A new way to make quadratic equations easy | Hacker News Reader

A new way to make quadratic equations easy

technologyreview.com

177 pointsuniqueid94 comments

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Let's get the criticisms of the article out of the way:

--It has terrible formatting and typos.

--It puffs up something more important than it is.

--It is more about pedagogy than a mathematical idea.

--It suggests the idea is original, when it almost certainly is not.

--It doesn't link to the original (and better source).

Okay, here is the good things about the approach:

--It is good to shift your thinking about mathematical derivations and proofs and think about them as code that runs on people's brains. You input a derivation into someone's brain and they return a boolean value (this is true, it makes sense, etc.). Pedagogy is trying to optimize the code for less powerful architectures. Just like when you are optimizing code tiny little details of instruction orders matter, the same with mathematical derivations.

--Fundamentally, algebraic manipulations are uncomfortable and nonintuitive for students. They feel like tricks. Going forwards from (x+a)^2=x^2+2ax+a^2 makes sense but going backwards as in the case of completing the square is hard. It's not the same case for x^2=a vs sqrt(x)=a. This is kind of a similar case to math students feeling confused by adding and subtracting the same quantity when doing calculus limits. For any trained mathematician, this is obvious, but it really feels like a trick at first. The nice thing about this approach is that it avoids this issue and gives you a good reason WHY the -b/2 term shows up. Additionally, it avoids the problem of substituting, which tends to bog students down (try teaching the chain rule someday).

Students should still understand completing the square but I don't think this is a bad way to introduce them to the quadratic formula. It highlights the symmetric of the roots (at least for real values), which makes sense if you plot a quadratic.

I agree. This is nothing novel and a fairly standard way of solving the equation.

I have never understood "completing the square". I mean, I get why and how it works, but it's completely unintuitive to me, or why you'd do it that way.

I think it is a historical hangover. It makes more sense viewed geometrically, and the technique is attributed to the same al-Khwarizmi for whom algorithms are named. But it's less intuitive as part of algebra, and we focus a lot more on algebra today than in the medieval times, when geometry ruled.

Algebraically, my intuition is that the x^2 + bx part of x^2 + bx + c = 0 "looks" pretty close to the expansion of (x+b)^2. If you relabel b so this becomes x^2 + 2b, it's even clearer. So you can try to fiddle with the constant term to make the LHS exactly that square, and you get completing the square.

This is similar to how you solve a first order linear DE, y' + f(x) y = g(x). The idea is that the LHS "looks" like the derivative of a product, (hy)' = h y' + h' y, so you fiddle with an integration factor to make the LHS exactly that derivative.

It's not intuitive for someone new to mathematics, but as someone with two math degrees, it is just the strategy of "do stuff to something until something about it looks similar to something you've seen before" that mathematicians use ALL the time. The additional part of this strategy is to dream optimistically about how you can make the thing you are dealing with "nice" after doing stuff to it. In the case of completing the square, you are hoping you can just straight up take a square root. That would be easy. Turns out if you try enough stuff, you can.

Once you get to quadratic forms you can use it again.

This is bizarre. It’s no less a mathematical trick than completing the square, and it doesn’t seem to be any simpler to use.

The example given is to find the roots of x² - 2x + 4 = 0.

Completing the square gives (x - 1)² + 3 = 0, from which you can immediately see that the roots are 1 ± √3 i. If anything this seems easier than the method of the article.

Am I missing something?

Added: The argument seems to be that young students will find this method easier to understand than completing the square. I have no experience of teaching mathematics to children, so this may be true for all I know. It would be interesting to test this hypothesis experimentally, because I don’t think it’s obviously true.

Was going to reply precisely this. Completing the square is, I believe, already taught in standard curricula (at least, it was when I was in grade school), and is a trivial way to derive the quadratic formula.

You are correct. It is generally expected to be part of an Algebra I curriculum (as per Common Core Appendix A's Traditional Pathway).

> Common Core High School: Algebra » Reasoning with Equations & Inequalities » Solve equations and inequalities in one variable. » 4 » a

> Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)^2 = q that has the same solutions. Derive the quadratic formula from this form.

http://www.corestandards.org/Math/Content/HSA/REI/B/4/a/

Depends. I took a mid-level calculus class recently because it had been awhile since I'd integrated anything, and the professor had to spend a day reviewing how to factor and divide polynomials when they got to integrals which used trig identities. "Completing the square" was a new phrase to most students.

Everyone knew the quadratic formula, but most people don't pay attention to how formulas are derived, much less remember that sort of thing a few years later. If you want to guess whether an average student knows something, ask yourself, "would it be on the test?"

Programming is full of these. Every day I see some new technique or style appear. Is it any good? We've got easier access to the biggest collection of free source code, the biggest and most diverse set of programmers, and the fastest automated testing of all time. Just do an experiment. You don't need to sell me on how cool it is. Show me how it's better.

- Does it lead to smaller programs? Don't show me a 10-line example. (I've heard medical researchers say: "Anyone can cure cancer in mice.") Convert a 10,000- or 100,000-line program.

- Does it lead to humans writing fewer bugs? Have a bunch of them try it, and measure their speed and defect rates. Everyone has designed a system that they themselves love, and nobody else can understand.

If you promote a new method which is logically equivalent to old ones, and not obviously much better, and without any data showing specific metrics that it improves, I'm forced to assume your metric is simply "I happen to like it better". Experimentation is cool, but it doesn't cause lasting long-term shifts.

I’ve taught elementary algebra for many years at a community college. The method described in the article is truly, in my opinion, very nice. I will be using this method from now on in elementary algebra. Students in elementary algebra don’t know what parabolas are. They haven’t been taught to complete the square and don’t know how to work with radicals. They have just been taught what square root is and basic binomial and trinomial factoring.

