Quadratic Formula Lore

January 27, 2009 in Cool Tricks, Math 148

I never learned the doggone thing until I was the teacher and had to, for one thing. I was trying to cop some math-geek attitude (“Never memorize what can be understood instead!”—it turns out this is sort of a damfool commitment). I knew I could derive it (by Completing The Square, of course) and that was by golly good enough for me—how often was I going to need to solve a quadratic equation, after all (many a thousand times, of course … but who knew?)?

Of course I knew I could derive it because I’d practice it from time to time (even the most exercise-phobic math major must occasionally solve a quadratic equation!).

Well, this week I got to practice some more, with live audiences. Had a great time of course. This is some of the world’s best material.

I’ve never studied the history in a systematic way (and don’t intend to now … but I’ll probably look at a few references along the way so I don’t make too much of a fool of myself … I try to keep things like dates pretty vague in lectures …).

Certain Babylonian texts, then, dating from about 1700 BCE, give procedures for finding (what we would now call) the roots of quadratic equations. But it wasn’t until the European Renaissance—the “rebirth of learning” after the so-called Dark Ages—that Algebra had its first flowering and it became possible to express such procedures as “formulas”. One crucial step along the way seems to have been learning to treat (the now-familiar) negative numbers on the same footing as positive ones: this eliminates the need for certain case-by-case breakdowns (as I was remarking the other day).

Anyhow, once variables and other enormous improvements in the notations were introduced, it became possible to write out the Quadratic Formula (QF). And to a certain kind of a person, that’ll be all it takes: give ’em a Quadratic Formula and some free time, and the next thing you know, they start asking questions like “What about a cubic formula?”. And so, with one heck of a lot of hard work by some really talented guys (mostly all guys doing math back then, I’m afraid … no genderbias intended) … they found it. And, dammit, it’s too unwieldy to actually set down as a single formula. The procedure is spelled out in a sidebar (“Historical Feature”) in the text. The general fourth degree equation was solved not too much later … and there the situation stayed for a few hundred years. Finally, in the early 19th Century, with the birth of Modern (“Abstract”) Algebra, it became possible to prove that there is no “Quintic Formula”—no procedure involving only roots, powers, multiplications, and additions (“algebraic” operations) that solves every polynomial equation of degree five.

Returning to QF. It’s worth remarking that we don’t need the “complete the square” technique to prove it. If once we have it in front of us, we can simply “plug in” the whole shebang on Ax^2 + Bx + C and perform a certain brute-force computation (and darn good exercise) … out pops zero. But this procedure gives no insight on where QF “comes from” (anyway, not immediately, not to me … though I can at least imagine working through the computation, backwards maybe, looking for some such insight [“now where does the “4AC” come from again?”]). It’s also worth remarking that, when f(x) = Ax^2 + Bx + C has real roots, we can literally see (on a graph) that the line of symmetry runs halfway between (the vertical lines)

and
; this accounts for the fact (also derived by me and the text in two other ways) that the x-co-ordinate of the vertex of f is -B/(2A).

And the rest of QF also has its own story to tell. The most-commonly-used properties of the discriminant B^2 – 4AC are spelled out in the text of course; I won’t rehash them here. Except to mention that the case of a negative discriminant points the way to the theory of Complex Numbers. And it was learning to take these seriously (i.e., to quote myself, “learning to treat them on the same footing” as the [so-called] Real Numbers [this eliminates the need for certain case-by-case breakdowns …]) that made it possible to state the Fundamental Theorem of Algebra (“every polynomial factors”). I’ll have much more to say about that.

Oh. One more thing. It has the scansion of “Pop Goes The Weasel”.

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