A new way to make quadratic equations easy • Hacker News Board

Re: A new way to make quadratic equations easy

#21

Post by fyp » Fri, Dec 06, 2019, 12:34 PM UTC

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.

Re: A new way to make quadratic equations easy

#22

Post by shikoba » Fri, Dec 06, 2019, 12:35 PM UTC

(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

Re: A new way to make quadratic equations easy

#23

Post by LatteLazy » Fri, Dec 06, 2019, 12:56 PM UTC

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 :)

Re: A new way to make quadratic equations easy

#24

Post by dooglius » Fri, Dec 06, 2019, 1:11 PM UTC

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.

Re: A new way to make quadratic equations easy

#25

Post by boxfire » Fri, Dec 06, 2019, 1:14 PM UTC

dooglius wrote: post #24

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.

Re: A new way to make quadratic equations easy

#26

Post by spicymaki » Fri, Dec 06, 2019, 1:24 PM UTC

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

Re: A new way to make quadratic equations easy

#27

Post by dooglius » Fri, Dec 06, 2019, 1:24 PM UTC

boxfire wrote: post #25

dooglius wrote: post #24

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.

Re: A new way to make quadratic equations easy

#28

Post by LatteLazy » Fri, Dec 06, 2019, 1:40 PM UTC

fyp wrote: post #21

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 clai…

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...

Re: A new way to make quadratic equations easy

#29

Post by jsjolen » Fri, Dec 06, 2019, 1:44 PM UTC

verandacoffee wrote: post #19

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 ).

Re: A new way to make quadratic equations easy

#30

Post by 6gvONxR4sf7o » Fri, Dec 06, 2019, 1:49 PM UTC

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.