oh fuck I forgot the quadratic formula
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Conversation
still have FOIL in there though, phew
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having a berenstain/stein moment, couldve sworn it was ever so slightly different
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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
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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
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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)
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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
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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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i interpret atiyah to mean that when you take an algebra-first approach to math you give up doing math in a geometry-first way, which means e.g. a lot of diagrams, strong reliance on visual intuition, that kind of thing
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