Recently, archaeologists unearthed a tablet of an ancient civilization in which the followingproblem was posed:Solve the equation 3x2 − 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 notationis used.

I have no idea on how to begin. Could someone please provide some pointers as to how to solve the problem. Thanks!

2. ## Re: Base of Quadratic

well let's plug the values in and see what we get

I assume the equation is actually this polynomial equals zero.

assume they use base $b$

$3x^2 - (2b+5)x + (6b+6) = 0$

Now plug in $4$ and $9$ (d means a decimal number where it could be confusing)

$3(16d) - (2b+5)(4) + (6b+6) = 0$

$48d - 8b - 20d + 6b + 6 = -2b + 34d = 0$

$b = 17d$

$3(81d) - (2b+5)(9) + (6b+6) = 0$

$243d - (18d)b - 45d + (6b+6) = 204d - (12d)b = 0$

$204d = (12d)b$

$b = 17d$

So it certainly looks like they use base $b=17$