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The Quadratic Formula You've Never Seen

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Tags: quadratic-calculus-derivation-formula

A derivation walkthrough of an alternative version of the quadratic formula I came up with. Meant for enthusiastic high schoolers, and even undergraduate students.

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6.35 Overall score*

62 Rank

17 Votes

10 Comments

Comments

This was a very interesting video. I think the target audience would be able to follow most things in the video. The topic is interesting, and pretty novel, in my opinion. To me the weak point was in the description of where the formula comes from. Yes, I could follow what was done, but I didn’t really feel like I got any insight into what this formula is telling me and why it works. Maybe that is my shortcoming, but I don’t think so. Still, I felt like this was a very good video overall.

5.6

I would’ve liked to see stronger motivation for the topic. Is this alternate quadratic formula actually used anywhere or is it just a curiosity? The original quadratic is written in terms of a b c. Yours is written in gamma f(beta) and f’(beta). So in my mind, there are 3 variables in both cases so it felt like it’s just converting one form to another. It was another way to think about it I guess but it also wasn’t too shocking.

7.5

This is a very nice video, with smooth animations and a pleasant speed and tone of voice. It also provides an interesting geometric interpretation of the two quadratic roots. However, I was a bit off-put by the claim that this could change the way we see the well-known formula, first because it can actually be derived from the standard formula (see below), but also because the example does not fully convince me of its power (needing to know the unique minimum/maximum is quite limiting, and finding the position and slop at a given point usually relies on knowing the parameters $a$ , $b$ , and $c$ anyway). I would have personally preferred a video that simply re-prove the quadratic formula using your geometric arguments. Especially since you start by shifting the triangle so that $\gamma=0$ and then use $\beta=0$ to come back to the standard form, I feel a single shift could have given the standard formula using only geometric interpretations.

Overall it was quite pleasant and well-explained, but not as life-changing as claimed :).

Proof of your formula with the standard quadratic solution. Given the polynomial $ax^2+bx+c$ , by letting $y=x-\beta$ we obtain the new polynomial $ay^2+f'(\beta)y+f(\beta)$ . Further using that $f'(\beta)=2a(\beta-\gamma)$ and replacing $a$ using this formula, the roots of the polynomial in $y$ are given by

\frac{-f'(\beta)\pm\sqrt{f'(\beta)^2-\frac{2f(\beta)f'(\beta)}{\beta-\gamma}}}{\frac{f'(\beta)}{\beta-\gamma}}=\gamma-\beta\pm\sqrt{(\beta-\gamma)^2-\frac{2f(\beta)\cdot(\beta-\gamma)}{f'(\beta)}},

which then leads to your formula for the roots of $x$ .

8.4

Great video! Clean presentation, animations, and on something that I didn’t know! This is a perfect example of why I love math, how so much of it is an extension of itself.

6.5

7:00 would be good to highlight the parts that changed for each step

7:25 needs an extra step where the square brackets contents are replaced with N. Also the notation should be matched exactly to the top right to make the equality more visually obvious.

7:25 the ⇒ isn’t worth it, just use another line as in all the other steps

I’d mention all the related maths about how small sections of functions contain information about the entire function, so anything from taylor expansion up to analytic continuation. Even a simple namedrop could ring a bell for a lot of watchers.

Very nice video. Well explained and a topic everyone knows but maybe hasn’t thought deeply about

This is an interesting generalization of the quadratic formula. I can follow how it all works, but it would help to highlight the intuition. What would lead someone to choose those three points and look at the integrated areas?

4.5

In terms of how easy it is to teach it to a student, your proof of this more general quadratic formula is not as straightforward as deriving the usual quadratic formula. And since I fail to see the motivation behind this more general quadratic formula, I am not sure that this explanation would support a student’s understanding of the topic…

4.5

Many formulas fill the screen and there are no moments to build diagrams; many students would be left out. Additionally, to cover this in-depth exploration of a formula useful in high school, you use notations more suited to undergraduate students.

6.4

The audio is a bit meh. The insight seems to be plucked out of nowhere.