Consider the quadratic equation $f(x)=ax^{2}+bx+c$. Where $a, b, c\in R$ and $a\neq 0$ such that $f(x)=f(2-x)$ for all real number $x$. The sum of the roots of $f(x)$ is :

The condition $f(x) = f(2-x)$ for all real numbers indicates that the quadratic function has a vertical axis of symmetry. The axis of symmetry for a parabola is located exactly halfway between its roots, meaning it can be found by taking the average of the roots. Setting the axis equation $x = (root 1 + root 2) / 2$ equal to the midpoint derived from the symmetry condition, we have $(root 1 + root 2) / 2 = (x + 2 - x) / 2$. Solving this simple equation shows the sum of the roots is 2.

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