Algebra students are often compelled to memorize the following jumble of symbols:
Every adult I meet seems to remember this equation. Some quote it proudly. Others recall it grudgingly, fists clenched. Some people sing it (to the tune of “Pop Goes the Weasel,” though the rhythm’s a little forced). Many believe it captures the essence of mathematics: mystical formulas, taught to everyone, comprehensible to few.
I’m ambivalent. I can’t deny the quadratic formula’s usefulness, but I can’t let its tyranny go unchallenged, either. This formula has planted itself, like a virus, in the minds of otherwise healthy adults. Our society demands that its members know this equation, but makes no mention of its context, its uses, its colorful history.
I come not to praise the quadratic formula, nor to bury it, but merely to shed a few rays of light. What is it for? What is its story? And where on Earth does it come from?
***
We begin with the polynomial – a special type of mathematical expression that looks something like this:
Mathematicians love polynomials. Over the centuries, they’ve spent hours inspecting them, toying with them, bending them different ways to see how they change. They’ve also devised a classification system (based on the highest power of x that appears):
•1st-degree (or “linear”) contain x1
•2nd-degree (or “quadratic”) contain x2
•3rd-degree (or “cubic”) contain x3
•4th-degree (or “quartic”) contain x4
•5th-degree (or “quintic”) contain x5
And so on (although the polynomials of degree 6 and higher don’t get cute nicknames).
One of mathematicians’ favorite games with polynomials is trying to make them equal zero, by picking the right value of x. They call this “solving” a polynomial. It can be tricky. Most polynomials have lots of terms. To get the whole expression to equal zero, you’ve got to pick just the right value for x.
Nevertheless, mathematicians have devised foolproof ways to solve such puzzles – at least, for the lower degrees.
1st-degree polynomials have only one solution. It’s so easy to find that most people, given a little training, can do it in their heads.
2nd-degree polynomials have two solutions. They can be found using the quadratic formula that you know and love (or don’t love).
3rd-degree polynomials have three solutions. Again, they can be found using a formula, though the cubic formula makes its quadratic cousin look quaint and simple by comparison.
4th-degree polynomials have (can you guess?) four solutions. The quartic formula, though, is prohibitively complicated: it would fill two chalkboards, and bore everyone in the audience to the point of salty, math-induced tears.
Anyway, that’s what the quadratic formula is for: solving 2nd-degree polynomials. We memorize it because it occupies a pleasant middle ground between the simple 1st-degree case (which is too easy to bother memorizing) and the dizzying 3rd– and 4th-degree scenarios, whose daunting formulas would exhaust and perplex even the most patient souls. The quadratic formula is hard enough to demand effort, but simple enough to master.
Besides, while 2nd-degree polynomials may not appear on your tax forms or credit card bills, they pop up all the time in math, physics, economics, and other sciences – often enough, at least, to make the formula worth committing to memory.
***
You may notice that I mentioned quadratic, cubic, and quartic formulas, but not a quintic formula. Is it even more complicated?
Not exactly. 5th-degree polynomials do have 5 solutions, as you’d expect. But there’s no similar formula for finding them, and there never can be. Quintic polynomials present an unwinnable game, a puzzle whose answer is somewhere out there, but which our old methods can never quite pinpoint.
The discovery of this fact – that quintic polynomials cannot be solved by any algebraic formula – is one of the great tales of mathematical history. It stars an irresistible protagonist: a boy genius, drawn into French revolutionary politics, spurned by his one true love, and slain in a duel before he reached the age of 21. His name was Évariste Galois.
Evariste was no head-in-the-clouds intellectual. As a teen, he joined a group of radicals, defied the monarchy, even served time in prison. Along the way, he fell for a woman named Stéphanie-Félicie – a French heartbreaker’s name if I’ve ever heard one – but found his affections unrequited. Even so, one day he found himself challenging a rival to a duel in order to defend the lady’s good name.
