancient Babylon peoples, only positive, negative or complex solution, Value. Virtual+real with i. vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_b%E1%BA%ADc_hai diendantoanhoc.org/topic/169871-gi%E1%BA%A3i-ph%C6%B0%C6%A1ng-tr%C3%ACnh-b%E1%BA%ADc-2-b%E1%BA%B1ng-ph%C6%B0%C6%A1ng-ph%C3%A1p-h%C3%ACnh-h%E1%BB%8Dc/
Understanding the basics is a critical part of learning; unfortunately, not many people know that. We need way more people like you that not only understand that statement, but also willing to teach it. Keep doing what you are doing!!!!
At the time (1973), I thought my 8th grade Advanced Math and my 9th grade Algebra teachers did a fabulous job of presenting the quadratic. I'm willing to concede that this is better. Very well done. 5 stars.
I really enjoyed this video. Years ago, I was in college, majoring in Mathematics, with the goal of becoming a math teacher. My ultimate goal was going to be the teacher who spotted the "hole" in a student's understanding of mathematics, transforming them from a math-phobic student to a math-master student (as was done for me by a wonderful teacher in high school.) Since the field of math is cumulative, knowledge wise, any area not completely understood by a student in the past, will ultimately become an abyss in the future. Sadly, a TBI ended my progress in college, but with the help of RUclipsrs like you, I am re-discovering my love of mathematics. Who knows? Maybe I can tutor some students one day. Thanks!
Frank De Mascio I hate how teachers use acronyms and such. What they do is give you a shortcut without actually getting the deep understanding you need in the future.
I love how simple he makes it all look, by using a calm voice and great representations. And Desert Cactus you are right, this should be shown in every high school and middle school algebra class. I teach adult ed, and they need to understand algebra to get their HiSet or prepare for the test to go into college, and this is yet another excellent video from him.
I never liked mathematics when I was in my younger age because most of my teachers presented it arbitrarily and non-chalantly, devoid of passion and wonder. Now, in my late 30's, I'm hooked on it because of great math channels like you. Can you do a series on millenium prize problems? Also: Hilbert's problems, Continuum Hypothesis. Thank you!
I’m 14 rn and I’m watching this cuz I have to memorize the proof and write it on the board and I didn’t even know what completing the square is. This video made me realize it actually makes sense. I was taught in class you do this and get this.
Ok this is awesome. I've read on school books that the proof of the quadratic formula was too hard, but this is very simple and ties in with the concept of completing the squares. And you can even show it with pictures, for people that easily loses track of numbers.
They gotta understand the concept that 'x' here doesn't refer to length but rather simple algebraic, since this is just a geometric illustration of quadratic formula to enlighten us
They either don't know it themselves or they don't know how to teach. But, as far as my knowledge goes... Kids usually don't understand math because they find it boring as kids usually don't have to use math very often in a concentrated way and that requires a little thinking (outside of school grounds). So, whenever they teach this, you are a 10 year old kid with almost no knowledge of algebra. So, despite trying to understand the visual part, you miss the opportunity because you lost yourself in algebra. And there is that game kids are playing so.. You probably paid zero attention. Why am I saying it? Because it is in curriculum and in almost every math book for kids in age ranging from 8 to 12. I, of course, can't guarantee this considering the USA. In my country at least, every kid is supposed to know this despite they don't: too poor to study (should be a motivation), no interest, no teachers.. But the books do cover it.
Wow, more than 30 years since I left high school and finally I understand what completing the square really means! You're an amazing teacher! Thank you so very much! You are a true champion 💯
I was struggling to teach some of my students the quadratic formula and I started thinking about it in similar terms such as this (minus the part about negative lengths). I knew some of my students might find this confusing. However, I took a chance and tried explaining the quadratic formula to my students in this way. For several of my students, I could see light bulbs going off--they got it! However, for many of my students, this only confused them further. I like your video. You explained it much better than I think I did (the colorful virtual manipulatives you used are extremely useful), so I think I'll be showing my students this in the future. I am a new math teacher and I try so hard to teach my students an intuitive way to think about math, like this for example. But sometimes, it seems that I only confuse them more. Also, if you've never been taught to think about math intuitively, then thinking intuitively requires a different orientation--kind of an undoing of just memorizing stuff to get answers. Keep up the great, great work!
Another piece of this is relevance to students' lives. Sure, "real world" applications utilizing the quadratic equation are everywhere, but in most cases students will counter with, "Yeah but I won't be using this in my own life, so why learn it?" For example: lobbing a basketball and modeling its motion with a quadratic might seem an ideal motivator, but how often do you see LeBron James out on the court with paper and pencil calculating the ball's trajectory? My own solution to this was to create a 2-D basketball simulator in the Scratch programming environment; players input a "best guess" angle and initial velocity and watched the results. This can springboard into a discussion of the math behind why the ball did or didn't go into the basket. We need to really put our backs into the relevance aspect of math. It's important.
Fantastic video, thank you! Not sure I ever understood exactly what completing the square was until today. Programming the quadratic formula into my HP-15C took me about 60 steps. My fellow engineering students were not impressed and showed me a brilliant "folded" version which was closer to 30 steps. Their version ran the same code twice, once for the positive root and the then again for the negative root. I was humbled but at the same time wanted to make a contribution. They assured me that this code had been gone over with a fine tooth comb by dozens of engineering students over years and that I stood zero chance of saving even one single step. I nearly gave up because the code was really good but in the end I did save them a step. Their code recalled a value from memory that was already present in the stack of the HP15C so that step was not required. The value made it to the top of the stack but never got pushed out so it was still available for use without needing to be recalled from memory.
You are a much much better teacher than the current average math teacher or any normal math book. I've been following your vids for quite a while now, and they really motivate me to learn and understand more about math. Thank you!
I can think of a lot of people who could've benefited from this means of demonstration; much more straightforward and powerful than just forcing them to memorize the quadratic formula.
We were taught this visual method to explain the formula in Algebra 2, but I don't think a lot of people made the connection or preferred to think in terms of numbers instead. I like this method the most.
+AngieMyst I knew about the algebraic derivation, but didn't connect it to anything geometrically. While geometry is much more intuitive in most cases, I think we can learn a lot from the ancient Greeks when it comes to this, as they connected nearly all mathematics to geometry.
+Cubik What kind of math teacher does that? In school all of my math teachers always gave the full explanations of all formulas we learned. We were not forced to remember the explanations as for the test only the formula was required. Most of my classmates memorized only the formulas, but I assure you, we had always the complete explanations. For me, memorizing a formula with no logic was a greater effort than remembering the (logic) way to get there. In a lot of cases because of poor memory, I did not remember formulas for the exam and I had to reconstruct them. That was easier for me that way. I was always in the 3 best students of my class in math... never knowing the formulas, always able to reconstruct them. So I guess in your class, with your teachers, I'd been in the 3 worst math student.
The complete explanation for the completing square which all high form students of the secondary should learn deeply by heart and digest the logics and inference. Excellent!
This was truly brilliant. I learned to complete the square 54 years ago, but I never realized how it was derived. This should be presented in every high school algebra class. Thank you so much for this and your other videos.
I did 'completing the square'in school over 15 years ago. I went on to do an engineering degree, and a degree in pure mathematics. I wish my maths teacher in school (who was a really good teacher) had explained completing the square and the quadratic formula this way!
Doc Daneeka Many students wish the same thing. Often times, approaching a problem with a visual example like this can blow open doors for students who "just don't get it."
7:47 if anyone's confused about what he skipped, here are the steps: He multiplies the (-c)/a by 4a/4a (which is simply equal to 1, so it doesn't change the value) in order to get a common denominator (which is 4a^2, because a × 4a = 4a^2) so he can combine the fractions. From there we have (-4ac)/(4a^2) + (b^2)/(4a^2), which we can simplify to (-4ac + b^2)/(4a^2) Finally, we can rearrange the fraction to be (b^2 - 4ac)/(4a^2) Hopefully that helps. TL;DR: he just simplifies the two fractions into one with algebra.
