When two quadratics seems to output the same values are they necessarily the same?

Joel is thinking of a quadratic and Eve is thinking of a quadratic. Both use $x$ as their variable. When they evaluate their quadratics for $x=1$, they get the same number. When they evaluate their quadratics for $x=2$, they both again get the same number. And when they evaluate their quadratics for $x=3$, they again both have the same result. Are their quadratics necessarily the same?

If $x=1$ results in $k_1$, $x=2$ in $k_2$ and $x=3$ in $k_3$ then three equations can be made by inputting these values in $ax^2+bx+c=k_i$

$a+b+c=k_1$

$4a+2b+c=k_2$

$9a+3b+c=k_3$

Using these equations we find the quadratic coefficients in terms of $k_i$:

$a=frac{k_1-2k_2+k_3}{2}$

$b=frac{-5k_1+8k_2-3k_3}{2}$

$c=3k_1-3k_2+k_3$

Now I’m stuck on what to do, hitherto I’ve followed hints but the answer still eludes me.