The quadratic equation is completing the square while inexplicably avoiding all the intuition of completing the square. For example, to solve x^2 + 6x + 5 = 0,

The quadratic equation is completing the square while inexplicably avoiding all the intuition of completing the square. For example, to solve x^2 + 6x + 5 = 0, you would re-write it as (x^2 + 6x + 9) - 4 = 0, which is (x + 3)^2 - 4 = 0 and hence equivalent to (x + 3)^2 = 4, so that x + 3 = ±2 and hence x = 3 ± 2 is 1 or 5. Euclid thought of things this way, though his language is, of course, very different to modern language; see, for example, Proposition 6 of Book II (http://aleph0.clarku.edu/~djoyce/elements/bookII/propII6.html http://aleph0.clarku.edu/~djoyce/elements/bookII/propII6.htm...). That's the same answer as the quadratic formula, but makes a lot more sense to me! Of course I've cooked the numbers so that you don't wind up with surds in the answer, but those are just complications in bookkeeping, not in concept.