<s>[INST] Roots of $abc^2x^2 + 3a^2 c x + b^2cx-6a^2 -ab +2b^2 = 0$ are rational We have to show that roots of $$abc^2x^2 + (3a^2 c  + b^2c)x-6a^2 -ab +2b^2 = 0$$ are rational. 
This can be possible if the discriminant is a perfect square. SO I tried converting it into perfect square but failed:
$$\text{Discriminant}=(3a^2c+b^2c)^2-4abc^2(2b^2-6a^2-ab)\\
c^2(9a^4+b^4+10a^2b^2+24a^3b-8ab^3)$$
I cannot proceed please help! Thanks!
 [/INST] The product of the roots is
$$
\frac{-6a^2 -ab +2b^2}{abc^2}
= \frac{(b - 2 a) (3 a + 2 b)}{a b c^2}
$$
which immediately suggests what the roots are.
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