Quadratic function

**alexmahone** · October 24th 2008, 12:30 AM

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.

**flyingsquirrel** · October 24th 2008, 01:32 AM

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

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

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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