Ive realized that my whole life i havent been learning math, i’ve merely been memorizing it. It made no sense to me why i was learning it until i had it explained to me in an illustrative and comprehensive way, where now i understand why math is so important.
We often think that maths is just a subject and for many students it's the scariest subject to study but most of us don't know that maths is the language in which nature is written and it is alot more than what we just think of it.
What this video clearly demonstrates, is what I've been saying for years and years. Generally, mathematics isn't difficult - well at this level. It's more the fact that you've been taught something that you commit to memory instead of understanding it. So, you see, it's not that you're bad at it - more like the education system and the way it's taught is useless. Once you understand something, as opposed to just memorizing it, it makes problem solving using this knowledge much easier.
Paul G68 Actually, none of what you said demonstrates mathematics is not difficult. Yes, mathematics is taught very poorly, but so is every single subject in school. Also, homeschoolers generally agree that mathematics is still a more difficult subject than other subjects. So while education contributes to its difficulty, it is not the cause. Mathematics is difficult because it is by its very nature counterintuitive. Humans are not logical beings. Mathematics is based on understanding definitions, axioms, and using logical arguments and consistency, all things which humans are bad at. Humans have historically always used their emotions to think, never actual logic. Everyone has logically inconsistent beliefs for this reason. History does not behave like a formal logic system, and neither do arts or culture. Those are things we adapt to. Math behaves like a formal logic system because it is one by definition, and as such, it resides outside the first nature and second nature to humans. For the record, geometric visualizations aren't any more intuitive than algebraic ones. Plenty of students struggle with those too. The only reason it looks like there aren't any is because students who struggle with them are always accomodated for by the education system, since curriculums are almost always purely algebraic visualization. But in reality, geometric visualizations don't make intuitive, it only makes it somewhat more accessible to students. Math is not intuitive. The vast majority of mathematics simply cannot be learned by visualizations like these, and the vast majority of mathematics isn't intuitive. These are an outlier and can be explained visually only because, in the grand scheme of things, it's one of the easiest subjects in mathematics. It's literally the second easiest thing after arithmetic. That's why they teach it after arithmetic.
Angel Mendez-Rivera : well you have your opinion and I disagree. Maths is taught in a way that makes it difficult. Want an example. Add up in your head, like you were taught as school these numbers. 1467+1667. Remember do it in your head. What you’ll find is, you’ll struggle because of the way you were taught. Now watch this video: ruclips.net/video/hesKQ_y1P7k/видео.html So, you see after you watch this video it’s much easier and intuitive. Theres the demonstration of what I was saying. So once again, I respectfully disagree with what you are saying. Once you build up a basic understanding of maths, it’s a case of building on top of that to further your learning. Nothing hard in that.
I all ways knew that the -B formula (the formula you showed at the end) came from factorising/simplifying ax^2+bx+c but I never could do this by myself ,my school never thought us the visual method I find this extremely helpful thank you
8:47 Can someone explain this step please? I figured it out. if anyone reads this in the future then here's the explanation: To make this easier to type out in RUclips format I'm going to call the term b^2-4ac: "v1" and the term 4a^2: "v2". 1. We start by moving the b/2a term from the left side over to the right side: x = - b/2a +- sqrt(v1/v2) 2. Since we have a fraction inside the sqrt on the right side we can simplify it according to the rule: sqrt(a/b) = sqrt(a)/sqrt(b): x = - b/2a +- sqrt(v1)/sqrt(v2) 3. We now have two fractions on the right side. To combine the two fractions into one they have to have the same denominator. Luckily for us they already have because the sqrt(v2) (4a^2) is equal to 2a. The equation now looks like this: x = - b/2a +- sqrt(v1)/2a 4. We can now combine the two fractions since they have the same denominator and we have the final equation (v1 is still equal to b^2-4ac): x = (-b +- sqrt(v1)) / ( 2a )
This is such a clearer way of illustrating 'completing the square'! I never knew this is why it was called this, or, for the life of me, remember how to do it as I rarely used it. Thank you so much for this!
This is one of the best videos on youtube. I am in grade eight and had no problem keeping up due to your amazing visuals and your simple explanation of the concepts. I look forward to using the quadratic formula with the understanding of how it came to be!
Sir I've studied quadratic equations today in my online classes but they told me to remember these 2 formulae so I remembered them and I follow ur channel randomly I found this video and understood how it comes thanx sir
I like the video, as a teacher I would have just liked the part at 2:50 explained, "why do we need to use a certain trick?" the idea that they have a common side is great, but the shapes could be combined just by smushing them together to get x(x+2) on the left, but it would be good to explain why making each side a square will simplify the solving. A simple statement such as "If we make a square we could easily determine the side length of the square once we know it's area. Getting the side lengths of a rectangle given it's area could lead to many different answers." or something like that. Otherwise, very great video! I will use this in my class. Thank you.
I have a degree in math and use the quadratic formula ALL THE TIME when tutoring in Algebra... I have NEVER visualized it like this but instead used a song tune to get it stuck in students heads (to the tune of a jack in the box). I was extremely happy to see this though and will DEFINITELY be using this from now on when I explain it!!!!
@@laikaperraespacial6025 how does 2a * root 2a^2 become 2a again in the final formula? wouldn't the 2a become another 4a^2 under the radical meaning it would end up as a 4a^4 after it's square rooted in the final form? based off his explanation
BROOOOOOOOOOOOO HELL YES!!! Finally understood the intuition behind it. You know how school is they just tell you to memorize it without even bothering to teach you where it comes from !!
I taught the derivation of the general quadratic formula 25 years ago to Intermediate Certificate students (c 15-year-olds) here in Ireland. Later, a revamp of the syllabus no longer required that students be able to *establish* the formula at IC Level, merely to be able to *apply* it in cases where the constants had specific values and the discriminant was > 0 or = 0 . We did not use the rectangles and squares method but rather completed the square by taking half the co-efficient of x , squaring it then adding this squared quantity to both LHS and RHS etc.etc.
We did it exactly as you (and PT) said at secondary school in England in the late 70s - but I had forgotten how to derive it, so I'm pleased to see this !
Because you have to teach students how to complete the square in order to do it and completing the square is only used for this and converting quadratic equations into the vertex form. Consequently, you can save a lot of time to do other more useful things if you're not required to cover the vertex form.
Wonderful geometric interpretation that brings satisfaction to those who wonder "why the hell I struggle learning with equation?" . This explation SHOULD be part of education to get interested in mathematics. Good job.
Just letting you know, once again, how much I love your videos. This was so well done. Empowering! I hope I get the chance to teach the quadratic equation to my grandchild using exactly this video!
The name "algebra" comes from al-khwarizmi book. In that book, he shows quadratic solutions in that way. So that is the way it has been done since the beggining =P But mr Khwarizmi doesn't considered negative values, so he split the quadratics in 6 different types. I didn't like the way you does it. I'm gonna show that example for my students though. Sorry for bad english :)
I'm honestly just sending this to my Maths teacher as something to show people in the future even if it isn't my class as we have moved on topic wise, but still as something to help others in the future
i'm a math teacher of junior high school teaching an honor's math class, presh makes my job easier with excellent videos like this, presh; i thank you; the math community thanks you and most of all my students thank you.
Can you similarly 'complete the cube' for the general cubic equation? And, although it's difficult, if not impossible to visualize, 'complete the tesseract' for the general quartic equation? Do any of these visualizations... those that are possible... provide any insight into why you can't solve the general quintic equation and higher with radicals?
I think the hard part there is that, once you get to 4th degree polynomials, visualizations kinda break down. Maybe someone could think of a clever way of still showing it intuitively, but at that point in kinda just comes down to proofs Also, I think it's probably a mistake to ever be using the quartic equation... that thing is a mess
***** but you could still use it since it is possible to do math on objects of any dimension. The difference is that you use geometric methods instead of algebraic ones to prove a point, even if the geometry requires more than 3 dimensions.