To me it is obvious that the method in the article is far superior than teaching completing the square. I’m teaching a pre-calculus course this semester and many of my students still can’t complete the square. Pre-calculus is 3 math courses beyond elementary algebra.

All of this is just my opinion of course and I have no data or studies to back up my opinion. I will be using the method described in the article from now on in my elementary algebra courses.

> To me it is obvious that the method in the article is far superior than teaching completing the square.

I disagree. I would need some convincing that "two numbers that multiply to C and sum to B must have an average of B/2, so they must be B/2 + z and B/2 - z, so (B/2 + z)(B/2 - z) = C" is by any means obviously superior to completing the square. Neither is immediately intuitive; both will require prompting and teaching by the teacher. Completing the square has uses beyond proving the quadratic theorem; this does not.

I should say: I find this an incredibly cool and level-appropriate proof of the quadratic equation, but I think its merits as an improvement in pedagogy are dubious.

I doubt I can convince you. I’m just going by my experience teaching the topic. At the time students first learn solving such equations they have just been taught factoring and what it means to factor a trinomial. They know the product of the constant terms in the binomials must be c. It’s also easy to explain that the average of two numbers is the midpoint. And thus if I start with the midpoint then to get to the numbers I took the average of I add and then subtract some number from the midpoint. The geometry makes this easier to explain over using completing the square.

I’ve seen a shocking number of calculus students struggle with completing the square. The merits of the approach in the article are entirely obvious to me but like everyone else I’ve had my share of obvious beliefs turn out to be false.

>but like everyone else I’ve had my share of obvious beliefs turn out to be false.

Refreshing candor! Wish it held true that more people saw it through to discover their obvious truths didn't hold up.

It's phrased in a funny way, but this: ""two numbers that multiply to C and sum to B must have an average of B/2, so they must be B/2 + z and B/2 - z" is pretty obvious.

If x+y=B then the average(x+y) = (x+y)/2 = B/2

B/2 is then the number in between x and y so you can represent x and y as B/2 + z and B/2 - z (where z is just half the distance between x and y, or |x-y|/2)

Apologies in advance for the harshness of my tone, but I nearly spit up my coffee...

> Students in elementary algebra don’t know what parabolas are.

The mind boggles. Have you never shown them a sliced cone? ( https://en.wikipedia.org/wiki/Conic_section )

I mean, they know what parabolas are: they live on a planet, with gravity. Every ball, every jump: parabolic motion, yeah?

"Michael Jordan exerts his muscular power to enter a low-altitude earth orbit..." ~some Nike commercial in the 90's.

They know what parabolas are, you just have to connect the dots.

All of the world’s best scientists didn’t know that projectiles moved on parabolic paths until Galileo’s experiments on inclined planes in the 17th century. 21st century schoolchildren who haven’t been taught about it likely don’t either.

Here’s Tartaglia’s picture from the 16th century, drawn from his experimental research: https://www.maa.org/sites/default/files/images/upload_librar...

And here’s another similar diagram tossed up by image search: http://ej.iop.org/images/0143-0807/33/1/149/Full/ejp405251f1...

If you look for even earlier diagrams they show a projectile moving in a straight line for some distance and then suddenly dropping straight down.

Cf. https://en.wikipedia.org/wiki/Aristotelian_physics

Maybe Tartaglia just didn’t like quadratics. He did, after all, find the cubic formula. ;)

I deduce from your post that you have very little experience in teaching people at the level of beginning algebra. And while one might know geometrically what a parabola is there is a lot one must know before dealing with parabolas algebraically. I suggest that these things appear easy and obvious because you already know them and that you no longer remember what is hard for people learning this stuff for the first time.

I don’t show conic sections in elementary algebra. One typically really mentions the phrase “conic section” is pre-calculus which is 3 courses after elementary algebra. Over the past few centuries the order in which concepts are introduced has been developed. It’s not perfect but one should not be so quick to discount the way things are done without knowledge/experience in presenting these ideas to beginning students.

You've got me. I wish I could delete the comment, or at least edit it.

It's a bit worse than just having no experience teaching algebra, I didn't have the same experience as most kids trying to learn it. I was kind of a freak. I was the weird quiet kid in the back of the class who always knew the answer to every question. (Other kids tended to not like that, but I'm also very disarming (in person) and so I did alright.) One year, I was misplaced into a basic geometry class, and the teacher very kindly let me pick out some calculus textbooks and sit in the back of the class teaching myself calculus. (Some bureaucratic reason for why I couldn't transfer, or the calc classes were full, or we didn't have any, or something, I forget.) Learning math for me feels like remembering things I always knew.

So, yeah, maybe I should keep my mouth shut when it comes to teaching normal people how to do math.

Or maybe we should try to figure out what my brain is doing and how to teach people to do that too?

Maybe I have a normal brain and I'm just using it differently than most people?

I like that idea better, because then, instead of a freak, I'm a front-runner. And, in addition, there's hope for a great improvement in didactic technique and humanity's general "numeracy" level, eh? If we could teach people to math like i do we could compress basic math education (up to calculus) into just a year or so.

"Something self-referential and hopefully unsnarky about how this is a thread on a how-to-math-better article." ~me, failing at rhetoric.

And for kicks, here's Iconic Math: http://iconicmath.com/ (No affiliation with me. I'm just trying to end on an upbeat, constructive note.)

