As a background I know nothing about how the golden ratio is used in actual mathematics or any formulae and such (only seen it used in a few examples I've seen online).
But then while messing on the Desmos graphing calculator with just graph intersections/simultaneous equations, i see that:
$x^2 + x = x^3\, $ when $\, x = 1.618033988749894\, $ (i.e: the golden ratio)
I'm really curious because I've never used the golden ratio in maths before, and now like the mathematicians of the past saw it come up randomly.
Edit: feel a bit stupid now, considering that if I just did a tiny bit more searching I would have gotten it probably very quickly. But thanks everyone for the simple and easy answers!
In the given case, the short aswer could be that it is well known that the only positive solution of $\phi^2-\phi-1=0$ is given by $\frac{1+\sqrt{5}}{2}$ so that a solution of $\phi^2-\phi-1=0$ is also a solution of $\phi^3-\phi^2-\phi=0 \Rightarrow \phi\cdot(\phi^2-\phi-1)=0$.
Now, we observe that the golden ratio appears often in graph theory optimization problems, as an example just take a look at what happens in this paper of mine, pp. $163-165$: Golden ratio in a random optimization problem.
Basically, the problem there was as follows: Let the grid $\{\{0,1\} \times \{0,1\} \times \{0,1\}\} \subset \mathbb{R}^3$ be given. Our goal is to find a polygonal chain characterized by the minimum link-length (i.e., with $6$ edges) which subtends the minimum volume Axis-Aligned Bounding Box.
The volume of the optimum AABB is "surprisingly" given by $\left[\frac{1-\phi}{2}+\epsilon, \frac{1+\phi}{2} \right] \times \left[\frac{4\cdot \phi \cdot \epsilon}{1-\phi+2 \cdot \epsilon}, 1+\phi\right] \times \left[0, \frac{1+\phi}{2}\right]$, where $\mathbb{R} \ni \epsilon \rightarrow 0$.