Recently, archaeologists unearthed a tablet of an ancient civilization in which the followingproblem was posed: Solve the equation $3x^2 − 25x + 66.$ Farther down the tablet, the solutions $x = 4$ and $x = 9$ were offered. What is the base for this civilization’s number system? You can assume that the symbols $2, 3,4, 5, 6,$ and $9$ have the same meaning for this civilization as for us, and that positional notation is used.
I am unable to solve this problem. I believe the answer is $17$, but that was found through guess-and-check. What is the algebraic and concrete way to proceed with this problem.
Hint: let $\,b \gt 9\,$ be the ancients' base, then $\,25_{b}=2b+5\,$ and $\,66_{b}=6b+6\,$. Substituting $\,x=4\,$ in the equation then gives $\,3\cdot16-4\cdot(2b+5)+(6b+6) = 0 \iff b=17\,$. Note that the other root $\,x=9\,$ has been neither used nor verified thus far.