If the equations x2+2x+3=0 and ax2+bx+c=0, \[a,\,b,\,c\, \in \,R,\] have a common root, then \[a\,:b\,:c\,\] is
Solutions · 1
Official solution
Given equations are
\[\,\,\,\,\,\,\,\,\,\,\,\,{x^2} + 2x + 3 = 0\,\,\,\,\,...\left( i \right)\]
\[\,\,\,\,\,\,\,\,\,\,\,\,a{x^2} + bx + c = 0\,\,\,...\left( {ii} \right)\]
Roots of equation (i) are imaginary roots.
According to the question (ii) will also have both roots same as (i).
Thus \[{a \over 1} = {b \over 2} = {c \over 3} = \lambda \left( {say} \right)\]
\[ \Rightarrow a = \lambda ,b = 2\lambda ,c = 3\lambda \]
Hence, required ratio is 1:2:3