I think a big part is that many (most?) children find school – i.e. lectures, textbooks, homework exercises, exams – unmotivating/boring at best, and often extremely stressful/frightening, which means that they aren’t fully focused on it and can easily miss important details. Beyond that, schools often fail to provide meaningful feedback or support when people suffer serious misconceptions or are missing fundamental prerequisite knowledge/skills, which makes it easy for students to fall behind and have great difficulty recovering. Someone who spends the exact same amount of time on academic work but for whatever reason (external help outside school, internal motivation, some insightful introspection, ...) manages to pay closer attention, stay more relaxed, think about things ahead of where the class is expected to be, connect new learning to material learned before and build a better-connected mental map, etc. can end up pulling far ahead.

A whole lot of this has to do with level of preparation before ever arriving at school. Some kids read with their parents for hours every day from age 1–5+, learn to play a variety of strategic games (and games involving basic arithmetic practice), build structures or mechanisms or electronics, practice making art, work through books of logic puzzles, etc. Other kids are left alone and bored without learning materials at a reasonable level, plonked down in front of developmentally inappropriate or just badly produced TV, or handed over to unthoughtful video games.

Then consider how many kids and parents have serious problems at home, with confrontational or even abusive relationships. Pile on work stress, financial stress, poor diet or even hunger, poor sleep, environmental toxins, illness, etc.

I couldn't agree more.

I’m surprised that these students learn to solve quadratic equations before they learn what parabolas are.

If I were teaching quadratics, I would probably start with squares and square roots, and I would draw pictures. I would present some motivating examples (from kinematics, maybe? just the pictures may be okay, especially if the fun game where you try to zap the targets by hitting them with graphs is still around.). Then I would teach translations. After that comes the distributive law and polynomials written ax^2 + bx + c.

And now you can solve them! As far as I’m concerned, solving equations (polynomials, systems of equations, integrals, ODEs, PDEs, etc) is a puzzle, and learning a bag of tricks to solve them is just that: learning a bag of tricks. Quadratics are nice because the tricks always work. In more complicated math, it’s important to understand that the tricks can be very hard or even probably nonexistent, and accepting that is important.

But I don’t see why we should teach people to solve quadratics before teaching what they are.

edit: it’s not clear to me that the method in the article is dramatically different from completing the square. Assume a=1 for simplicity. Given the knowledge the the average of the roots is -b/2 (which one can deduce by any number of means), you can solve the equation in quite a few ways. One is the way in the article. Another is to say “the average is -b/2, so the vertex of the parabola is at x=-b/2, so the polynomial can be written (x+b/2)^2 + something”. Another is to just write down the solution x = -b/2 ± something and solve for “something” (which is more or less the same thing as in the article).

In high school, I used to have fun solving quadratics in my head by seeing which technique gave a quick answer.

I believe the objection is that deriving the formula itself from completing the square is the challenge, not the direct use of completing the square as you show in your example. (Although I wonder where you learned that, because I was certainly never taught that so directly.) The problem with deriving the quadratic equation that way is that for most people, that is a lot of symbols to keep track of, and you have to not only do an unintuitive "completing the square" step but you have to unintuitively do it fully generically.

For a professional mathematician this is barely a warmup, of course, but for the average person in middle school or early high school, this is distinctly nontrivial work.

(I'm not taking a side here, just trying to describe what the issue is, since you asked. I'm ambivalent. Neither of my kids are quite this far yet, so I'm not quite here yet.)

> The problem with deriving the quadratic equation that way is that for most people, that is a lot of symbols to keep track of, and you have to not only do an unintuitive "completing the square" step but you have to unintuitively do it fully generically.

"Unintuitive" depends entirely on your introduction to the topic. If you're already completing the square, using it to solve quadractic equations you cannot factor is not unintuitive.

As far as doing it fully generically, well, how else do you get a generic formula? When teaching this, we would do some completing the square to solve quadractics, and then tell our students:

"You know, this is annoying to have to complete the square EVERY TIME. What if we just decided to solve it the really hard way once, with A, B, and C in the equation instead of the numbers, and see what we get?"

I think you're speaking from the perspective of someone very casually comfortable with symbol manipulation. This does not describe the average middle school student. When I said "unintuitive", I was speaking from the perspective of an average middle school student, for whom this is all either at the edge of the ability, or, often, a bit past it, and for a non-trivial number of them, way past it.

In high school, I was a tutor for a non-accelerated, non-honors class that was about at this level in high school. After years of being in the accelerated course, it was a bit of an eye-opening experience. There's a lot of people who are just passed through this stuff with a C-, and I'm not even sure that's wrong, because there's a lot of people who just aren't ever going to get to the point where they can fluidly derive any of these equations. What you, and probably a great deal of the HN commetariat experience as "average" is actually way above average.

(And the students I was tutoring for, in the parlance of the day, would still mostly be considered "privileged". I would still not be calibrated for the mathematical skill of the truly disadvantaged.)

I teach mathematics and write math curriculum professionally, and have literally taught the lesson I describe above for the better part of a decade. I say "we" in that post because in more than half of those classes, I did not teach it alone, but in a co-taught inclusion class for special education and general education students together. None of the classes where I used this lesson was an "honors"/"accelerated"/"pre-AP"/etc. class.

Every single one of those students was able to derive the quadratic equation by completing the square. It was not easy for some of them, but every single one did it.

Your parenthetical also implies to me that you think that the "truly disadvantaged" have less "mathematical skill". I would encourage you to reflect on that.

I thought the article was trying to compare to the "typical" fractional formula, not to completing the squares?

The old (Baudhayana, Omar Khayyam) geometric approach to completing the square (and so solving [some] quadratics) is very nice because it's visual, and it's what I show students in many classes. It explains why it's "completing the square".

