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Thread: Base of Quadratic

  1. #1
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    Base of Quadratic

    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 civilizationís 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!
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  2. #2
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    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 $b=17$
    Thanks from HallsofIvy
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