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Math Help - Quadratic function

  1. #1
    MHF Contributor alexmahone's Avatar
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    Quadratic function

    If the quadratic ax^2 + bx + c takes rational values for more than two rational values of x, then show that a, b, c are all rational numbers.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi,
    Quote Originally Posted by alexmahone View Post
    If the quadratic ax^2 + bx + c takes rational values for more than two rational values of x, then show that a, b, c are all rational numbers.
    We know that there exist three distinct rational numbers x_0,x_1,x_2 such that

    \begin{cases}<br />
ax_0^2+bx_0+c=r_0 & (1)\\<br />
ax_1^2+bx_1+c=r_1 & (2)\\<br />
ax_2^2+bx_2+c=r_2 & (3)\\<br />
\end{cases}

    where r_0,r_1 and r_2 are also rational numbers. If you solve these three equations for a, b and c you'll see that these numbers are rational too.
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