Suppose you want to complete the square for x^2 + 6 x. Represent this as an x-by-x square and a 6-by-x rectangle:

       x
     .....
  x  .....
     .....
     *****
     *****
     *****
  6  *****
     *****
     *****

Cut the 6 x rectangle into two 3-by-x rectangles:

       x
     .....
  x  .....
     .....

     *****
  3  *****
     *****
  
     *****
  3  *****
     *****

Move the lower 3-by-x rectangle up next to the square. The L-shaped figure still has area x^2 + 6 x.

       x      3
     .....  *****
  x  .....  *****
     .....  *****
  
     *****
  3  *****
     *****

What do you need to add (what is the size of the small square on the lower right) to complete the (large) square? The small square is 3-by-3, so it has area 9:

       x      3
     .....  *****
  x  .....  *****
     .....  *****
  
     *****  +---+
  3  *****  |   |
     *****  +---+

You get x^2 + 6 x + 9 = (x + 3)^2. If the original x^2 + 6 x was on one side of an equation, you add 9 to both sides.

This kind of approach is familiar to me from competition math. Back in middle school math club we were taught this exact approach (under the name of "Vieta's formulas"), i.e. that thinking about the sum and product of the roots could be faster in some cases. Po-Shen Loh is the director of the US IMO team, so it makes sense he would like this approach.

However, I don't think it makes logical sense to teach it only this way. Here you start by assuming that a quadratic has two roots, which is not at all obvious the first time a kid sees a quadratic equation. (Especially because those roots can be complex numbers!) Completing the square tells you why there are two roots, and also naturally leads you to the necessity of complex numbers, i.e. when the "square" you end up making is negative. You can use the nice Vieta's formula tricks only after establishing that.

I am also familiar with Vieta from math competitions but I think this is more than that.

Given two numbers, r1 and r2, if you know their arithmetic mean:

  m = (r1 + r2) / 2

and geometric mean:

  g = sqrt(r1 * r2)

then following Loh's derivation you get a very cute formula:

  r1 = m - sqrt(m^2 - g^2)

  r2 = m + sqrt(m^2 - g^2)

Vieta gives an easy way of finding those means from a quadratic equation: r1+r2=-b and r1*r2=c. So you can plug in m=-b/2 and g=sqrt(c) in the equation above.

The fact that you can state roots in terms of their means is the more novel insight to me. (note: Loh doesn't talk about geometric means but I thought using just product of the roots isn't as "symmetric")

People are criticizing this because it is still the same quadratic formula. But of course it is! Math is consistent.

But representation matters. A good chunk of mathematics is just about rewriting the same mathematical fact in a different way. For example the equation of a line could be written with coefficients or in slope/intercept form or in polar coordinates or in homogeneous coordinates or etc etc.

Here the claim is that explicitly giving a name to the variable -b/2 makes the equation easier to think about. I see nothing wrong with that.

Full disclosure: I critized the article in a comment below...

Respectfully, that's not the reason people are critiquing the article.

I fully agree mathematics is what works and many methods use identical underpinning logic, just expressed in different ways. I'm fine with that.

But that doesn't mean all methods are equally good. This method is no quicker or easier or less error prown than the quadratic formula it "replaces". Even in the authors chosen example, it's no better. In many other cases it's harder (if B or C are not divisible by A, dividing by A to force A=1 just spreads and increases the complexity).

That makes it a bad method because now, a user has to not only know both methods but also pick the right one. And for this extra time and risk, the gain nothing the standard Quadratic Formula didn't give them.

We could equally "simplify" the quadratic formula by forcing B=1 or C=1. Are those methods new and useful? No. They're trivial and have limited use cases. They're never better than just using the full formula.

My issues with the article are a bit wider: this is not new. I was taught this as a limited version of quadratics in 2000 in a run of the mill school in London. I also think it's derivative. Anyone smart enough to be solving quadratics should also be smart enough to apply basic algebra to simplify quadratics. But the article presents this, assuming the audience knows no better, like it's a breakthrough. That feels dishonest to me...

I think it's useful to think of this as refactoring code to make it more readable (and therefore more teachable).

If you're a memorizer, your code might as well be obfuscated code:

    def quadratic_formula(a, b, c):
        return [
            (-b - math.sqrt(b ** 2 - 4 * a * c)) / (2 * a),
            (-b + math.sqrt(b ** 2 - 4 * a * c)) / (2 * a)
        ]

If you understand the importance of the expression under the square root (i.e. the sign of the discriminant) you can rename one subexpression:

    def quadratic_formula(a, b, c):
        discriminant = b ** 2 - 4 * a * c
        return [
            (-b - math.sqrt(discriminant)) / (2 * a),
            (-b + math.sqrt(discriminant)) / (2 * a)
        ]

Of course you can refactor it further with pointless stuff like denom = 2a, but that doesn't add much semantic value. So the above is more or less the vocabulary we have about quadratic equations today.

Loh's contribution is a specific way of refactoring the code by first dividing by a:

    def quadratic_formula(a, b, c):
        b = b / a
        c = c / a
        discriminant = b ** 2 - 4 * c
        return [
            -b / 2 - math.sqrt(discriminant) / 2,
            -b / 2 + math.sqrt(discriminant) / 2
        ]

Which then unlocks the ability to talk about the subexpressions in relation to the roots (a la Vieta's formula):

    def quadratic_formula(a, b, c):
        b = b / a
        c = c / a
        sumOfRoots = -b
        productOfRoots = c
        averageRoot = sumOfRoots / 2

        # Want roots [averageRoot - delta, averageRoot + delta]
        # such that:
        #   productOfRoots == (averageRoot - delta) * (averageRoot + delta)
        #                  == averageRoot ** 2 - delta ** 2
        delta = math.sqrt(averageRoot ** 2 - productOfRoots)

        return [
            averageRoot - delta,
            averageRoot + delta
        ]

Your code is no longer using single-letter variable names!

