How to Derive the Quadratic Formula

How to Derive the Quadratic Formula

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[armchairtin-kicker]

Having failed calculus II twice over forty years ago, I returned to college and earned a mathematics degree in 2020. Along the way, I tutored college mathematics and physics for five semesters. However, before returning to college, there were three lessons that I had to absorb: (1) never walk into a course with the expectation of learning anything; the trick is to know the material before one hits the door. In other words, one is merely there to be graded on what one knows. As Leo Szilard said, “If you want to succeed in the world, you don't have to be much cleverer than other people; you just have to be one day earlier.” (2) do not overload oneself with courses. For my third—and last—attempt at calculus II, although it was not necessary, it was the only course I took, living a breathing the subject for the entire semester. And (3) focus on composing derivations. Many believe learning formulas is the key, but they are wrong. Learning to derive the formulas, learning the patterns in various states is the key. Reading “Chess Chunking and Skills, ” Chapter X: Levels of Description, and Computer Systems, Gödel, Escher, Bach: an Eternal Golden Braid (GEB) by Douglas R. Hofstadter, will convince you. For example, chess grandmasters win by internalizing thousands of board patterns, not by “seeing” many moves ahead. Indeed, in the book, “Outliers, ” by Malcolm Gladwell, the author asserts that it takes 10, 000 hours (or approximately 10 years) to become an expert. Over that course of time, our would-be expert is internalizing patterns and how to properly respond to them. In many cases, knowing the patterns and how they relate, understanding the architecture, provide inferences on how to respond to novel situations. Having worked as a system software developer for nearly thirty years, having had to reverse engineer operating system changes, I know this to be true. With that in mind, let’s return to the derivation of the quadratic formula and its relevance.

For instance, when working with Calculus II students that are struggling, I will often asked them the derivatives of the trigonometric functions, logarithm, and exponential functions as well. If they hesitate or stumble, I will assign them the task of deriving each one, using the limit definition of the derivative. But the first task I assign is deriving the quadratic formula from the general form of the quadratic equation. And I am generally shocked that so few of my tutees, at that level, even knew it could be done. Of course, I give them other assignments like deriving all the trigonometric formulas, squeeze theorem, integration by parts formula, trigonometric reduction formulas, trigonometric arcus formulas, polar formulas, etc. Once my tutees have researched and struggled to compose these derivation for the first time, I tell them, as practice, to compose these derivation from memory until the can do it from a dead sleep and then keep doing it. Doing this solves to objectives: (1) one never forgets a formula and if one does they can always derive it; and (2) because one is always doing integrations from various stages, knowing formulas, unlike previous math courses, is not good enough, offering one reason why the failure rate is so high for the course. Yes, mathematics, like many other disciplines, is all about pattern recognition and one’s ability to compose derivation of formulas from memory is the key. Finally, composing derivations is also the steppingstone for writing proofs, an essential skill for higher level mathematics.

[nellwackwitz]

You are making my days happier! Thank you!

[laluramasarya]

I've been done making video about this topic, but i created it with my own language (Indonesia) in explaining. glad that the way i explained it almost the same as your, sir. we're looking forward for another brilliant video.sir.

[juanitamutisa]

I have a test in a few days and I was about to lose my mind after trying 4 different channels and not understanding their explanations. This video was a great help. Thanks a lot!

[mocks]

always wondered where that equation came from, turns out it's not too bad.

[yadunandankarthi]

OMG. THank sososososo much i got an A because i had to present this in front of my class and i only watched ur video ten times to remember it and it helped a lo once again thanks a lot

[jermainerace]

Personally I always move c over to the right side first, then divide by a when I'm working through this (I do it as an exercise every year or whenever I feel like my algebra needs practice).

Is there any principled reason why dividing by a first might be better in general? It does seem like if you had zero on one side of any equation you could divide out an inconvenient common factor, and that would generally be a good start to simplifying just about anything.

[mixeri]

Thank you very much, it helped me a lot

[God-ldll]

a*x^2 + b*x + c = d(x+e)(x+f)=dx^2+d(e+f)x+def
a hypothetically good start imo.

[navamoore]

how do you know when to complete the square?

[jeremybobbin]

I thought you were joking with your “completing the square” nonsense. On further inspection, it appears to work, though it’s still a cheap hack in my opinion.
Otherwise, good video 👍

[bigdog]

holly shit why the fuck should I know this? I mean is this going to help me defend an innocent black person accused of murder? Spoiler unless I bore the jury into submission like this video bored me it ain't going to help. So why the fuck do I have to learn this?