If `a ,b ,c` are non zero`(a b c^2)x^2+3a^2c x+b^2c x-6a^2-a b+2b^2=0` are rational.

Explanation

`(abc)^(2) x^(2) + 3x^(2) cx + b^(2) cx - 6a^(2) - ab + 2b^(2) = 0`
For roots to be rational, D sholud be a perfect square. Thus ,
`D = (3a^(2) c+b^(2)c)^(2) + 4(abc)^(2) (6a^(2) _ ab - 2b^(2))`
`= c^(2) [9a^(4) + b^(4) + 10a^(2)b^(2)+24a^(3) b - 8ab^(3)]`
`= c^(2) [9a^(4) + b^(4) + 16a^(2)b^(2)-6a^(2)b^(2)+24a^(3) b - 8ab^(3)]`
`= c^(2) (3a^(2) - b^(2) + 4ab)^(2)`
`= [c (3a^(2) - b^(2) + 4ab)]^(2)`
Hence proved .