> this is not new

While the mathematician claims to not find historical evidence of this, it is suuuuper similar to something we went over in high school in Calculus. It was related to finding the vertex of a parabola and noting the roots will be equally distant to both sides of the mid point. At the time, it was used as a "see? Neat. It all works out" type lesson.

This was around 2001.

*error prone

Perhaps people are skeptical about the claim of originality for a straightforward algebraic manipulation that can be carried out and understood by middle school students. (As far as I can see, that is what the article claims.) Bright people "reinvent the wheel" all the time. The rediscovery of the trapezoidal rule for numerical integration is a recent popular example.

Also, unrelated and probably unfair ... maybe a few cynical and overly skeptical people become a little cautious when they see a link to Tech Review.

There is an error in the (pixelated) example that is given int the MIT tech review article (z² = 3 instead of -3). I think the fact that the author of the news missed this, that he probably took a screenshot of the formulas rewritten in Word, and that he was compelled to write such a long article on such a simple topic speaks for his level on the topic.

The fact that he only lists the formal article as a reference instead of the announcement, video, and accessible blog post by Po-Shen Loh really baffles me.

The original "disclosure" by Po-Shen Loh [0] is much less sensational and gives some context for his work (teaching middle school students). In the formal article, he is also stating that the method is very likely not __new__, but that he wants to popularize it in teaching.

I think, as many other commenters pointed out, that there is no great breakthrough here. However "his" method may have the advantage of training the intuition of young students, by helping them understand the concepts of average and "deviation" (I'm not really sure how to call it in that case), and maybe visualizing them.

[0]: https://www.poshenloh.com/quadratic

And the third pixelated equation has an error - the final term on the left side should be C, not Cx^2.

I think the point of the arXiv article is this is a more straightforward /proof/ of the quadratic formula. That article has some interesting historical commentary that shows exactly where the author thinks his contribution fits - he is not naively coming up with something "new".

As far as computation of the roots goes, it is a slightly streamlined approach:

First, put the quadratic into canonical form by dividing by A, if necessary.

Then, take B/2 into a new variable, call it F. Get F^2.

The roots are then -F +- sqrt(F^2 - C)

Why is MIT technology review on the front page so often. It's literally a student newspaper with the expected quality of one.

Are you perhaps thinking of https://thetech.com/ ? I'm not sure in what sense the Technology Review is a student newspaper - to my knowledge (and checking a few bylines), the journalists are not students.

Oh wow I just read their about page and I stand corrected. I'm not sure if the fact that they're a (supposedly) hundred year old newspaper makes me feel any better about the quality of their articles though.

This seems more complicated and roundabout than completing the square. The average of the roots shows up that way too:

We want to write x^2 + bx + c = 0 in the form (x+m)^2 + n = 0, so there's only one x left and we can rearrange for it.

Expanding, (x+m)^2 = x^2 + 2mx + m^2, so we get the x^2 we want, and the coefficients of x tell us b = 2m, so m = b/2. We also get an m^2 (= b^2/4) we don't want, so let's take it away:

(x + b/2)^2 - b^2/4 = x^2 + bx

That x + b/2 is x - (-b/2), x minus the average of the roots, which is the x value the parabola is centred on. Then we add c:

(x + b/2)^2 - b^2/4 + c = x^2 + bx + c

To find the roots, set it to 0 and rearrange for the one x that's left:

(x + b/2)^2 - b^2/4 + c = 0

(x + b/2)^2 = b^2/4 - c

x + b/2 = ±√(b^2/4 - c)

x = -b/2 ± √(b^2/4 - c)

Note that this is the average of the roots ± the article's z. Then combine:

x = -b/2 ± √((b^2-4c)/4)

x = -b/2 ± √(b^2-4c)/2

x = (-b ± √(b^2-4c))/2

If you have ax^2+bx+c = 0, divide the equation by a first, then do the same steps and you get the normal quadratic formula:

x = (-b ± √(b^2-4ac))/(2a)

I think the linked post misstates the purpose of the article: it's not about new maths, but about pedagogy and ways of explaining the quadratic formula.

Having taught elementary algebra for many years at a community college I think your perception of what is complicated and easy are not correct. The method in the article is far easier for elementary algebra students than what you describe.

You’ve proved the quadratic formula. This is a formula students in elementary algebra will struggle with. It’s a formula whose proof will be lost on them. What is shown in the article is an easy to apply mechanism for finding the roots. The method in the article is one that I can use in the classroom. There’s no way I’d attempt your explanation in an elementary algebra class.

I wouldn't try to teach it to elementary algebra students either; I was explaining for maths-weak HN readers. It would be worth trying with some of my first year university students (but definitely not all). What's shown in the article is also a derivation of the quadratic formula. I'm not convinced the less general solving method at the end of the article is any easier than completing the square, and it's certainly more mystifying, as completing the square can be checked by expanding, but this is a bit out of nowhere. I will try the article's method if I get an opportunity, but I think the introduction of an extra variable z is quite an obstacle, and that the fact that -b/2 is the average of the roots will be quite a leap for most students.

I have trouble understanding why this deserves a paper. This is what I learned in middle school back in Vietnam (although I did take advanced Math).

