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I briefly looked at the author's paper. I find the account of the traditional method there questionable. He suggests it starts by going from: x^2 + bx + c t

I briefly looked at the author's paper. I find the account of the traditional method there questionable. He suggests it starts by going from: x^2 + bx + c to x^2 + bx + c + b^2/4 - b^2/4 This step is freaking weird, I've never seen it before, and I don't think this way. The process I follow is more like I observe that for general k: (x + k)^2 = x^2 + 2kx + k^2 so I observe that for k = b/2 I get (x + b/2)^2 = x^2 + bx + b^2/4 and then I figure out what I need to add to both sides of that equation to make the RHS equal the original quadratic, which gives me: (x + b/2)^2 - b^2/4 + c = x^2 + bx + c and I then find the roots of the RHS by instead finding them for the LHS. We say that the LHS is the result of "completing the square" on the RHS. Question: Does anyone seriously learn to solve quadratics by rewriting x^2 + bx + c to x^2 + bx + c + b^2/4 - b^2/4. Because that seems like a bad way of learning it.

HarryHirsch 5y ago
We did in school, but the original equation x^2 + bx + c was rewritten as x^2 + bx + (b/2)^2 - (b/2)^2 + c, and now you see the binomial. Our teacher said it felt like pulling a rabbit out of the hat.
- ogogmad 5y ago
  Where did you go to school? I'm from the UK. > Our teacher said it felt like pulling a rabbit out of the hat. This is something you do when you prove something. It's the shortest way but it's also totally lacking in motivation. It's a Deus Ex Machina, which is a terrible way of explaining mathematics to people. When I was taught "completing the square", it was presented as another method for solving a quadratic equation, and not just a proof of the quadratic formula. If people in the US are taught this way, that's not good for US-ians. And this whole discussion is weirdly US-centric.
  - HarryHirsch 5y ago
    This was in Germany, in 9th grade. Should have added that this was the proof of the quadratic formula.
BeetleB 5y ago
It's only as bad as memorizing the usual quadratic formula. While most people are taught the derivation of the quadratic formula, few remember it, opting instead to memorize it. Likewise, you've just derived the b^2/4 aspect. Most people don't want to rederive it every time they solve it, so they just memorize that they should add and subtract b^2/4 and complete the square. BTW, if the notion of adding and subtracting the same amount bothers you: Yes, this is done all the time.
- ogogmad 5y ago
  > BTW, if the notion of adding and subtracting the same amount bothers you: Yes, this is done all the time. I'm a mathematician. I'm not ignorant; I have taste. I've seen a similar technique used two prove the product rule in analysis. I always find ways of avoiding such tricks because I see them as Deus Ex Machinas. I don't think of mathematics as an exercise in ill-motivated tricks.
  - Twisol 5y ago
    > I've seen a similar technique used two prove the product rule in analysis. I always find ways of avoiding such tricks because I see them as Deus Ex Machinas. I don't think of mathematics as an exercise in ill-motivated tricks. Just to expand on this thought... in logic, "cut elimination" is a desirable property to have in a logical system. The "cut" rule states that, if you've proved "A -> B" (A implies B) and "B -> C", then you're licensed to deduce "A -> C". But in the statement "A -> C", B doesn't ever have to appear on its own, so if you're working backward, the "B" that splits the problem into two pieces can appear deeply unmotivated and appear out of nowhere. A deductive system with the "cut elimination" property means that, whenever you have a proof that uses "cut", there's always a way to transform the proof into one that doesn't use "cut". Put less precisely, there's always a proof that doesn't use any unmotivated variables. (In computer science terms, "cut" is like implementing a function in terms of an existing function; you can always inline the original and then simplify the resulting code.) "Cut elimination" is desirable because it makes automatic reasoning tractable. If you want to prove "A -> C", rather than trying to discover some arbitrary "B" that produces more tractable goals "A -> B" and "B -> C", there are (usually) only finitely many deductive steps that could terminate in "A -> C", and you can simply try each one. Although humans aren't machines, creativity is still difficult in fields we're not accustomed to. While adding and subtracting a term formally doesn't change the result, there are infinitely many choices of term that produce this result. Why that one? It's entirely unmotivated; it clearly works, but there's no obvious way you could have plucked that particular term out of thin air yourself. > The process I follow is more like I observe that for general k: Your derivation is awesome! Rather than deducing forward from the raw materials (the given quadratic), you deduce backward from the goal. This lets you leverage known information to pick values that get you visibly closer to the starting point, term by term. In the classic case, the goal here is to "complete the square", which is still of the unmotivated "cut" flavor -- it's not clear to the unskilled eye why that ought to be helpful. I don't think the article's approach is any better motivated in that respect; we still have to introduce the idea that all quadratics can be factored into two first-order terms. This is not a trivial theorem, and it assumes access to the entire complex field (otherwise it's simply untrue in general). But I think both approaches are valuable, as reliant on some "unmotivated" magic as they are. They all lend some insight into what's going on. Minimizing those moments is definitely beneficial though.
ogogmad 5y ago
For clarity: The geometric view of "completing the square" by splitting a parabola in two is not really necessary here. In my view, the algebra is well-motivated on its own.

