# 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 $a + b = 1$