| 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. [*] Because we read it in baby Rudin, ofc |