For equation Ax^2 + Bx^2 + C = 0, the roots are:

x1 = (-B + sqrt(B^2 - 4AC))/2A

x2 = (-B - sqrt(B^2 - 4AC))/2A

Now the author looks at a special case where A = 1. The equation becomes x^2 + Bx^2 + C = 0. Of course the roots simply become:

x1 = -B/2 + sqrt(B^2 - 4C)/2 = -B/2 + sqrt((B^2)/4 - C)

x2 = -B/2 - sqrt((B^2)/4 - C)

> "The author would actually be very surprised if this approach has entirely eluded human discovery until the present day, given the 4,000 years of history on this topic, and the billions of people who have encountered the formula and its proof. Yet this technique is certainly not widely taught or known (the author could find no evidence of it in English sources)"

I certainly don't think it has eluded humany discovery until the present day. It's known to middle school students in Asia that multiplying the original equation so that A == 1 would greatly simplify the roots formula.

> I have trouble understanding why this deserves a paper. This is what I learned in middle school back in Vietnam (although I did take advanced Math).

> [...]

> It's known to middle school students in Asia that multiplying the original equation so that A == 1 would greatly simplify the roots formula.

I think the same having attended a regular grammar school in Germany.

The a = 1 is not the important bit - that's probably nearly universal.

Yeah, half the elegance of the linked expressions just comes from setting A = 1.

This is definitely well-known. I learned this under the name of "Vieta's formulas" back in middle school. Vieta lived in the 1500s.

Isn't this the same as the pq-formula? https://www.matteboken.se/lektioner/matte-2/andragradsekvati... (Link in Swedish).

Yes, and apparently it's also used in Indian schools (see comments found here: https://www.facebook.com/poshenloh ).

It would be interesting to have math historian compare this with other derivations of quadratic roots to assess its originality.

My favourite bit of knowledge about quadratic equations is that its roots can always be visualised as the intersection between a simple parabola (x^2) and a straight line (m*x + c).

In fact, the above is why imaginary numbers did not arise from needing to solve quadratic equations. Because in the case of complex roots, the line and the parabola simply do not interesect. So it was originally thought that there was no worthwhile solution anyway. The real 'need' for complex numbers arose from solving cubic equations. [1]

[1] https://www.goodreads.com/book/show/19161684-a-friendly-appr...

/r/math discussion: https://www.reddit.com/r/math/comments/dzbmbu/a_new_way_to_s...

Original source: https://www.poshenloh.com/quadratic

I disagree that this proof is better pedagogically; it assumes the quadratic case of the Fundamental Theorem of Algebra (the correspondence between factorization and solutions), which at this stage would have to be taken on faith by students, whereas completing the square is fully justified.

Not sure why you think it's easier to take that on faith than complete the square, but I'm sure you never tutored new students to mathematics. At this point we are taking all of abstract algebra and some number theory for granted in anyone's education. Pedagogically it's easier to understand, the thesis of the article.

Granted, I have not tutored new students. I am making the assumption that someone encountering this would be familiar with basic algebraic manipulations, and the solutions to x^2=a. This is all that is necessary to justify completing the square. The assertion that quadratic equations must have at most two solutions R and S, and that it can be equivalently written (x-R)(x-S) is what has to be taken on faith.

> it assumes the quadratic case of the Fundamental Theorem of Algebra

No, it doesn't; it proves that case. The key step in the proof is the fact that, over the complex numbers, every number has a square root: from that it follows that the factorization used in the proof must always exist.

Is it just me, or are the images in the article extremely pixelated?

Yes, they're terrible. Katex or MathJax would be both easier and better. There's also at least one formula mistake: a cx^2 shows up briefly.

One would think an MIT technical publication (PR arm?) aimed at showing off their prowess would know how to typeset an equation.

I think this is nice. It's a method of calculation derived from an important fact—that quadratic equations have two solutions and those solutions can be written in such a form.

I don't have the experience or interest to debate whether it's better or worse than any other method pedagogically. I can definitely imagine that I would have appreciated learning this method as well as completing the square as a child.

But honestly, the real interesting thing here is the (a) gathering information about what you think the solution ought to look like and (b) working backwards from there. That's a good general trick and worth having in your back pocket. It also emphasizes the importance of visualizing and examining your belief about what should occur.

People should suspect the fundamental theorem of algebra long before they prove it.

I must be missing something here...

The "standard" quadratic formula at the top of the article is just as quick and painless to solve the equation he uses as an example. And its easier for many other versions (basically any time B/A or C/A are not integers).

Plus, this isn't new: I was taught to do exactly this IF it simplified the whole equation. That was in 2000 in London in a pretty standard secondary school.

(Also, what's with taking "z^2 = 3" at font 10, treating it as an image and then displaying it 20 times the original size?! Is it meant to look more maths-ey?)

I am excited to reveal something though: I recently discovered a whole new way to make a percentages! Instead of multiplying the number by 100, you just multiply it by 10 twice! So much better :)

The focus is not on getting students to be able to solve quadratics (if it were, then the quadratic formula is a great choice!), but rather to understand how solving a quadratic works. That is, how to derive the quadratic formula for themselves. Generally this is done by completing the square. This is another method which some students may find more intuitive.

I have been using a technique that's almost identical to this for years, based simply on observing that the roots are symmetric around the min/maximum: differentiate, set to zero, then difference of squares.

This cannot be an unrecorded technique can it?

No, it's basically what the quadratic formula does. (Well, all methods solve the same equation and get the same roots, so they have to be equivalent.)

I don't know that it was ever hard.

It is hard for the people that isn't comfortable doing symbolic algebra. I don't see how this method improves it for those.

They are https://www.youtube.com/watch?v=o_zoHxC4y-U

>> to make the quadratic equations easy

They mean: to make the quadratic equation easy to remember.

However, I don't think this will have any impact on the average high school student.

The key is:

>> Loh points out that the two roots, R and S, add up to -B when their average is -B/2.

