I have never understood "completing the square". I mean, I get why and how it works, but it's completely unintuitive to me, or why you'd do it that way.
I think it is a historical hangover. It makes more sense viewed geometrically, and the technique is attributed to the same al-Khwarizmi for whom algorithms are named. But it's less intuitive as part of algebra, and we focus a lot more on algebra today than in the medieval times, when geometry ruled.
Algebraically, my intuition is that the x^2 + bx part of x^2 + bx + c = 0 "looks" pretty close to the expansion of (x+b)^2. If you relabel b so this becomes x^2 + 2b, it's even clearer. So you can try to fiddle with the constant term to make the LHS exactly that square, and you get completing the square.
This is similar to how you solve a first order linear DE, y' + f(x) y = g(x). The idea is that the LHS "looks" like the derivative of a product, (hy)' = h y' + h' y, so you fiddle with an integration factor to make the LHS exactly that derivative.
It's not intuitive for someone new to mathematics, but as someone with two math degrees, it is just the strategy of "do stuff to something until something about it looks similar to something you've seen before" that mathematicians use ALL the time. The additional part of this strategy is to dream optimistically about how you can make the thing you are dealing with "nice" after doing stuff to it. In the case of completing the square, you are hoping you can just straight up take a square root. That would be easy. Turns out if you try enough stuff, you can.
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I have never understood "completing the square". I mean, I get why and how it works, but it's completely unintuitive to me, or why you'd do it that way.
I think it is a historical hangover. It makes more sense viewed geometrically, and the technique is attributed to the same al-Khwarizmi for whom algorithms are named. But it's less intuitive as part of algebra, and we focus a lot more on algebra today than in the medieval times, when geometry ruled.
Algebraically, my intuition is that the x^2 + bx part of x^2 + bx + c = 0 "looks" pretty close to the expansion of (x+b)^2. If you relabel b so this becomes x^2 + 2b, it's even clearer. So you can try to fiddle with the constant term to make the LHS exactly that square, and you get completing the square.
This is similar to how you solve a first order linear DE, y' + f(x) y = g(x). The idea is that the LHS "looks" like the derivative of a product, (hy)' = h y' + h' y, so you fiddle with an integration factor to make the LHS exactly that derivative.
It's not intuitive for someone new to mathematics, but as someone with two math degrees, it is just the strategy of "do stuff to something until something about it looks similar to something you've seen before" that mathematicians use ALL the time. The additional part of this strategy is to dream optimistically about how you can make the thing you are dealing with "nice" after doing stuff to it. In the case of completing the square, you are hoping you can just straight up take a square root. That would be easy. Turns out if you try enough stuff, you can.
Once you get to quadratic forms you can use it again.