A wrinkle that came my way via my excellent friend @realityminus3 (Elizabeth A. Williams in real life); I paraphrase slightly:

Show that $\frac{12x^2 + x - 16\sqrt{x}}{4x\sqrt{x}} = 0$ (for $x>0$) can be rearranged as $x = \br{\frac{4}{3} - \frac{\sqrt{x}}{12}}^{2/3}$.

Now, Elizabeth is the best godsdamned maths tutor in her geographical area ((Cardiff)), and is perfectly capable of doing that rearrangement – the question is why that’s the chosen rearrangement.

For the sake of form, let’s go through it:

Since $x > 0$, I can multiply both sides by $4x\sqrt{x}$ to get $12x^2 + x - 16\sqrt{x}=0$. Here is where the paths diverge:

  • You can divide by $\sqrt{x}$ to get $12x^{3/2} + x^{1/2} - 16 =0$, so $x^{3/2} = \frac{16 - x^{1/2}}{12}$, or $x = \br{\frac{4}{3}-\frac{\sqrt{x}}{12}}^{2/3}$
  • Or you can rearrange in place to get $12x^{2} = 16\sqrt{x} - x$, giving $x^2 = \frac{4}{3}\sqrt{x} - \frac{x}{12}$ and $x = \br{\frac{4}{3}\sqrt{x}-\frac{x}{12}}^{1/2}$.

The first is what they ask for in the question; the second is what felt obvious to Elizabeth – and, when put through the iteration hand-wringer, converges on the correct root.

So why does the question prefer the first?

There are a couple of reasons I can point to ((and I stress that I don’t find Elizabeth’s preferred answer at all unreasonable)) here:

  • It makes sense to remove as much mathematical clutter as possible: since there’s a factor of $\sqrt{x}$ throughout, we may as well remove it;
  • If you plot the graphs, the paper’s right hand side (in blue) is much flatter than Elizabeth’s (in red), with the consequence that it converges more quickly.

Now, I don’t like fixed-point iteration as a method at all (give me Newton-Raphson any day, much less guesswork and jiggery-pokery), and I’d be hard pushed to explain why it works, let alone why it’s supposed to be interesting. I suppose the interesting thing here is that there are multiple rearrangements that work. I wonder if there are others? ((But not enough to go looking for them.))

* Many thanks to Elizabeth A. Williams, Ben Sparks (@sparksmaths) and Paul Harrison (@singinghedgehog) for their contributions to the conversation about this.

* Edited 2021-06-14 to put some missing powers back in. Thanks, Adam!