I myself never learned the quadratic formula, I always just used completing the square, i.e. just using the fact that x^2+ax, and that you can split the ax in t

I myself never learned the quadratic formula, I always just used completing the square, i.e. just using the fact that x^2+ax, and that you can split the ax in two put the resulting rectangles on sides of the square, to make a square with side x+a/2, by only adding an a^2/4 which is independent of x. Then you know that you can just remove the little a/2 square from the big square to relate it to x^2+ax. I think that was the first case where I showed good taste in how to do mathematics.