If equations `ax^(2)+bx+c=0` (where `a,b,c epsilonR` and `a!=0`) and `x^(2)+2x+3=0` have a common root, then show that `a:b:c=1:2:3`

Explanation

Given equations are
`ax^(2)+bx+c=0`…….i
and `x^(2)+2x+3=0`………..ii
Clearly, roots of Eq. (ii) are imaginary, since Eqs (i) and (ii) have a commoni root. Therefor, common root must be imaginary and hence both roots will be common.
Therefore, Eqs. (i) and (ii) are identical.
`:.a/1=b/2=c/3` or `a:b:c=1:2:3`