Hi,
Originally Posted by
alexmahone If the quadratic $\displaystyle ax^2 + bx + c$ 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 $\displaystyle x_0,x_1,x_2 $ such that
$\displaystyle \begin{cases}
ax_0^2+bx_0+c=r_0 & (1)\\
ax_1^2+bx_1+c=r_1 & (2)\\
ax_2^2+bx_2+c=r_2 & (3)\\
\end{cases}$
where $\displaystyle r_0,r_1$ and $\displaystyle r_2$ are also rational numbers. If you solve these three equations for $\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ you'll see that these numbers are rational too.