Why is this approximation of polynomial root so accurate?

I have an engineering problem where I have to find the smallest positive real root of a polynomial in $x$: $$Ax^5+Bx^3 - C = 0$$ Instead of solving numerically, I want simple approximative formulas ("design equations") that describe the behaviour. For that matter, I split the problem into two regimes:

• Large $x$: $\ \ \ \ x^5$ is dominant $\Longrightarrow \ \ Ax^5 \approx C \ \ \Longrightarrow \ \ x \approx \sqrt[5]{C/A} =: x_A$
• Small $x$: $\ \ \ \ x^3$ is dominant $\Longrightarrow \ \ Bx^3 \approx C \ \ \Longrightarrow \ \ x \approx \sqrt[3]{C/B} =: x_B$

The approximations work well below/above a certain threshold on $x$, but require an ugly case distinction. In order to avoid that and having something smoother than $$x \approx \min\{x_A, x_B\}$$ I tried Pythagorean-style combination of the inverses (inspired by the resistance of parallel circuits in electrical engineering) and found that $$x \approx 1\Big/ \ \left\|\left(\begin{matrix} 1/x_A \\ 1/x_B \end{matrix}\right)\right\|_4 = 1 \Big/ \sqrt[4]{1/x_A^4 + 1/x_B^4}$$ is a really, really good approximation. Pretty much perfect. That leads me to my actual question: Is it possible to argue why that 4-norm is such an amazing approximation? Does my initial polynomial have a certain structure that explains the high accuracy of my approximation?

Since I want to present/defend that stuff, I'd appreciate some sophistication.

Your equation can be rewritten

$$\left(\frac x{x_A}\right)^5+\left(\frac x{x_B}\right)^3-1=0.$$

With $y=\dfrac x{x_A}$ and $r=\dfrac{x_A}{x_B}$, a single parameter remains:

$$y^5+r^3y^3-1=0.$$

Then $$r=\sqrt[3]{\frac{1-y^5}{y^3}}=\frac1y\sqrt[3]{1-y^5},$$ can be compared to your approximation $$y=\frac1{\sqrt[4]{1+r^4}},$$i.e. $$r=\sqrt[4]{\frac1{y^4}-1}=\frac1y\sqrt[4]{1-y^4}.$$ The agreement is indeed excellent on a wide range [abscissa $y$, ordinate $r$]:

For small $y$, both behaviors are identical, $\dfrac1y$. For $y$ close to $1$, behaviors are similar, approximately $\sqrt[3]{5(1-y)}$ and$\sqrt[4]{4(1-y)}$, and a blend in between.

The good agreement is explained by the fact that the functional relations are similar, with the exponent $4$ intermediate between $3$ and $5$ (but the value $4$ has nothing "magical").