I'm learning how to convert quadratic equations from general form to standard form, in order to make them easier to graph. We know the general form is , and the standard form is . To help with the conversion, we can expand the standard form, and see that it turns into the general form. I totally get how to go from standard to general. I can easily memorize what h and k are, and use them to consistently derive standard forms.
What I'm curious about is how to, a priori, go from the general form to the standard form? Is there a way to see that can turn into without knowing that form ahead of time? How was the form discovered in the first place? How are alternate forms of equations discovered in general? I honestly wouldn't know where to begin.
I ask out of curiosity, and because I believe knowing how to go in the other direction will help really solidify this concept for me. Even if that knowledge is above my skillset at the moment, at least an overview of what kind of math is involved may supplement this concept for me.