***** I agree, it could still be used as an alternative to an algebraic proof but it will loose some of it's pedagogical value. Maybe it will be easier than the algebraic proof, maybe it will be harder. But it certainly won't be as easy as the n=3 or n=2 case.
OMG! I can't talk English very well, but I got it throughout the video and I could understand to solve that problems! Thank you very much... +1 subscriber
OH MY GOD MAN!!!! THAT. WAS. AWESOME. I think we should forget the normal quadratic formula explanation, and use this instead! It's much more -easier -understandable -and makes people like math!! Congratulations, that was beautiful ❤️
x²+2x-15=0 x²+2x=15 Now we add a term that is given by the following formula: (b/2)² (2/2)²=1 By the way we need to have a=1, if not then divide both sides of the equation by "a" end then do this process. x²+2x+1=16 If you did this correctly you will always be able to factor this out like this: (a+b)², in this case :(x+1)² (x+1)²=16 Let's take the root on both sides: ✓(x+1)²=✓16 Now when we have this situation ✓y² , this is equal to |y| To understand this let's see an example: If we have y=2 then ✓2²=✓4=2 And if we have y=-2 ✓(-2)²=✓4=2 We always end up with the positive number, the absolute value of y, |y| . So ✓y²=|y| |x+1|=4 This means x+1=4, or x+1=-4 Because |-4|=4 So we have x1=3 and x2=-5 ax²+bx+c=0 4a(ax²+bx+c)=4a.0 4a²x²+4abx+4ac=0 4a²x²+4abx+4ac+b²=b² 4a²x²+4abx+b²=b²-4ac (2ax+b)²=b²-4ac 2ax+b=+-√b²-4ac 2ax=-b+-√b²-4ac x=(-b+-√b²-4ac)/2a This is how you can prove the formula
"I'm going to skip a couple of steps" - you have no idea how painful it is to hear that for someone trying to understand this. If you want people to really understand your subject NEVER skip any steps or assume people know what you are talking about.
The steps he skipped arent too complex. He just simplified the expression. He multiplies the (-c)/a by 4a/4a (which is simply equal to 1, so it doesn't change the value) in order to get a common denominator so he can combine the fractions. From there we have (-4ac)/(4a^2) + (b^2)/(4a^2), which we can simply to (-4ac + b^2)/(4a^2) Finally, we can rearrange the fraction to be (b^2 - 4ac)/(4a^2) Hopefully that helps, but to put it simply in case you're still confused, he just simplifies the two fractions into one with algebra.
this kind of video may be helpful for those who interested in maths and always seek for the truth . textbook from school only gives us the equation straightforwardly but never show the proving steps so most of student could just understand how to use the equation to solve question but never understand how does the equation came from
So here is my Solution (an easy one): First I added 15 on both sides, leaving me with the equation x^2+2x = 15 Then, I added +1 -1 on the left side, so it said x^2+2x+1-1 = 15 Then I used the binomic formular (is that how you call it in english?) (x^2 + 2×1×x +1^2)-1 = 15 --> (x+1)^2 -1 = 15 Then I added one on both sides (x+1)^2 = 16 I took the root x+1 = 4 or x+1 = -4 And finally, I subtracted 1, which lead me to x=3 or x=-5 I think this is an easy solution, using basic mathematic formulars. I'm sorry for my bad english, i'm from Germany (ze state where zey can't pronounce ze words correctly) 😉
Add 1 to each side. Factor left side as ( x+1)^2 then use square root property. Try my channel mathfullyexplained. Full unit on solving quadratic equations plus more units
That's awesome to have a graphical representation of completing the square. I learnt the algebraic way to solve quadratic equation in my high school but never have thought this could be done using geometry. I wonder if this method also works on solving cubic equation.
I thought of it this way. if you have an expression of the form a^2 + 2ab + b^2, it can be rewritten as (a+b)^2. so basically it all comes from forcing a quadratic equation into this form by adding a constant term on either side to 'fill in the gap' in the original quadratic expression so that it becomes a square number, thereby completing the square. the motivation to find the form (a+b)^2 is that if (a+b)^2 = k^2, then a+b is equal to both positive and negative k, thereby giving you the two solutions.
Impresionante!! Se siente tan bien entender de una forma más profunda a una ecuación que a primera vista parece bastante complicada, es tan simple que no entiendo porqué no lo enseñan así en el colegio, de verdad, todo es memorizar formulas resolver ejercicios y dar examen, pareciera que están fabricando robots. Gracias por esta explicación! Saludos desde Perú.
Translation: (Esta es una traducción del español al inglés) Awesome! It feels so good to understand in a more profound way to an equation that at first glance seems quite complicated, it is so simple that I do not understand why they do not teach it thus in the school, really, it is all memorizing formulas to solve exercises and to give examination, it seems that they're making robots. Thank you for this explanation! Greetings from Peru.
Albanovaphi7 Eso es muy cierto, siento que el sistema educativo no está actualizado con las necesidades actuales de educación y crecimiento para los humanos. Algo tiene que cambiar Saludos desde América.
Más profundo no se traduce more profound, se traduce deeper. En do not teach it thus in the school, el thus sobra y es this, no it. En "examination", eso es incorrecto, lo correcto es test para referirte a un examen escolar. Se que es muy tarde ya xD pero mejor tarde que nunca.
Históricamente, las escuelas existen para fabricar trabajadores humanos que operarían como si fueran robots. El que te dijo que uno va al colegio a aprender no sabe nada de historia.
Negative quantities have physical meaning. Think negative displacement, negative velocity and acceleration. Going further, even "imaginary numbers" have real meaning in the physical world.
Imagine the trajectory of a cannonball. The quadratic formula tells you the positive answer, when it will hit the ground, and the negative answer, when and where in the past it was shot out of the cannon.
What is a box with a side of -4 ?? -4 is not wrong ! The use of the equation, IS wrong ! As noted by others, it could mean: Before your determined start time !
@@francistan4674 yes. but length or distance is a scalar quantity so cant be represented by negative numbers. this is why separation of algebra and geometry is important.
I was just realizing that I have followed every explanation you’ve every given on your channel with a minimum of effort. That’s what makes you a great teacher, and that explains why I keep coming back.
this video was MOST EXCELLENT..!!!... I've never seen this kind of explanation for the Quadratic Equation....... it's amazing how the Visual of the Geometry can really Convey the logic of math proofs... THANK YOU!!...
Overall a fantastic video; what I appreciated most was the derivation of the quadratic formula using completing the square. I often wonder why students are asked to solve quadratic equations via completing the square when that process, as applied to the general quadratic form, yields the quadratic formula. In the past I've given a single, high point value extra credit question on exams: _show me how to get the quadratic formula._ I would tell students in advance that it might be on there. Students really liked those points of course (major damage repair if they'd had a rough time on the rest of the exam), and even if they forget about the process in the future, they'll never quite be able to say, "that danged formula was a total mystery."
I was taught this 45 years ago as completing the square without the geometric representation. It of course leads directly to the quadratic equation and that was why we were taught the method rather than simply using a formula with no idea how it came to be. Education used to be called teaching and teachers were effective. Now we have "educators" and all sorts of theory about teaching and no common sense. Oh dear, I'm getting old.
+984francis At the risk of igniting internet rage, I am going to propose that partly this is an unintended consequence to employment equality between men and women. Once upon a time, women were very restricted in employment opportunities. For white collar work, it was pretty much: secretary, nurse, teacher. Smart women probably went into those fields (rather than into unskilled work). Smart women are as smart as smart men. Thus you have women with doctorate-level intelligence doing these jobs. I suspect that we used to have hidden high-achiever women in admin assistant, nursing, and teaching positions that now have moved out to better paying positions. It is good for them, but bad for the rest of us.