>> “So we seek two numbers of the form -B/2±z, where z is a single unknown quantity,” he says. We can then multiply these numbers together to get an expression for C.

But you still have to derive the formula and their first example assumes A is fixed.

> their first example assumes A is fixed

This is usually how completing the square is taught, in my experience. The leading coefficient is handled later with "just divide everything by a".

That's very cool. To the best of my knowledge the only purpose of learning how to complete the square in high school math is to be able to understand the derivation of the quadratic formula, so if we can use this alternative proof then students don't need to know "completing the square" anymore. Good riddance!

The only other application of "completing the square" I can think of is the trig substitution in calculus, see https://en.wikipedia.org/wiki/Completing_the_square#Integrat... or worked example here https://www.khanacademy.org/math/ap-calculus-ab/ab-integrati...

In sum, completing the square is definitely a cool trick (see https://en.wikipedia.org/wiki/File:Completing_the_square.gif ), but maybe we can skip it... or present it as extra/optional material? I'm going to think about dropping it from my books. It will save 5+ pages of suffering for readers, which is a clear win.

Nope, completing the square is primarily used to translate a standard quadratic function into a vertex form quadratic so it can be graphed using transformations instead of guessing.

It is an essential multi-use tool in precalculus. The thing we should dump is the quadratic formula.

If you're going to rename the constants to make a=1, you might as well also replace b with b/2. The solution to x^2 + 2bx + c = 0 is

    x = ± √(b^2 - c) - b

I have read the paper and it gives a good educational context from which the quadratic equation may be looked at. I have also worked out simpler way to solve quadratic equations using function evaluation without much memorization of symbolic formula:

for the equation : f(x) = ax^2 +bx + c = 0

X = Z +- Sqrt [-f(Z)/a] ; Z = -b/2a

for the equation f(x) = x^2 +bx+c = 0 it is even a bit simpler:

X = Z +- Sqrt[ -f(z) ] where Z = -b/2.

Example 1: f(x) = 3x^2 -8x-35 = 0

Z = -b/2a = - (-8)/(2.3) = 4/3

F(Z) = -121/3

X = 4/3 + or - Sqrt ( -(1/3)*(-121/3) = 4/3 + or - 11/3 = {5, -7/3)

Example 2: (Simpler form): f(x) = x^2 - 4x+ 3 = 0

Z = -b/2 = -(-4)/2 = 2

f(Z) = 2^2 - 4.2 + 3 = -1

X = 2 + or - Sqrt( -(-1) = 2 +- 1 = { 3, 1}

For detail of this method, please see the following pre-print

https://www.researchgate.net/publication/337829551_A_simple_...

Um... what's the difference?

  x = (-b ± √(b^2 - 4ac)) / 2a

Ok, remove a assuming it's 1 - well, yeah, less general math is simpler.

  x = (-b ± √(b^2 - 4c)) / 2

Move around 2 & 4... voila.

  x = -b/2 ± √(b^2 / 4 - c)

Like, whatever. :)

I think there's an unacknowledged premise here that is fundamentally misguided: that the point of teaching math is to minimize the complexity of the formulas and recipes the students have to memorize, or that skill in finding the roots of a quadratic is so important that we must optimize the experience of learning it. I work with mathematics teachers and engineers and I know of no one who has willingly factored a quadratic by hand since we put our slide rules away.

The goal, fundamentally, is not the skill of factoring a quadratic. The goal is understanding of the relationships of numbers, operations, and domains that allow algebra to be a powerful set of tools for solving huge classes of problems. I _never_ teach the quadratic equation. I teach completing the square, because it's an illustration of a useful way of algebraically manipulating a relationship into a form (square of a linear binomial) that they recognize and can easily factor. I usually do a quick proof of the quadratic equation using completing the square, but generally as an illustration that if you properly understand the associated algebra, you don't need to memorize formulas and algorithms. The use of the field axioms to manipulate polynomials, and the goal of manipulating them to a tractable form, is what they need---not the quadratic equation. This 'new' technique (which appears to be a simple riff on the standard technique of looking for a pair of numbers whose sum is B and whose product is C that is taught as a matter of course in every middle school on the planet) is missing the actual goal of the lesson.

In particular, the reason I even teach completing the square is because it's a precursor of me bringing up the roots of $x^2 + 1$, which takes us in to an introduction to complex numbers, the fundamental theorem of algebra, and the discovery that the same old algebra over the Reals they learned in high school can now be used to do things like solve differential equations. Later, we introduce matrices and I can return to our old friends the algebraic field axioms to solve whole systems of equations. If they're lucky, they also get modular arithmetic and can be shown Galois fields and start using their understanding of algebra for problems in logic and set theory. All of that is easily within the reach of students in their first or second year of college, and I think if we bothered to try, we'd discover it's easily in the reach of secondary school students---or would be if we'd stop pretending that teaching them that "mathematics" is making change and pushing numbers from a word problem into the blanks of some generic formula they've memorized.

The task isn't teaching them to factor quadratics. The task is teaching them algebra.

To me this seems like a special case of the Vieta formulas (https://en.wikipedia.org/wiki/Vieta%27s_formulas) which in my case I learned in highschool

This still seems way more complicated than it needs to be to teach it. I always used a much simpler way to avoid memorizing anything.

Imagine you have a parabola y - c = k x^2 and want to solve for y = 0. Dead easy, right?

To turn any other parabola into this form, you only need to scroll left or right on x until the minimum is at x'=0 (algebraically, this means eliminating any b*x' term). Teach students how to do change of coordinates and how to solve this trivial problem, and they don't need to memorize any formulas.

It also sets students up for the useful math mindset of solving new problems by reducing them to previously solved ones and relies on conceptual understanding. Seems way better than the "memorize this formula" approach.