Legend has it that the night before his fatal duel, Galois glimpsed his own mortality. He feared departing our world without sharing his mathematical discoveries. So he worked until dawn, his quill scribbling furiously, committing to parchment all his ideas and breakthroughs. The next morning, a bullet pierced his belly. His last words, to his brother: “Don’t cry, Alfred! I need all my courage to die at twenty.”
Those notes – the ones he scrawled out by the early-morning lamplight – live on. Among them was the answer to the riddle of the quintic polynomial. We call those notes Galois Theory, and teach them to mathematicians across the world.
It’s beyond me why I memorized the quadratic formula as an 8th grader, but didn’t hear the tale of Galois until college. If you ask me, the latter might just enrich the mind more than the former. Don’t let anyone ever tell you that math isn’t romantic.
***
Here, then, is why I wish death to the quadratic formula: Because mathematical truths are not handed down by decree from on high.
The quadratic formula has become a sad emblem for how most people see math – formidable, alienating, and beyond the common man’s patience or intelligence. In the popular imagination, mathematics is at once tedious and bewildering, a tree that’s nearly impossible to climb and which bears no fruit even if you do. This quadratic formula – an inscrutable symbol of academic oppression; an ugly, pointless relic of years spent bored in math classrooms – this quadratic formula must die.
And here, too, is why I wish long life to the quadratic formula: Because mathematical truths do come from somewhere.
Mathematical truths emerge naturally, from the patterns of logic, from the laws of nature, and from mortal men and women who, amidst this tumultuous life of duels and revolutions and scorned love, carve out an evening or two to solve puzzles by candlelight. The quadratic formula is a part of history like any other, a piece of technology perfected a millennium ago in India, and which no one in the intervening centuries has managed to improve. Long live this quadratic formula – a fundamental truth; a product of labor and logic; a slice of the human experience.
Hi Ben, I generally like your blog posts a lot and I enjoyed the discussion of Galois here, but I was surprised after reading the first half of this (which talks about shedding some light on the quadratic formula) that there isn’t a derivation of the quadratic formula in the second half. The derivation comes out of completing the square, which is a really great (and more intuitive) strategy for factoring quadratic polynomials in general. Plus you get to use the word “discriminant”, potentially. I demand a sequel!
A fair point! My original draft actually did include the derivation, but it felt a little technical for the blog, and I think a lot of Alg. 1 teachers do show it to their students in class. But perhaps I’ll reconsider.
I found the quadratic formula strange and annoying until I found the proof which satisfied me and allowed me to use it without relying on faith. I had a similar issue with trig functions which took 6 years before I understood them to the same level of satisfaction.
Well enough that I could calculate sin(15 degrees) on a test to 3 sig figures by hand. (Because the test had loads of time and I didn’t feel like admitting that I had forgotten my calculator. (I used the first two terms of the Maclaren series).
I quibble with the wording “But there’s no formula for finding them, and there never can be. Quintic polynomials present an unwinnable game, a puzzle whose answer is somewhere out there, but which we can never quite pinpoint.”, for the reasons you might expect. There are formulas for the roots of quintic polynomials, and one can calculate the zeros to any level of precision you like. They just aren’t formulas solely in terms of addition, subtraction, multiplication, division, and radicals. [In the same sense, mind you, there’s no formula to solve 2^x = c or cos(x) = c for x in terms of c, either…]
That’s a valid quibble.
I’d maintain that calculating a value to arbitrary levels of precision doesn’t quite qualify as “pinpointing” it. But, as you say, we CAN pinpoint a quintic’s roots with a formula, if we allow ourselves some slightly less pedestrian operations.
I’ll work on editing the language. I could perhaps more defensibly say that “winning the game” of quintic polynomials requires changing the game (i.e., expanding our range of possibilities beyond the arithmetic operations and radicals) – almost like a game of chess that you can only win by inventing new pieces.
The game is the game, Ben. Playas change, game stay the same. The Wire + math mashup soon?