Mark M. That is a point I've never thought about. I grew up homeschooled and now work a job as a math tutor, so I get questions regarding my opinion of the public school system all the time, and why it now sucks. This makes sense as a contributing factor: the majority of teachers are actually just dumber than in times past. Thank you.
BTGTB Smarter teachers don’t equal better teachers though. I’ve been taught be people with PhDs and those with bachelors and the PhDs have never been very good. Because it’s easy to them, and they would rather be at a university.
@Fester Blats I do find that the beauty of maths isn't the solving part, but it is understanding the logic behind every step. Appreciating logic is found in maths, and this same appreciation of logic can apply to other aspects of life beyond maths as well. For example, if you have a strong foundation and appreciation of understanding mathematical proofs, Surely, you wouldn't be as gullible to propaganda and conspiracy theories in real life as you would be meticulous towards the proofs and evidences behind them, skills you've honed through scrutinizing mathematical proofs.
The fact that this type of explanation was missing from my education is exactly why I gave up on math. But now that I know there are answers out there I am finding my way
Wow. I swear to god I would NEVER expect to see Bhaskara being formulated at the end. Initially, it seems such a surreal formula, but when you look at step-by-step you realize why it makes so much sense.
When I was a young student, my Father’s boss would quote the quadratic to me, I was very unimpressed. With the benefit of 56 years experience, I am now impressed. But, thanks to your vid, I now understand the term completing the square, and will remember this.
Amazing explanation! Thank you! It's better to use "completing the square" than using "quadratic formula" in an "automatic way". Your superb explanation should be teached in school. :-)
Pranav Kumar S. I think of X times X as adding X number of X spaces I think of -X times -X as removing X spaces of -X. basically like if there was some object taking up space, you are removing this "negative space". in other words multiplying positive numbers is like adding another room to you house, while multiplying negatives is like clearing out a full room so you have more space. either way, both give you more space.
This is also worth checking out: "New" Way To Solve Quadratic Equations That Everyone Is Talking About
: ruclips.net/video/oWFRU-ula-A/видео.html
Professor Po-Shen Loh devised a different way to solve quadratic equations that could be helpful to many students.
No bro factor method is so good and if he not work quadratic formula always worke
You're so good in math, thank you for the explanation.
X^2 +2X+1 = (X+1)^2
The "couple of steps" you skipped in the second half of the video is what I came here to understand. It's been a long time since I studied math, and even my arithmetic is rusty when it comes to subtraction of fractions that include variables. Can you please point me to where I can learn why
(-c/a)+(b²/4a²) = (b²-4ac)/(4a²),
step by step? I remember it has to do with cross-multiplication, but I'm not getting it. :-(
You just multiply the term to make both have a common denominator, so here it would be by 4a
I've been using that formula for over 40 years and until now I've never wondered where it came from. Thanks for the enlightenment.
Yes. 38 years for me. While the (negative lengths) kind of throw a wrench in, overall seeing it as matching up equal areas makes it easier to grasp.
i used it for about 2 years and i knew it a year ago!
half way through an engineering major and didn't have a clue until now, I just used it until I got used to it
28 years here... and I am completely mind-blown. So amazingly intuitive!
the way it's usually derived is pretty simple though.
Everyone made fun of me using completing the square in 12th grade, whilst they were using the quadratic formula - they said I was wasting my time. I believed it was a lot more elegant. So I decided to prove to them that it was the same thing during one of my classes near the end of the year... took me forever - but I eventually got it. IF ONLY I knew this. SUCH an elegant proof - great video explaining it!
Yeah more elegant but y consume more time
I prefer factorization more than the formula or the completing square method
@@sunjidasultana4648 Well I mean, most good teachers will show you that all three methods are best depending on the numbers you are working with. Basically you go through and see if it can easily solved with factorization/completing the square. If the answer to both is they can't (too tedious/"not nice" numbers), you just go straight to the quadratic formula.
@@sunjidasultana4648 Factorization doesn't work for all quadratics though
@@sunjidasultana4648 Factorization is mainly situational. It doesn't work when x,a,b,or c are surds or extremely large numbers
This video should be presented in every high school and middle school algebra class.
+Benny Doe I agree, I do hope students share this with one another. I taught high school math years ago, I never made this as clear as the video does in less than ten minutes! Bravo.
+Bill Stokes I thought I'd just tell my maths teacher, so that when he teaches gcse he can show this to his students so they can hopefully understand it better and not think 'its magic' or similar things.
This is how math should be taught.
I see too many math teachers who just present formulas out of nowhere without actually deriving them and explaining them, and this is pretty much like giving an instruction booklet to a beginner and ask him/her to use that booklet for everything.
Yes, there are situations when certain topics might be useful and important but also too complicated to actually derive at that point, but if something *can* be derived without too much trouble then it *should* be derived.
One rule of thumb is that students should be able to reprove those formulas to themselves with absolutely no extra help or reminders - everything should come from logical step-by-step reasonings.
As a new math teacher, I'll definitely be using this.
+alejandro cartes you can also tell your teacher that this is the same ancient method that Al-khwarismi used in around 820 ad in his compilation of algebra techniques. this geometric method has been around for millennia.
5:10 you could've explained this as when square rooting both sides you will go from (x+1)²=16, to x+1 = [+/-]4 because when square rooting, you get a negative and postive value. I just thought giving the length a negative value is quite confusing as that's quite impossible..
Definitely true. Ancient Babylon peoples, only positive, negative or complex solution, Value.
Virtual+real with i.
vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_b%E1%BA%ADc_hai
diendantoanhoc.org/topic/169871-gi%E1%BA%A3i-ph%C6%B0%C6%A1ng-tr%C3%ACnh-b%E1%BA%ADc-2-b%E1%BA%B1ng-ph%C6%B0%C6%A1ng-ph%C3%A1p-h%C3%ACnh-h%E1%BB%8Dc/
Understanding the basics is a critical part of learning; unfortunately, not many people know that. We need way more people like you that not only understand that statement, but also willing to teach it. Keep doing what you are doing!!!!
At the time (1973), I thought my 8th grade Advanced Math and my 9th grade Algebra teachers did a fabulous job of presenting the quadratic. I'm willing to concede that this is better. Very well done. 5 stars.
At least they showed you, my teachers just said dodged the question.
I really enjoyed this video. Years ago, I was in college, majoring in Mathematics, with the goal of becoming a math teacher. My ultimate goal was going to be the teacher who spotted the "hole" in a student's understanding of mathematics, transforming them from a math-phobic student to a math-master student (as was done for me by a wonderful teacher in high school.) Since the field of math is cumulative, knowledge wise, any area not completely understood by a student in the past, will ultimately become an abyss in the future.
Sadly, a TBI ended my progress in college, but with the help of RUclipsrs like you, I am re-discovering my love of mathematics. Who knows? Maybe I can tutor some students one day.
Thanks!
Frank De Mascio it's never too late to live a dream!
Frank De Mascio I love your comment
I hope you get the chance to tutor some kids, you sound like the ideal math teacher!
Frank De Mascio wow! You moved me to tears. Thanks for your inspirational comment!
Frank De Mascio I hate how teachers use acronyms and such. What they do is give you a shortcut without actually getting the deep understanding you need in the future.
I love how simple he makes it all look, by using a calm voice and great representations. And Desert Cactus you are right, this should be shown in every high school and middle school algebra class. I teach adult ed, and they need to understand algebra to get their HiSet or prepare for the test to go into college, and this is yet another excellent video from him.
I never liked mathematics when I was in my younger age because most of my teachers presented it arbitrarily and non-chalantly, devoid of passion and wonder. Now, in my late 30's, I'm hooked on it because of great math channels like you. Can you do a series on millenium prize problems? Also: Hilbert's problems, Continuum Hypothesis. Thank you!