I teach mathematics at a community college. This includes teaching a lot of elementary algebra courses. These are pre-college level math courses. In elementary algebra we introduce solving quadratic equations by factoring. In the next course intermediate algebra we teach the quadratic formula.

Students in elementary algebra are not equipped to understand change of coordinates. This is too hard of a concept at that stage in my opinion. The method described in the article is a very nice one and is appropriate for elementary algebra. It ties in nicely with factoring and makes solving quadratic equations quite easy. I will be using this method in the future.

How aren't they equipped for it? Do you mean change of coordinates in its full generality? I'm talking about simple substitution of x' = x + k, connected with the intuition sketched below. They know how to perform simple substitution by then, and they can surely follow this logic:

    x       0---1---2---3-->
    x'  0---1---2---3---4-->

    x' = x + 1
    ==>
    x = x' - 1

    y = f(x)
    ==>
    y = f(x' - 1)

If you can talk about Napa being an hour north of San Francisco and San Francisco being an hour north of San Jose, this kind of student should be able to tell how far north of San Jose Napa is.

Can they not even do that? If not, I'd questioning why we teach them to solve quadratic equations before they can do substitution.

Well for one thing the simple substitution you mention is a hard concept. They certainly will have a hard time with f(x-1) and introducing a new variable is a mental block at this stage. At my college we don’t introduce functions until the next course. I suggest your viewpoint is clouded by the fact that you know this stuff so well that you no longer remember what the pain points are for students learning it for the first time.

After learning to solve simple quadratic equations and then the quadratic formula we typically introduce variable substitutions to solve things like x^4 + 5x^2 - 6 = 0.

I freely admit that my viewpoint is clouded by fluency, but I didn't come up with this today. This is how I've done it since I was still in school and I remember being annoyed by the opaque "just memorize this formula" approach from early on, where the concepts just seemed much clearer. I admit to not being the typical math student. But I think my approach was my competitive advantage, not something that makes my experience inapplicable.

To be clear, you're saying that if you tell a student

    y = x^2 + b*x + c

and you tell them

    x = z + 2

they aren't yet equipped to learn to combine those into

    y = (z+2)^2 + b*(z+2) + c.

Is that right? Why do you teach quadratic equations at this stage? I'd consider those to be much more advanced than simple "replace x with (z+2) everywhere you see it" plus some "alice's house to bob's house to carol's house" problems along a single axis for the concepts.

It seems like you're saying they're taught quadratic equations before they're equipped to poke around with them, which seems to be setting up for the black box/memorize-the-formula version of math.

In elementary algebra students can do this for the most part:

Evaluate x^2 + 2x + 4 for x = 3.

Quite a few will struggle with:

Evaluate x^2 + 2x + 4 for x = –3

Almost all will struggle with:

Evaluate -x^2 - 2x + 4 for x = -3

I don't think they'd handle replacing x with z+2.

In elementary algebra they hate fractions. Many struggle with 4 – (–5). I like the approach in the article because of how it relates to factoring and difference of squares. This reinforces those concepts and shouldn't be too great of a leap at this stage. Also, there is a nice geometry behind the (b/2 + z)(b/2 - z) idea. The approach is nice precisely because it isn't a memorization approach. It's an approach that says, "Hey, let's analyze what factoring trinomials is all about and what the relationship between b, c, r, and s are.". It says, we know that r and s have to have the property that r+s is b and rs is c. We know this because of our analysis of multiplying binomials and our experience with factoring trinomials. We are using patterns and pattern recognition and from this we are constructing solutions to an equation that we can't solve by isolating the x like we did with linear equations. This to me, is true mathematics and it's nice to show students this exploration. To show how mathematicians think and approach problems. It's not a black box. It builds upon previous ideas and uses them to solve problems we weren't able to before.

Note that I'll be using this method in elementary algebra from now on but will not be using it to prove the quadratic formula. Indeed I will not even tell them what the quadratic formula is. I will save that for the next class.

'sykick you have my sincere appreciation for your tireless efforts in this thread to communicate both the existence and particulars of real differences in math aptitude among humans. Those on HN who have never attempted to help someone else with math would have had no idea...

Thank you. That’s very nice of you to say.

I remember discovering this for myself when tutoring math in college. It's amazing how difficult it is to explain a lot of "basic" math without using "advanced" math. Trying to solve limits without L'Hospital's rule is excruciating.

Here is a link to the paper the story was based on: "A Simple Proof of the Quadratic Formula" https://arxiv.org/pdf/1910.06709.pdf

Note that he critisize the classical formula but that he, then, suppose that A=1 when he gives his formulation (making it simpler). Adding A back gives :

-B/2A ± sqrt(B²/4A - C/A)

which is not obviously simpler than :

(-B ± sqrt(B² - 4AC)) / 2A

In Nepal, you are taught around the sixth grade that you can solve ax^2 + bx + c = 0 by splitting b into m and n such that mn = ac and m + n = b ( or m -n = b, if a and c have opposing signs). This "discovery" the author claims and some other corollaries were quite commonly known.

I find it absurd and, frankly, laughable that this is being heralded as something new!

There is nothing new about this formula or derivation. In fact this precise formula (including that derivation) is taught in Germany.

One of my favorite things to break the ice with a younger gamer and the Quadratic Formula.

https://www.youtube.com/watch?v=ewAHYVzMobw

Super Mario Quadratics.

(r-s)^2=r^2-2rs+s^2=(r+s)^2-4rs=b^2-4c

r=((r-s)+(r+s))/2=(sqrt(b^2-4c)-b)/2

Surely this is a prank?

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