Quadratic Confusion

I have a horrible memory. I don’t mean just that I misplace things or forget names; it takes a lot of effort to commit arbitrary facts, figures, dates, etc., to my long-term memory. So throughout my school years, most of my studying was for things like History, trying hard to remember dates and statistics that I would quickly eject from my mind after my next exam. I seldom had to study for Math or Science though, because I figured out something that worked for me there: learning how and why things work rather than just memorizing formulas. This worked well for those subjects, but I do remember stumbling in algebra when I was not able to factor quadratic functions. There was a handy Swiss army knife of sorts for this, of course, in the Quadratic Formula.

I avoided this formula as much as possible, usually by spending way too much time trying to guess the factors myself, or by converting from “standard form” to “vertex form”, or guessing, or skipping that question. This was almost entirely because I could not bring myself to memorize the formula. Call it laziness, or foolishness, or whatever you’d like.

Well, recently I decided to brush up on my math skills. After yet again encountering the need for this equation, I decided enough is enough. Since I can’t memorize the equation, I will instead learn where it comes from by deriving it from the standard form of a quadratic function. This is my attempt to do so.
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While I was watching a video from one of my favorite YouTube channels, I decided I wanted to try the math(s) that Professor Merrifield was describing to explain Gamma / the Lorentz factor. I’m very excited that my new blog supports \LaTeX and MathJax so the formulas actually turn out looking the way they should.

While this math might look difficult at first, it is really easy. It is essentially just using the Pythagorean theorem to isolate the relationship between t and \tau, which is the factor by which time (and length, and mass) changes for a moving object.

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