I’m 14 rn and I’m watching this cuz I have to memorize the proof and write it on the board and I didn’t even know what completing the square is. This video made me realize it actually makes sense. I was taught in class you do this and get this.
Ok this is awesome. I've read on school books that the proof of the quadratic formula was too hard, but this is very simple and ties in with the concept of completing the squares. And you can even show it with pictures, for people that easily loses track of numbers.
Gosh why didn't the school teacher explain this theorem in this illustrative manner. God bless you Sir 🙋☺👍.
Because the school teacher doesn't have a strong math background; but you probably knew that.
I was going to ask the same question.
They gotta understand the concept that 'x' here doesn't refer to length but rather simple algebraic, since this is just a geometric illustration of quadratic formula to enlighten us
Coma White nn
They either don't know it themselves or they don't know how to teach. But, as far as my knowledge goes... Kids usually don't understand math because they find it boring as kids usually don't have to use math very often in a concentrated way and that requires a little thinking (outside of school grounds). So, whenever they teach this, you are a 10 year old kid with almost no knowledge of algebra. So, despite trying to understand the visual part, you miss the opportunity because you lost yourself in algebra. And there is that game kids are playing so.. You probably paid zero attention.
Why am I saying it? Because it is in curriculum and in almost every math book for kids in age ranging from 8 to 12. I, of course, can't guarantee this considering the USA. In my country at least, every kid is supposed to know this despite they don't: too poor to study (should be a motivation), no interest, no teachers.. But the books do cover it.
Wow, more than 30 years since I left high school and finally I understand what completing the square really means! You're an amazing teacher! Thank you so very much! You are a true champion 💯
I was struggling to teach some of my students the quadratic formula and I started thinking about it in similar terms such as this (minus the part about negative lengths). I knew some of my students might find this confusing. However, I took a chance and tried explaining the quadratic formula to my students in this way. For several of my students, I could see light bulbs going off--they got it! However, for many of my students, this only confused them further. I like your video. You explained it much better than I think I did (the colorful virtual manipulatives you used are extremely useful), so I think I'll be showing my students this in the future. I am a new math teacher and I try so hard to teach my students an intuitive way to think about math, like this for example. But sometimes, it seems that I only confuse them more. Also, if you've never been taught to think about math intuitively, then thinking intuitively requires a different orientation--kind of an undoing of just memorizing stuff to get answers. Keep up the great, great work!
Legend. Hope you had 7 good years since you commented this.
vi.wikipedia.org/wiki/Ph%C6%B0%C6%A1ng_tr%C3%ACnh_b%E1%BA%ADc_hai
Another piece of this is relevance to students' lives. Sure, "real world" applications utilizing the quadratic equation are everywhere, but in most cases students will counter with, "Yeah but I won't be using this in my own life, so why learn it?" For example: lobbing a basketball and modeling its motion with a quadratic might seem an ideal motivator, but how often do you see LeBron James out on the court with paper and pencil calculating the ball's trajectory? My own solution to this was to create a 2-D basketball simulator in the Scratch programming environment; players input a "best guess" angle and initial velocity and watched the results. This can springboard into a discussion of the math behind why the ball did or didn't go into the basket.
We need to really put our backs into the relevance aspect of math. It's important.
Fantastic video, thank you! Not sure I ever understood exactly what completing the square was until today.
Programming the quadratic formula into my HP-15C took me about 60 steps. My fellow engineering students were not impressed and showed me a brilliant "folded" version which was closer to 30 steps. Their version ran the same code twice, once for the positive root and the then again for the negative root.
I was humbled but at the same time wanted to make a contribution. They assured me that this code had been gone over with a fine tooth comb by dozens of engineering students over years and that I stood zero chance of saving even one single step. I nearly gave up because the code was really good but in the end I did save them a step.
Their code recalled a value from memory that was already present in the stack of the HP15C so that step was not required.
The value made it to the top of the stack but never got pushed out so it was still available for use without needing to be recalled from memory.
Wow. 'Completing the square'. Well, I'll be damned, that's exactly what we're doing.
thats what i was thinking
100th like. Completing the 10th square ;)
1 year since this comment.. Did it work?
Andrew Goering Proof of your statement?
Duh
You are a much much better teacher than the current average math teacher or any normal math book. I've been following your vids for quite a while now, and they really motivate me to learn and understand more about math. Thank you!
I can think of a lot of people who could've benefited from this means of demonstration; much more straightforward and powerful than just forcing them to memorize the quadratic formula.
We were taught this visual method to explain the formula in Algebra 2, but I don't think a lot of people made the connection or preferred to think in terms of numbers instead. I like this method the most.
I find that u just remember it and then it is impossible to forget it
+AngieMyst I knew about the algebraic derivation, but didn't connect it to anything geometrically. While geometry is much more intuitive in most cases, I think we can learn a lot from the ancient Greeks when it comes to this, as they connected nearly all mathematics to geometry.
+Cubik What kind of math teacher does that?
In school all of my math teachers always gave the full explanations of all formulas we learned.
We were not forced to remember the explanations as for the test only the formula was required.
Most of my classmates memorized only the formulas, but I assure you, we had always the complete explanations.
For me, memorizing a formula with no logic was a greater effort than remembering the (logic) way to get there. In a lot of cases because of poor memory, I did not remember formulas for the exam and I had to reconstruct them. That was easier for me that way.
I was always in the 3 best students of my class in math... never knowing the formulas, always able to reconstruct them.
So I guess in your class, with your teachers, I'd been in the 3 worst math student.
ydela1961 Congrats, I guess
The complete explanation for the completing square which all high form students of the secondary should learn deeply by heart and digest the logics and inference. Excellent!
I've never thought about understanding quadratic equation in this way.Thank you so much for making this video.
watch my useful mathematics videos.
This was truly brilliant. I learned to complete the square 54 years ago, but I never realized how it was derived. This should be presented in every high school algebra class. Thank you so much for this and your other videos.
that makes a whole lot more sense now. thank you
I did 'completing the square'in school over 15 years ago. I went on to do an engineering degree, and a degree in pure mathematics. I wish my maths teacher in school (who was a really good teacher) had explained completing the square and the quadratic formula this way!
Doc Daneeka
Many students wish the same thing. Often times, approaching a problem with a visual example like this can blow open doors for students who "just don't get it."
And often times it can just confuse them further..
@@docdaneeka3424 how do you get a degree in pure math without knowing this? XD
@@donlansdonlans3363 no, they knew how to complete the square, they just didn't know the proof behind it
7:47 if anyone's confused about what he skipped, here are the steps:
He multiplies the (-c)/a by 4a/4a (which is simply equal to 1, so it doesn't change the value) in order to get a common denominator (which is 4a^2, because a × 4a = 4a^2) so he can combine the fractions.
From there we have (-4ac)/(4a^2) + (b^2)/(4a^2), which we can simplify to (-4ac + b^2)/(4a^2)
Finally, we can rearrange the fraction to be (b^2 - 4ac)/(4a^2)
Hopefully that helps.
TL;DR: he just simplifies the two fractions into one with algebra.
The first time I saw the algebraic derivation of the quadratic I thought I was gonna cry, but this was even more sublime. Nice work keep it coming.
x=3
Ive realized that my whole life i havent been learning math, i’ve merely been memorizing it. It made no sense to me why i was learning it until i had it explained to me in an illustrative and comprehensive way, where now i understand why math is so important.
We often think that maths is just a subject and for many students it's the scariest subject to study but most of us don't know that maths is the language in which nature is written and it is alot more than what we just think of it.
MATHS IS NOT TAUGHT CORRECTLY.
I am giving you the biggest like I have ever given on RUclips!!! Thank you for enlightening one great frustration from an elementary school.
Petrhrabal
What this video clearly demonstrates, is what I've been saying for years and years. Generally, mathematics isn't difficult - well at this level. It's more the fact that you've been taught something that you commit to memory instead of understanding it. So, you see, it's not that you're bad at it - more like the education system and the way it's taught is useless.
