Finding the unknown base

Statement

The number $81$ (in base ten) is written $213$ in another base $b$. What is the base $b$?

Solution

The string $(213)_b$ equals $2b^2 + b + 3$. Set it equal to $81$: $2b^2 + b + 3 = 81 \implies 2b^2 + b - 78 = 0.$ The discriminant is $1 + 624 = 625 = 25^2$, so $b = \frac{-1 + 25}{4} = \boxed{6}.$ Check: $2\cdot 36 + 6 + 3 = 81$. $\checkmark$