Using a number system other than the decimal

Question

Travelling in the Kingdom of Crystal Skull, Indiana Jones discovered a small box with notation $$ 3 x^2 - 25 x + 66 = 0 \implies x_1 = 4,\; x_2 = 9, $$ which seems to be incorrect. However after some reflection he realised a number system other than the decimal was used to determine the roots of the quadratic equation. What is the base of that number system?

Answer 1

If the base is $b>9$ (assuming the digits from $0$ to $9$ have the same meaning as in our standard decimal system), we have $$ 3\cdot 4^2-(2b+5)\cdot 4+6b+6=0 $$