Gotta confess: I haven’t watched the Wire yet. Which is pretty stupid, I realize. Pretty sure I could get deported for that.
Where do you stand on the “when will I ever use this in real life?” query? I mean, yeah, I do use Pythagoran (and lots of algebra and stuff), but I went into science.
I mean, I use the quadratic formula all the time… but I teach high school math, so bad example.
Strictly speaking, I suspect most of my students will never use most of the content I teach them after they finish schooling. (“Say you’re in the dairy aisle at a grocery store, and you find yourself taking the secant of an unknown angle whose sine is equal to its cosine and whose cotangent is positive…”) The valuable takeaways, I hope, are the cognitive skills – logic, reasoning, grit, drawing connections. Memorizing a formula might ease your way in the next class, but I bet it won’t be “useful” in the “real world” for >90% of them.
Do you have good instances of quadratic formula being useful “on the job”?
As a high school senior, most of the times I’ve used the quadratic formula have been in my physics classes (currently AP C Mech/E&M) and during the design process during build season on my robotics team.
There was a point a summer or two ago, when I was helping out with some renovation work, where I ended up having to use it for something (more trig than quadratics though, but nevertheless).
Hi,
I had expected a link to the quintic formula, although it is not algebraic. 🙂
And yes, I have used the quadratic formula many a times in Robotics, Programming and Physics; though I agree that’s not the case with most others.
Hmmm… good point. That will take a little digging. I can’t even remember if my Galois Theory class presented such a formula.
In the meantime: http://en.wikipedia.org/wiki/Quintic_function#Beyond_radicals
I wonder if Galois hadn’t stayed up all night working on the math, if he would have had faster reflexes, and better aim, and survived the duel?
Always a possibility, though I’ve heard he was a terrible shot anyway and his opponent almost never missed. However, it’s a fixed point in time now, so we couldn’t go back and change it even if we wanted to. 🙁
Certainly possible. I think the story as traditionally told is a little embellished. Ian Stewart has a good review of the actual history in his book on Galois Theory… as I recall, it was customary to miss on purpose, but his opponent didn’t…
There are also interesting applications of the quadratic formula in high school mathematics to solve challenging quadratic functions problems that go beyond the usual finding the roots, the discriminant, the vertex, the axis of symmetry, etc.
https://docs.google.com/file/d/0B6lw97EHbvfHM3hjVVhHYnBuQ1E/edit
Doesnt seventh become septic? :p
Exactly! Like a septic tank. Perhaps this is why we leave off at quintic…
Thank you for this, Ben! I hadn’t really thought about what treating these “mathematical greatest hits” like they were just found on a stone tablet somewhere – without any of the human story of how they were developed – does to our concept of math.
How many people have learned the proof that the square root of two is irrational, without hearing the story of how Hippasus was drowned for it? (Possibly apocryphal, but I think it’s a good enough story to be worth telling anyway)
One of my earliest experiences with the thrill of math was when I managed to derive the quadratic formula myself, completing the square on a general quadratic. I felt like Prometheus stealing fire from the gods and delivering it to earth. XD This wasn’t something you had to wait on elders to hand to you – you could go out and figure this stuff out for yourself!
As Katrina and Ben point out above, this derivation is taught in some classes. But I wonder if an even better approach would be to bring students just to the verge of the derivation, and encourage them to work it out on their own. I think this was my first time generalizing from step-by-step instructions to a universal formula, and it gave me a bit of a taste of what “real math” is like.
I love that story about the Pythagoreans drowning the man who would dare to suggest that not all numbers are rational. It nicely dramatizes how weird and sometimes intractable the irrationals can be.
I think you’re exactly right about the ideal way to teach the quadratic formula. First, you’d have kids complete the square until they gained comfort with that technique; then, perhaps recognize that the vertex of a quadratic is easy to identify; then, notice that the roots are symmetric about the vertex; then, derive the quadratic formula by completing the square, as you did.
