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having a berenstain/stein moment, couldve sworn it was ever so slightly different

Did You Know: the quadratic formula gets nicer if you realize it's trying to tell you that the coefficients of a quadratic should be written ax^2 + 2bx + c; then it becomes (-b \pm sqrt{b^2 - ac}}) / a and there aren't any 2's or 4's anymore

the next simplification you can do is to realize that the roots don't change if you divide by a, so you might as well assume that a = 1, and then it becomes -b \pm sqrt{b^2 - c} also if you don't like the -b you can write the coefficients of a quadratic as x^2 - 2bx + c

then it becomes b \pm sqrt{b^2 - c} and at this point it becomes a lot easier to remember the proof: complete the square to get x^2 - 2bx + c = (x - b)^2 - (b^2 - c)

also here's a fun fact i learned recently that i feel like someone should've told me a long time ago: "completing the square" refers to literally completing a literal square

beautifully demonstrated here:

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Visa

SF (Aug 2-14)

@visakanv

· Sep 15, 2017

Replying to @paulg

There's a great video of a math teacher blowing his students minds with "fill in the square" – you'd love it: youtube.com/watch?v=McDdEw

this is great algebra was a mistake

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· Oct 3, 2020

Replying to @QiaochuYuan and @selentelechia

truly it is said, "Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine."

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· Jul 16, 2020

Replying to @littmath and @SC_Griffith

I remember when I asked my AG professor "how do you imagine k^n for k an arbitrary field?" "what do you mean imagine"

the good news is that for a lot of purposes the behavior is independent of k so you can imagine any particular field you want, and for other purposes it only depends on the characteristic or whether k is algebraically closed etc. so again you have a lot of freedom