Once you understand something, as opposed to just memorizing it, it makes problem solving using this knowledge much easier.
Paul G68 Actually, none of what you said demonstrates mathematics is not difficult. Yes, mathematics is taught very poorly, but so is every single subject in school. Also, homeschoolers generally agree that mathematics is still a more difficult subject than other subjects. So while education contributes to its difficulty, it is not the cause. Mathematics is difficult because it is by its very nature counterintuitive. Humans are not logical beings. Mathematics is based on understanding definitions, axioms, and using logical arguments and consistency, all things which humans are bad at. Humans have historically always used their emotions to think, never actual logic. Everyone has logically inconsistent beliefs for this reason. History does not behave like a formal logic system, and neither do arts or culture. Those are things we adapt to. Math behaves like a formal logic system because it is one by definition, and as such, it resides outside the first nature and second nature to humans.
For the record, geometric visualizations aren't any more intuitive than algebraic ones. Plenty of students struggle with those too. The only reason it looks like there aren't any is because students who struggle with them are always accomodated for by the education system, since curriculums are almost always purely algebraic visualization. But in reality, geometric visualizations don't make intuitive, it only makes it somewhat more accessible to students. Math is not intuitive. The vast majority of mathematics simply cannot be learned by visualizations like these, and the vast majority of mathematics isn't intuitive. These are an outlier and can be explained visually only because, in the grand scheme of things, it's one of the easiest subjects in mathematics. It's literally the second easiest thing after arithmetic. That's why they teach it after arithmetic.
Angel Mendez-Rivera : well you have your opinion and I disagree. Maths is taught in a way that makes it difficult. Want an example. Add up in your head, like you were taught as school these numbers. 1467+1667. Remember do it in your head.
What you’ll find is, you’ll struggle because of the way you were taught.
Now watch this video:
ruclips.net/video/hesKQ_y1P7k/видео.html
So, you see after you watch this video it’s much easier and intuitive. Theres the demonstration of what I was saying.
So once again, I respectfully disagree with what you are saying.
Once you build up a basic understanding of maths, it’s a case of building on top of that to further your learning. Nothing hard in that.
Wow. I have been searching the exact reason behind this quadratic formula to work.
I finally found it
I all ways knew that the -B formula (the formula you showed at the end) came from factorising/simplifying ax^2+bx+c but I never could do this by myself ,my school never thought us the visual method I find this extremely helpful thank you
8:47 Can someone explain this step please?
I figured it out. if anyone reads this in the future then here's the explanation:
To make this easier to type out in RUclips format I'm going to call the term b^2-4ac: "v1" and the term 4a^2: "v2".
1. We start by moving the b/2a term from the left side over to the right side:
x = - b/2a +- sqrt(v1/v2)
2. Since we have a fraction inside the sqrt on the right side we can simplify it according to the rule: sqrt(a/b) = sqrt(a)/sqrt(b):
x = - b/2a +- sqrt(v1)/sqrt(v2)
3. We now have two fractions on the right side. To combine the two fractions into one they have to have the same denominator. Luckily for us they already have because the sqrt(v2) (4a^2) is equal to 2a. The equation now looks like this:
x = - b/2a +- sqrt(v1)/2a
4. We can now combine the two fractions since they have the same denominator and we have the final equation (v1 is still equal to b^2-4ac):
x = (-b +- sqrt(v1)) / ( 2a )
W , you are on the same level as the guy who asks 'music?' and gets a reply, except that u do it yourself 🗿🍷
good explaine
This is such a clearer way of illustrating 'completing the square'! I never knew this is why it was called this, or, for the life of me, remember how to do it as I rarely used it. Thank you so much for this!
This is one of the best videos on youtube. I am in grade eight and had no problem keeping up due to your amazing visuals and your simple explanation of the concepts. I look forward to using the quadratic formula with the understanding of how it came to be!
Did you?
Did you ?
Did you؟
I learnt about quadratic formula in 1983 and only today I can visualize where the formula come from. Thanks for the enlightenment.
This was really terrific. You did such a good job bringing us along and the visuals were super clear. Thanks for doing this!
I’m a 38 year old physicist and have never seen this explanation before.
I will teach this to my children.
Thank you
Bro, your channel is going to help so many unsure geniuses to unleash their geniusity 🤣 Love you, keep the good work!
Try my channel mathfullyexplained
This is the best video on 'completing the square' on all of internet both graphically and algebraically.
I am so glad I watched the proof at the end. Holy Cow! NOW I understand the HOW and WHY.
same här dude, :))) I had this quadratic completing thing a years ago, now i am really understanding it
Joe Edwards Same Mind Blow
Sir I've studied quadratic equations today in my online classes but they told me to remember these 2 formulae so I remembered them and I follow ur channel randomly I found this video and understood how it comes thanx sir
Interesting and amazing. And I had to memorize deriving the quadratic formula the hard way. Thank you, sir.
Good as akways but i asleep
I like the video, as a teacher I would have just liked the part at 2:50 explained, "why do we need to use a certain trick?" the idea that they have a common side is great, but the shapes could be combined just by smushing them together to get x(x+2) on the left, but it would be good to explain why making each side a square will simplify the solving. A simple statement such as "If we make a square we could easily determine the side length of the square once we know it's area. Getting the side lengths of a rectangle given it's area could lead to many different answers." or something like that. Otherwise, very great video! I will use this in my class. Thank you.
I can't thank you enough! The visual representations make it so much clearer!
Presh Talwalkar - Very well done. I'm retired, and with teachers like you on RUclips, I can learn faster than greased lightning.
This video blew my mind. I always wanted to know where the formula comes from, and know I got it!
Love your videos :)
I have a degree in math and use the quadratic formula ALL THE TIME when tutoring in Algebra... I have NEVER visualized it like this but instead used a song tune to get it stuck in students heads (to the tune of a jack in the box). I was extremely happy to see this though and will DEFINITELY be using this from now on when I explain it!!!!
me: cool wow im understanding
presh: im gonna skip a couple steps
me: what no
shy bound literally me, lmao
Itbwas multiplication of divisions CD
@@laikaperraespacial6025 how does 2a * root 2a^2 become 2a again in the final formula? wouldn't the 2a become another 4a^2 under the radical meaning it would end up as a 4a^4 after it's square rooted in the final form? based off his explanation
@@whalestailz nop
The Khan video doesn't skip the steps ruclips.net/video/mDmRYfma9C0/видео.html
BROOOOOOOOOOOOO HELL YES!!! Finally understood the intuition behind it. You know how school is they just tell you to memorize it without even bothering to teach you where it comes from !!
why the fuck have i never encountered a teacher who has taught this?
You have. When you watched this video.
Omar Goodman haha that's a good point
I taught the derivation of the general quadratic formula 25 years ago to Intermediate Certificate students (c 15-year-olds) here in Ireland. Later, a revamp of the syllabus no longer required that students be able to *establish* the formula at IC Level, merely to be able to *apply* it in cases where the constants had specific values and the discriminant was > 0 or = 0 . We did not use the rectangles and squares method but rather completed the square by taking half the co-efficient of x , squaring it then adding this squared quantity to both LHS and RHS etc.etc.
We did it exactly as you (and PT) said at secondary school in England in the late 70s - but I had forgotten how to derive it, so I'm pleased to see this !
Because you have to teach students how to complete the square in order to do it and completing the square is only used for this and converting quadratic equations into the vertex form.
Consequently, you can save a lot of time to do other more useful things if you're not required to cover the vertex form.
This is such a huge help to me, I never understood how to complete the square until now, thank you.
Why there's no one like you to teach these mathematical concepts so easly at schools?
Wonderful geometric interpretation that brings satisfaction to those who wonder "why the hell I struggle learning with equation?" . This explation SHOULD be part of education to get interested in mathematics. Good job.