HarryHirsch 5y ago

We did in school, but the original equation x^2 + bx + c was rewritten as x^2 + bx + (b/2)^2 - (b/2)^2 + c, and now you see the binomial. Our teacher said it felt like pulling a rabbit out of the hat.

ogogmad 5y ago
Where did you go to school? I'm from the UK. > Our teacher said it felt like pulling a rabbit out of the hat. This is something you do when you prove something. It's the shortest way but it's also totally lacking in motivation. It's a Deus Ex Machina, which is a terrible way of explaining mathematics to people. When I was taught "completing the square", it was presented as another method for solving a quadratic equation, and not just a proof of the quadratic formula. If people in the US are taught this way, that's not good for US-ians. And this whole discussion is weirdly US-centric.
- HarryHirsch 5y ago
  This was in Germany, in 9th grade. Should have added that this was the proof of the quadratic formula.

BeetleB 5y ago

It's only as bad as memorizing the usual quadratic formula. While most people are taught the derivation of the quadratic formula, few remember it, opting instead to memorize it. Likewise, you've just derived the b^2/4 aspect. Most people don't want to rederive it every time they solve it, so they just memorize that they should add and subtract b^2/4 and complete the square. BTW, if the notion of adding and subtracting the same amount bothers you: Yes, this is done all the time.

ogogmad 5y ago
> BTW, if the notion of adding and subtracting the same amount bothers you: Yes, this is done all the time. I'm a mathematician. I'm not ignorant; I have taste. I've seen a similar technique used two prove the product rule in analysis. I always find ways of avoiding such tricks because I see them as Deus Ex Machinas. I don't think of mathematics as an exercise in ill-motivated tricks.
- Twisol 5y ago
  > I've seen a similar technique used two prove the product rule in analysis. I always find ways of avoiding such tricks because I see them as Deus Ex Machinas. I don't think of mathematics as an exercise in ill-motivated tricks. Just to expand on this thought... in logic, "cut elimination" is a desirable property to have in a logical system. The "cut" rule states that, if you've proved "A -> B" (A implies B) and "B -> C", then you're licensed to deduce "A -> C". But in the statement "A -> C", B doesn't ever have to appear on its own, so if you're working backward, the "B" that splits the problem into two pieces can appear deeply unmotivated and appear out of nowhere. A deductive system with the "cut elimination" property means that, whenever you have a proof that uses "cut", there's always a way to transform the proof into one that doesn't use "cut". Put less precisely, there's always a proof that doesn't use any unmotivated variables. (In computer science terms, "cut" is like implementing a function in terms of an existing function; you can always inline the original and then simplify the resulting code.) "Cut elimination" is desirable because it makes automatic reasoning tractable. If you want to prove "A -> C", rather than trying to discover some arbitrary "B" that produces more tractable goals "A -> B" and "B -> C", there are (usually) only finitely many deductive steps that could terminate in "A -> C", and you can simply try each one. Although humans aren't machines, creativity is still difficult in fields we're not accustomed to. While adding and subtracting a term formally doesn't change the result, there are infinitely many choices of term that produce this result. Why that one? It's entirely unmotivated; it clearly works, but there's no obvious way you could have plucked that particular term out of thin air yourself. > The process I follow is more like I observe that for general k: Your derivation is awesome! Rather than deducing forward from the raw materials (the given quadratic), you deduce backward from the goal. This lets you leverage known information to pick values that get you visibly closer to the starting point, term by term. In the classic case, the goal here is to "complete the square", which is still of the unmotivated "cut" flavor -- it's not clear to the unskilled eye why that ought to be helpful. I don't think the article's approach is any better motivated in that respect; we still have to introduce the idea that all quadratics can be factored into two first-order terms. This is not a trivial theorem, and it assumes access to the entire complex field (otherwise it's simply untrue in general). But I think both approaches are valuable, as reliant on some "unmotivated" magic as they are. They all lend some insight into what's going on. Minimizing those moments is definitely beneficial though.

ogogmad 5y ago

For clarity: The geometric view of "completing the square" by splitting a parabola in two is not really necessary here. In my view, the algebra is well-motivated on its own.