Let f(x) = $x^2+ax+b,a,b \in R$. If $f(1)+f(2)+f(3)=0$, then the nature of the roots of the equation $f(x) =0$ is ..... Let $f(x) = x^2+ax+b$ for $a,b \in \mathbb{R}$. If $f(1)+f(2)+f(3)=0$, then the nature of the roots of the equation $f(x) =0$  is
(A) real
(B) imaginary
(C) real and distinct
(D) equal roots
My attempts:
\begin{align}
  f(1) &= 1+a+b \\
  f(2) &= 4+2a+b \\
  f(3) &= 9+3a+b \\
  f(1)+f(2)+f(3)
  &= 1+a+b+4+2a+b+9+3a+b \\
  0 &= 14+6a+3b
\end{align}
now how can we take it further about the nature of the roots , whether the roots of $f(x)=0$ is imaginary or real , please help, thanks...

Brute force,
\begin{align}
  \Delta &= a^2-4b \\
  &=a ^2-4\left( -\frac{14+6a}{3} \right) \\
  &= a^2+8a \color{red}{+16} +\frac{56}{3} \color{red}{-16} \\
  &= (a+4)^2+\frac{8}{3} \\
  &> 0
\end{align}
