<s>[INST] 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...
 [/INST] We have three real terms summing up to $0$.
They can't be all zero as a quadratic has at most two zeroes.
Hence at least one term is positive and at least one term is negative, hence the roots must be distinct real roots.
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