Base Number Arithmetic

Question

I got no idea how to solve that problem. What I have done so far is I have randomly guessed numbers. I wanted to know a faster way to solve it in the future, just in case of some even more complex cases.

Answer 1

I.e. solve $(b-1)^2=3b+1$, $\,b\in\{4,5,\ldots,9\}$ (because $3$ has to be a valid digit in base $b$).

$$\iff b^2-2b+1=3b+1\iff b^2=5b\iff b=5$$