This is so beautiful! I could have never imagined this. I learnt this formula by rote when I was in school. Didn’t realize that it had a geometric equivalent. Same thing with differentiation and integration. Couldn’t understand. Learnt that too by rote. Is there a geometric equivalent for most formulas if not all? Is there a website?
The notion of “completing the square” is often done as an algebraic trick (“just add and subtract this magic term”).
But it can also be viewed as translating the quadratic expression along the “X-axis” so that (at its new origin) it is left/right symmetric.
That is,
Q(x) = ax^2 + b x + c
With the right substitution, x’ = (x - B), the linear term vanishes. So when you re-write in terms of “x”, you get:
Q (x) = a (x - B)^2 + C
So the intuition is that the linear term in the original quadratic is the thing that shifts the “symmetry axis” of the quadratic.
I have found this helpful when “X” is a vector and you have a quadratic form. In this case, the coordinate shift centers the quadratic “bowl” about some point in R^n.
*
The chain rule for differentiation is another one with simple geometry but cumbersome notation. It’s like: we know[*] that
f(x) = g(h(j(k(x))))
must have a linear approximation about some point x0. The only possible thing it could be is the product of all the little local curve slopes of k, j, h, and g, at the “correct” point in each.
Thinking about little slopes also clarifies derivatives like
f(x) = g(x^3, x^2)
where g is an arbitrary function of two variables.
I have really enjoyed reading some of the better explained pages. I wouldn't recommend most people start on calculus on that site, but since you requested it specifically here's the overview: https://betterexplained.com/guides/calculus/
There is some geometric equivalent for any formula that can be built using some arbitrary combination of the four operations of arithmetic, plus square roots. If you allow for either 3D geometry or origami folds, you can extend this allowing for cube roots too. Note that the geometric equivalent is not going to be necessarily trivial or intuitive in most cases, though.
The geometric “proof” isn’t actually equivalent, because it assumes a > b, and doesn’t generalize geometrically to b > a. The algebraic proof, on the other hand, generalizes at least to commutative rings.
Geometric “proofs” like this are neat, but are no real substitute for the algebraic ones. I’d argue that in cases like the present one they also don’t provide any deeper insights. You’re just moving geometric shapes around instead of algebraic symbols. They might give you the feeling that the theorem isn’t as arbitrary as you thought, but it isn’t arbitrary in algebra either.
I’m putting “proof” in quotes here because there are many examples of incorrect geometric “proofs”, and there is generally no formal geometric way to verify their correctness.
> there is generally no formal geometric way to verify their correctness.
There are formal models of synthetic (i.e. axiom-and-proof based) Euclidean geometry where proofs can in fact be verified. This is accomplished by rigorously defining the set of allowed "moves" in the proof and their semantics, much like one would define allowed steps in an algebraic computation.