It’s a shame we tend to leap straight to formulas and applying them, rather than letting formulas arrive as the satisfying payoff of a journey. The main problem, I think, is compression for time. It’s so much FASTER to just supply the formula…
Good point.
I think The Art of the Infinite by Ellen & Robert Kaplan was my first exposure to math as an ongoing human search, rather than a collection of facts. Bary Mazur’s Imagining Numbers did a bit of the same for complex numbers specifically. And Godel, Escher, Bach for incompleteness. When I want to tell someone a weird math story with some human interest, those are usually my main sources. Would love to find more. 🙂
Ooh, I’ll have to check those out! Some other suggestions:
Fermat’s Enigma, by Simon Singh
The Man Who Only Loved Numbers, by Paul Hoffman
The Drunkard’s Walk, Leonard Mlodinow
Ian Stewart, David Berlinski, John Allen Paulos, and Steven Strogatz have all written multiple good books, too.
http://en.wikipedia.org/wiki/Niels_Henrik_Abel#Contributions_to_mathematics <- You seem to have missed this gentleman. 🙂
Abel! Yeah, point taken. He should’ve uttered cooler dying words if he wanted me to tell his story, too.
The quadratic formula has a *great* visual derivation. Start with a square whose sides have length x. Next to it, put a rectangle of the same height x, and call its width B. Off to the side, make a blob of area C. The total area of all three shapes is x^2 + Bx + C.
Split the B-by-x rectangle vertically, to make two rectangles of height x and width B/2. Attach one of them to the side of the square, and rotate the other and attach it to the top of the square. Now we *almost* have a bigger square, but we’re missing a little corner piece that is a B/2 by B/2 square.
Okay, so “cut out” a B/2 by B/2 square from the C blob, and slide it over to literally complete the new large square. The large square has sides of length (x + B/2), and the blob-with-a-square-hole has area C – (B/2)^2.
I’m sure you see where this is going.
We want the total area to be zero, which means the area of the big square has to be (B/2)^2 – C. But the big square’s area is (x + B/2)^2. So we can just take square roots and get x + B/2 = ±√((B/2)² – C).
Of course, we started from x² + Bx + C = 0, when the general form of a quadratic is ax² + bx + c. But we can get from one to the other easily enough, as B = b/a and C = c/a. And the quadratic formula follows.
I promise it makes sense when you draw it. (And I really hope the non-ASCII characters like curly quotes, superscripts, and surds I used show up properly here…)
via Tristan Peñarroyo
The quadratic formula is also useful numerically, unlike the cubic and quartic formulas. There are several good ways to get accurate approximations of square roots, and once you have a square root, you’re only an addition and a division away from getting the two roots.
The cubic formula, with its two cube roots, each containing a square root (mercifully the same square root, at least), is laughably impractical, particularly since it can introduce nonreal numbers when you’re only interested in a real result. Newton’s approximation method is probably much more practical in the vast majority of cases for getting an accurate approximation of one or more roots of a cubic.
On the other hand, with a quadratic, Newton’s method is probably not worth the trouble, since you have to figure out an inital guess, and why bother when virtually every computer setup has square root functions anyway? (Some basic environments may not have them built in, but you’ll probably want one if you’re doing serious computations on them anyway.)
The quadratic formula is also useful for teaching. Specifically, students can use it to remember or derive the formula for the sum of the roots of a quadratic, or a nice way to find the x-coordinate of the vertex of a vertical parabola. Furthermore, it also clearly illustrates what the discriminant can tell you about the roots of a quadratic. The effort students must invest in memorizing the quadratic formula is actually an advantage, because in doing so, they memorize this other information that’s hidden inside it. The cubic and quartic formula do not confer comparable advantages that I’m aware of.
So, the quadratic formula isn’t just easier to learn than the quartic and cubic formulas. It’s also much more useful, partly by the virtue of its comparative simplicity!
Thanks for this – that’s a nice discussion of the advantages of the quadratic’s simplicity.