Just letting you know, once again, how much I love your videos. This was so well done. Empowering! I hope I get the chance to teach the quadratic equation to my grandchild using exactly this video!
Wonderful. No teacher or Prof ever explained this. Just memorized and used it. Thank you.
Awesome video I came not understanding a thing and I go away with an intuitive understanding.
I agree with this 😃
The name "algebra" comes from al-khwarizmi book. In that book, he shows quadratic solutions in that way. So that is the way it has been done since the beggining =P But mr Khwarizmi doesn't considered negative values, so he split the quadratics in 6 different types. I didn't like the way you does it. I'm gonna show that example for my students though. Sorry for bad english :)
Arley Antes wasn’t it Euclid in the Elements?
I'm honestly just sending this to my Maths teacher as something to show people in the future even if it isn't my class as we have moved on topic wise, but still as something to help others in the future
One square to another:
"You complete me."
The Dark Knight? :D (I’m from the future :P)
Lobotomy kaisen
Seems like something I would see in a CGP book
@@nilefield8910 You're not one of them.
This is absolutely how math should be taught. Understanding the logic eliminates the need for memorization.
he made me understand it in 10 mins what 2 years in my school couldn't
But if you don't understand it then you haven't learned it, which goes against the very idea of schools.
How about 30s ?
i'm a math teacher of junior high school teaching an honor's math class, presh makes my job easier with excellent videos like this, presh; i thank you; the math community thanks you and most of all my students thank you.
Can you similarly 'complete the cube' for the general cubic equation? And, although it's difficult, if not impossible to visualize, 'complete the tesseract' for the general quartic equation? Do any of these visualizations... those that are possible... provide any insight into why you can't solve the general quintic equation and higher with radicals?
I think the hard part there is that, once you get to 4th degree polynomials, visualizations kinda break down. Maybe someone could think of a clever way of still showing it intuitively, but at that point in kinda just comes down to proofs
Also, I think it's probably a mistake to ever be using the quartic equation... that thing is a mess
+Daisy Bell and to top it off, show geometrically why this trick fails with the 5th degree polynomial using a 5 dimensional 'cube'.
***** but you could still use it since it is possible to do math on objects of any dimension. The difference is that you use geometric methods instead of algebraic ones to prove a point, even if the geometry requires more than 3 dimensions.
***** I agree, it could still be used as an alternative to an algebraic proof but it will loose some of it's pedagogical value. Maybe it will be easier than the algebraic proof, maybe it will be harder. But it certainly won't be as easy as the n=3 or n=2 case.
Imagine completing the hypercube.
I learned this formula at school when I was a child. 30 years later I've understood where it come from. Thanks a lot.
OMG! I can't talk English very well, but I got it throughout the video and I could understand to solve that problems! Thank you very much... +1 subscriber
OH MY GOD MAN!!!! THAT. WAS. AWESOME. I think we should forget the normal quadratic formula explanation, and use this instead! It's much more -easier
-understandable
-and makes people like math!!
Congratulations, that was beautiful ❤️
This was an amazing video! For the first time, I really understand the real meaning of the quadratic formula. Thanks
x²+2x-15=0
x²+2x=15
Now we add a term that is given by the following formula: (b/2)² (2/2)²=1
By the way we need to have a=1, if not then divide both sides of the equation by "a" end then do this process.
x²+2x+1=16
If you did this correctly you will always be able to factor this out like this: (a+b)², in this case :(x+1)²
(x+1)²=16
Let's take the root on both sides:
✓(x+1)²=✓16
Now when we have this situation ✓y² , this is equal to |y|
To understand this let's see an example: If we have y=2 then
✓2²=✓4=2
And if we have y=-2
✓(-2)²=✓4=2
We always end up with the positive number, the absolute value of y, |y| . So ✓y²=|y|
|x+1|=4
This means x+1=4, or x+1=-4
Because |-4|=4
So we have x1=3 and x2=-5
ax²+bx+c=0
4a(ax²+bx+c)=4a.0
4a²x²+4abx+4ac=0
4a²x²+4abx+4ac+b²=b²
4a²x²+4abx+b²=b²-4ac
(2ax+b)²=b²-4ac
2ax+b=+-√b²-4ac
2ax=-b+-√b²-4ac
x=(-b+-√b²-4ac)/2a
This is how you can prove the formula
"I'm going to skip a couple of steps" - you have no idea how painful it is to hear that for someone trying to understand this. If you want people to really understand your subject NEVER skip any steps or assume people know what you are talking about.
True bro, that's the main reason I don't get anything my teacher says at school 😭😭
The steps he skipped arent too complex. He just simplified the expression.
He multiplies the (-c)/a by 4a/4a (which is simply equal to 1, so it doesn't change the value) in order to get a common denominator so he can combine the fractions.
From there we have (-4ac)/(4a^2) + (b^2)/(4a^2), which we can simply to (-4ac + b^2)/(4a^2)
Finally, we can rearrange the fraction to be (b^2 - 4ac)/(4a^2)
Hopefully that helps, but to put it simply in case you're still confused, he just simplifies the two fractions into one with algebra.
I liked the challenge and succeeded in the excercise of figuring out the steps. It is not painful but gratifying.
this kind of video may be helpful for those who interested in maths and always seek for the truth . textbook from school only gives us the equation straightforwardly but never show the proving steps so most of student could just understand how to use the equation to solve question but never understand how does the equation came from
So here is my Solution (an easy one):
First I added 15 on both sides, leaving me with the equation x^2+2x = 15
Then, I added +1 -1 on the left side, so it said
x^2+2x+1-1 = 15
Then I used the binomic formular (is that how you call it in english?)
(x^2 + 2×1×x +1^2)-1 = 15
--> (x+1)^2 -1 = 15
Then I added one on both sides
(x+1)^2 = 16
I took the root
x+1 = 4 or x+1 = -4
And finally, I subtracted 1, which lead me to
x=3 or x=-5
I think this is an easy solution, using basic mathematic formulars.
I'm sorry for my bad english, i'm from Germany (ze state where zey can't pronounce ze words correctly) 😉
Your English is actually pretty good. It's just that you spelt a word wrong. It's formula, not formular.
You're missing the point. He explained how to derive the quadratic formula. He wasn't explaining how to solve for x, which is what you did.
He is explaining why we are doing this.
Its *BINOMIAL FORMULA* btw..
Add 1 to each side. Factor left side as ( x+1)^2 then use square root property. Try my channel mathfullyexplained. Full unit on solving quadratic equations plus more units
Very clear and precise teaching. This is by far the best explanation of completing the square method to solve quadratics on YT. Thanks brother!! ❤
That's awesome to have a graphical representation of completing the square. I learnt the algebraic way to solve quadratic equation in my high school but never have thought this could be done using geometry.
I wonder if this method also works on solving cubic equation.
It does, i forgot how but i've seen it done
I thought of it this way. if you have an expression of the form a^2 + 2ab + b^2, it can be rewritten as (a+b)^2. so basically it all comes from forcing a quadratic equation into this form by adding a constant term on either side to 'fill in the gap' in the original quadratic expression so that it becomes a square number, thereby completing the square. the motivation to find the form (a+b)^2 is that if (a+b)^2 = k^2, then a+b is equal to both positive and negative k, thereby giving you the two solutions.
Impresionante!! Se siente tan bien entender de una forma más profunda a una ecuación que a primera vista parece bastante complicada, es tan simple que no entiendo porqué no lo enseñan así en el colegio, de verdad, todo es memorizar formulas resolver ejercicios y dar examen, pareciera que están fabricando robots. Gracias por esta explicación! Saludos desde Perú.
Translation: (Esta es una traducción del español al inglés)
Awesome! It feels so good to understand in a more profound way to an equation that at first glance seems quite complicated, it is so simple that I do not understand why they do not teach it thus in the school, really, it is all memorizing formulas to solve exercises and to give examination, it seems that they're making robots. Thank you for this explanation! Greetings from Peru.
