The most memorable math problem I've ever done

A decade later, one problem and its solution still haunt me…

May 18, 2023

Sometime in early 2013, I came across the following question:

What are all the solutions to \sqrt{5-x}=5-x^2 ?

To save you the trouble of thinking, the standard approach would be to square both sides and collect the terms, resulting in a quartic equation:


You can try evaluating the polynomial at all the factors of 20; none of them are zeros, so the roots of this equation must all be irrational. What do you do?

By some divine guidance, someone came up with the following approach: square the original equation, but collect the terms differently:



We can now see the equation is a quadratic in 5, of the form a\cdot5^2+b\cdot5+c=0, with a=1 , b=-2x^2-1 , and c=x^4+x . We can then “solve for 5” with the quadratic equation:

\begin{aligned} 5 &=\dfrac{2x^2+1\pm\sqrt{(-2x^2-1)^2-4(x^4+x)}}{2} \\ &=\dfrac{2x^2+1\pm\sqrt{4x^2-4x+1}}{2} \\ &=\dfrac{2x^2+1\pm(2x-1)}{2}. \end{aligned}

This leads to two quadratic equations: 10=2x^2+1+2x-1 and 10=2x^2+1-(2x-1) . These can in turn be solved with the (normal) quadratic equation, leading to four roots, two of which satisfy the original equation: \frac{1}{2}(1-\sqrt{17}) and \frac{1}{2}(\sqrt{21}-1) .

There are a number of other, more “normal” or reasonable” approaches to the problem, which you can find if you Google “quadratic in five.” But even though those solutions have their own elegance, none of them, or for that matter, no other math I’ve seen in my life, have made as much a mark on me as the quadratic in five.

Ten years after seeing this problem and its solution, I still wonder, probably on a monthly basis: who comes up with this stuff?? Maybe Young Thug? I feel like only someone who could write lines like “I put that crack in my crack,” who buys a mansion purely because he liked the elevator, whose delivery can alternate from syrupy croons to guttural grunts (“I just bought a CUban \text{L}^{\text{I}^{\text{NK}}} ?? FLOODED!! with B I G rocks”), and all the above while coming off as spontaneous and effortless, could possibly make the galactic leap of thought to say, “let’s use the quadratic equation to solve for five.”

Will I ever create something of such beauty, of such surprise? I feel I’ve spent much of my life in search of the great beauty, hoping I could use my knowledge and my craft to make people laugh or gasp in shock.

It was once more an open question to me, whether I’d succeed. Back then, I was more preoccupied with the rarified nature of work in the technical fields; only those who’ve ensconced themselves in the same ivory tower of theory that you have, can really appreciate whatever exhibit you hold at its top. But over time, I began to believe in something that implied a far more fundamental problem: the most astonishingly beautiful works can never be fully appreciated by their creators, because by the time one has the intuition, skill, and inspiration to make something so great, it probably only feels quite natural and obvious to do so.

At least for myself personally, my appreciation of my own work hasn’t budged at all with regards to the amount of respect or praise my work has garnered from others. Maybe this is an obvious corrollary to the fact that your expectations adjust to your situation, that it’s very difficult to be very happy or very sad for too long of a time, but it was still a rather disappointing finding for me. The satisfaction Young Thug gets from spitting a golden bar probably isn’t so different than what we feel when we come up with some shitty pun that no one will ever remember. Claude Shannon discovering information theory was probably about as cool to him as one of us cracking a particularly tough homework problem.

I think this is also why, despite the extreme emphasis mathematics puts on generalization, I’ve never spent much time investigating the broader implications of the quadratic in five trick. Does a quadratic in six exist? What interesting structure is present in the discriminant that allows for this to happen? My math education stopped very early, and while I may not have the background to place the “quadratic in five” trick within the greater mathematical cosmos, I’m sure some people do, and to them, the solution is \text{“a specific instance} \text{of the AAA Theorem”} or a \text{“natural consequence} \text{of BBB} \text{applied to CCC."} I don’t want to be that person; if I don’t need that level of mathematical expertise in my life, I’d rather go without it and preserve the beauty that comes from my naivete. Meanwhile, for the things I create myself, I’m gradually acclimating to viewing creation as a more humanistic process whose ultimate rewards are the effects such work can bring to others. We continue to do what we do because we love it, and because we hope that someday our work can maybe move others the same way our heroes have done to us.