Find roots of a function

Question

$f$ is a function defined on the whole real line which has the property that $f(1+x)=f(2-x)$ for all $x$. Assume that the equation $f(x)=0$ has $8$ distinct real roots. Find the sum of these roots. I don't get this problem at all, just that if $x=\frac{1}{2}$, the property works. Can someone explain this to me?

Answer 1

Put $x\to1/2+x$,

$f(3/2+x)=f(3/2-x)$

This implies that function is symmetric about $3/2$. Hence, the root cannot be $3/2$ as then there would be odd roots.

Hence, roots are of form $3/2\pm c$