If ${\mathrm{x}}^2+3\mathrm{x}+5=0$ and ${\mathrm{ax}}^2+\mathrm{bx}+\mathrm{c}=0$ have a common root and $\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{N}$ the minimum value of a +b +c  is

Roots of the equation x²+3x +5 =0 are non-real. 
Thus, given equations will have two common roots. 

$\begin{array}{ll}\Rightarrow & \frac{\mathrm{a}}{1}=\frac{\mathrm{b}}{3}=\frac{\mathrm{c}}{5}=\mathrm{\lambda }\\ \therefore & \mathrm{a}+\mathrm{b}+\mathrm{c}=9\mathrm{\lambda }\end{array}$

Thus, minimum value of a + b + c =9      $[\because \mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{N}]$