math

# A Recurring Problem

A recurring problem I keep running into is solving

$$ax+b=x^c$$

As far as I can tell, there is no general closed-form solution for $x$, so I'm maintaining this page as a compilation of things I know about it.

## Closed-Form Solutions

The closed-form solutions for $a$ and $b$ are obvious. $x$ has a few closed-form solutions when $c$ is a particular fixed value:

 $c$ solution $$c=-0.5$$ some wonky cubic solutions $$c=0$$ $$x = \frac{1-b}{a}$$ $$c=0.5$$ $$\frac{1-2ab \pm \sqrt{1 - 4ab}}{2a^2}$$ $$c=1$$ $$x = \frac{b}{1-a}$$ $$c=2$$ $$\frac{a \pm \sqrt{a^2+4b}}{2}$$

## Responsiveness

We know that

$$x^c-ax-b=0$$

so

$$\left( c x^{c-1} - a \right) \delta x - \left( x \right) \delta a - \delta b = 0$$

so

$$\delta x = \frac{ x \cdot \delta a + \delta b}{c x^{c-1} - a}$$

In particular, at $x = 0$, we have

$$\delta x = \frac{\delta b}{a}$$

and at $x = 1$, we have

$$\frac{\delta a + \delta b}{c - a}$$

## Other Properties

• If $x=0$, then $b = 0$
• If $x=1$, then $a + b = 1$