## Base of Number

If the number $86$ in base $10$ is represented as $321$ in base $b$, what is the base-ten representation of the number $123$ in base $b$?
Source: NCTM Mathematics Teacher, November 2006

SOLUTION
$3b^2+2b+1=86$
$3b^2+2b-85=0$
Solving the quadratic equation for $b$ yields $b=5$.
$123_b=1\cdot 5^2+2\cdot 5+3$
$=25+10+3$
$=38_{10}$

Answer: $38$