Quadratic Formula Lore

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)
x = {{-B}\over{2A}}{\bf +} {{\sqrt{B^2-4AC}}\over{2A}}
x = {{-B}\over{2A}}{\bf -} {{\sqrt{B^2-4AC}}\over{2A}}; 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”.


  1. pdexiii

    In my K.I.S.S. method of creating tests, I’ve always felt that if an 8th grader taking algebra (we do that crazy stuff here in CA) can derive the solution to a quadratic they demonstrate a summation of basic computational skills (adding rational expressions, balancing algebraic equations, etc.) that you don’t need too many more questions on a semester exam. Lo and behold, those students who’ve done it successfully were the ones who succeeded throughout the year.

    Another reason why standardized tests never tell me anything I didn’t already know.

  2. i also like to write tests of the form
    “a few carefully-selected exercises”
    and quite agree that *grading* ’em
    is a very effective assessment …
    telling the (careful) grader much more
    than any multiple-guess deal ever could.
    but of course these are created
    for the needs of larger institutions
    than the individual class–and this
    is in the very nature of the case.

    the *point* of standardizing, if i can
    belabor the point for a few more lines,
    being to compare *different* classes
    with *different* instructors … and then,
    of course, up the ladder of abstraction
    (and intractability), different schools,
    different urban regions, cities, states,
    and so on … an endless smorgasbord
    of demographic divisions … & it’s not hard
    to see why those in the business of actually
    trying to communicate some basic truths
    of the universe tend to recoil in horror
    from these instruments of politics.

    part of the reason i’ve sometimes
    been known to endorse ’em
    is simply that once we admit the need
    for the politics — or simply allow as how they’ll
    probably never go away — it seems natural to prefer
    that they be done as honestly as possible
    (in the sense of “truth-telling” honesty;
    it’s probably best not even to consider
    honesty of the “not ripping people off”
    variety … we are talking about *politics* …).
    the more you can get away from “spin”,
    the better as far as i’m concerned.

    also it’s convenient in discussions of one’s own
    math background to say stuff like “my small
    sixth grade math class produced me,
    another guy with a doctorate in physics,
    an aerodynamical engineer, and the guy
    who, throughout public school, was always
    way ahead of the lot of us: this last guy
    got an 800 on the math SAT … nobody
    in the country did better … one day
    in grad school it dawned on me to my delight:
    “hey, i actually know *more math* than
    Bathtub Fish [not his real name]!”).
    recently i admitted in the blog somewhere
    how badly i’d done in gre and
    hmm. i appear to be rambling.

    first post-like entity created with
    my spanking-new macbook.
    (in the wifi cloud of the coffeehouse
    on my route to the bus-stop.
    it appears they’ve got a new regular.)
    ordinarily i’d be working but it’s a snowday.
    VME his mark this 28th of january 2009 CE

  3. IF you like the history of the quadratic equation,

    There must be 50 ways to leave your lover according to the old Neil Simon song, but I only found 18 (maybe 20) ways to solve a quadratic equation, with some notes about their history.


  4. i couldn’t open the (.DOC) file (the problem is
    very likely at this end … the college
    is so paranoid about “security” that
    they’ve made everything nearly unusable).

    “pat’s blog” rocks the math …
    thanks for the link!

    (of course you mean paul simon …)

  5. Yikes, Of course I mean Paul Simon (good thing I didn’t call it the Gilbert and Ed Sullivan Opera…. ;-}
    The link seems to work, so maybe it is a security issue…sorry

  6. i got it today just fine … one never knows …
    and an interesting document it is too.
    sure enough, some methods i’ve never seen
    (like it says in its subtitle). with notes on
    its own history, too, like some wikipedia page.

  7. I “refused” to memorize it, and completed loads of squares, too stubborn to admit that I actually did have it committed to memory.

    At some point I stopped taking pleasure from pretending.

    But, as I showed kids this term, completing the square an be awfully ugly, but if you don’t mind fractions, so what?


  8. carlbrewer

    @Pat Ballew:

    HE he… funny. I can’t stand quadratic equation though…. puh.

    fantastic blogpost.

    Carl Brewer
    Mobile marketing expert

  1. 1 Quadratic Formula « Social Mathematics

    […] I was reading an interesting article about the quadratic formula, which I highly recommend over at Vorblik’s Math Education.  This article is well within the understanding of anyone who reads this blog.  In fact, I think […]

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