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.

2. Hi,
Originally Posted by alexmahone
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

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

where $r_0,r_1$ 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.