Albanovaphi7 Eso es muy cierto, siento que el sistema educativo no está actualizado con las necesidades actuales de educación y crecimiento para los humanos. Algo tiene que cambiar Saludos desde América.
Más profundo no se traduce more profound, se traduce deeper. En do not teach it thus in the school, el thus sobra y es this, no it. En "examination", eso es incorrecto, lo correcto es test para referirte a un examen escolar.
Se que es muy tarde ya xD pero mejor tarde que nunca.
Históricamente, las escuelas existen para fabricar trabajadores humanos que operarían como si fueran robots. El que te dijo que uno va al colegio a aprender no sabe nada de historia.
Wow thanks, I don't recall that an instructor ever geometrically derived the quadratic formula in my algebra class. That is really genius.
4:55 the mystery length of -4
Negative quantities have physical meaning. Think negative displacement, negative velocity and acceleration. Going further, even "imaginary numbers" have real meaning in the physical world.
Imagine the trajectory of a cannonball. The quadratic formula tells you the positive answer, when it will hit the ground, and the negative answer, when and where in the past it was shot out of the cannon.
What is a box with a side of -4 ??
-4 is not wrong !
The use of the equation,
IS wrong !
As noted by others,
it could mean:
Before your
determined start time !
@@francistan4674 But this is completely different from what you've told. This one deals with negative length.
@@francistan4674 yes. but length or distance is a scalar quantity so cant be represented by negative numbers. this is why separation of algebra and geometry is important.
This is gold. I can't thank you enough. Now everything is clear and not boring
tbh, watching this make me realize how unreal or imaginative the negative numbers are. How do you even picture a negative number.
Just picture an overdrawn bank account :) .
Interestng. When I was at school 40 yrs ago we just learned the formula. Nobody explained where it came from. Thanks for this.
This makes me want to try the same thing with a 3rd degree polynomial to see how it turns out! ;)
Dont. I filled 80 pages of my A4 size exercise book doing that
you will need a 3d model for that. a 'cube' they say...
You will end up with what is called THE CUBIC FORMULA
Try it with a 5th degree polynomial
There is no general formula for roots of polynomials of degree higher than 4 (Galois' law)
I was just realizing that I have followed every explanation you’ve every given on your channel with a minimum of effort. That’s what makes you a great teacher, and that explains why I keep coming back.
this video was MOST EXCELLENT..!!!... I've never seen this kind of explanation for the Quadratic Equation....... it's amazing how the Visual of the Geometry can really Convey the logic of math proofs... THANK YOU!!...
kicsimoe there was a time we learned from books, not wiki! 😂 I've never had the privilege of seeing this done visually either.
Philip Y nice
बहुत अच्छा वीडियो है पूर्ण वर्ग बना कर हल करने का ज्यामितीय तरीका बहुत बहुत धन्यवाद आभार
Overall a fantastic video; what I appreciated most was the derivation of the quadratic formula using completing the square. I often wonder why students are asked to solve quadratic equations via completing the square when that process, as applied to the general quadratic form, yields the quadratic formula. In the past I've given a single, high point value extra credit question on exams: _show me how to get the quadratic formula._ I would tell students in advance that it might be on there. Students really liked those points of course (major damage repair if they'd had a rough time on the rest of the exam), and even if they forget about the process in the future, they'll never quite be able to say, "that danged formula was a total mystery."
I was taught this 45 years ago as completing the square without the geometric representation. It of course leads directly to the quadratic equation and that was why we were taught the method rather than simply using a formula with no idea how it came to be. Education used to be called teaching and teachers were effective. Now we have "educators" and all sorts of theory about teaching and no common sense. Oh dear, I'm getting old.
+984francis At the risk of igniting internet rage, I am going to propose that partly this is an unintended consequence to employment equality between men and women.
Once upon a time, women were very restricted in employment opportunities. For white collar work, it was pretty much: secretary, nurse, teacher.
Smart women probably went into those fields (rather than into unskilled work). Smart women are as smart as smart men. Thus you have women with doctorate-level intelligence doing these jobs. I suspect that we used to have hidden high-achiever women in admin assistant, nursing, and teaching positions that now have moved out to better paying positions. It is good for them, but bad for the rest of us.
as long as I am getting laid, I don't care about equality or inequality.
Mark M. That is a point I've never thought about. I grew up homeschooled and now work a job as a math tutor, so I get questions regarding my opinion of the public school system all the time, and why it now sucks. This makes sense as a contributing factor: the majority of teachers are actually just dumber than in times past. Thank you.
and your evidence for this is .....?
BTGTB Smarter teachers don’t equal better teachers though. I’ve been taught be people with PhDs and those with bachelors and the PhDs have never been very good. Because it’s easy to them, and they would rather be at a university.
I have a maths exam tomorrow and now I finally get everything! Thanks Mate!
This explanation would have been nice to have 15+ years ago back in Grade 8. I was just told the formula, and that was it.
@Fester Blats Only if he's a sheep like you. People go to school to understand, not just use things without knowing anything about them
You probably still wouldn't understand it at that age and level even if it was explained geometrically or non geometrically
@Fester Blats I do find that the beauty of maths isn't the solving part, but it is understanding the logic behind every step. Appreciating logic is found in maths, and this same appreciation of logic can apply to other aspects of life beyond maths as well.
For example, if you have a strong foundation and appreciation of understanding mathematical proofs,
Surely, you wouldn't be as gullible to propaganda and conspiracy theories in real life as you would be meticulous towards the proofs and evidences behind them, skills you've honed through scrutinizing mathematical proofs.
The fact that this type of explanation was missing from my education is exactly why I gave up on math. But now that I know there are answers out there I am finding my way
Wow, that's something our teacher never said to us :) Time to impress him :D
I'm one year late but still came back to watch this. Quadratic equations is a concept that keeps coming back in all you acadmic years.
Thanks lot now I understand much more about it and how it actually works! Thanks love it!
It's just amazing, 27 years after I've learned this I discover this video, just great job. Thank you!
So the quadratic formula is just compressing all the steps of completing the square into one equation.
That was fantastic! Extremely visual and clear to understand!
In our school:
They give the equation and say solve a few sums using it fast, so that we can complete syllabus for EXAMS.
my videos help your school works.
tatakau heroooooooooo!
You explained the quadratic formula so well it should be the standard of teaching.
Wow. I swear to god I would NEVER expect to see Bhaskara being formulated at the end. Initially, it seems such a surreal formula, but when you look at step-by-step you realize why it makes so much sense.
When I was a young student, my Father’s boss would quote the quadratic to me, I was very unimpressed. With the benefit of 56 years experience, I am now impressed. But, thanks to your vid, I now understand the term completing the square, and will remember this.
you should do it for the cubic and quartic equations formula
Hour and then a day long video, respectively.
lol
And then explain the proof as to why there is no analytic formula for degree 5 or more.
xapoliz vieira "complete the cube" and "complete the tesseract"? :P
x^3-6x^2=0
x^3-6x^2+12x-8=12x-8
(x-2)^3=12x-8
You have to add a x-term in the right side and because that it does not "works like a magic" with cubes.
Amazing explanation! Thank you! It's better to use "completing the square" than using "quadratic formula" in an "automatic way". Your superb explanation should be teached in school. :-)
The video was very intuitive. But can you please explain the geometrical meaning of negative value of x
Pranav Kumar S.
I think of X times X as adding
X number of X spaces
I think of -X times -X
as removing X spaces of -X.
basically like if there was some object taking up space, you are removing this "negative space".
in other words multiplying positive numbers is like adding another room to you house, while multiplying negatives is like clearing out a full room so you have more space. either way, both give you more space.
There is no geometrical meaning to having negative side lengths. There is a reason negative numbers are almost never used in geometry.
That is the best demostration of the quadratic formula I've ever seen.