# If f(7+x)=f(7-x), forall x in RR and f(x) has exactly three roots a,b,c, what is the value of a+b+c?

Oct 22, 2016

$a + b + c = 21$

#### Explanation:

Suppose $7 + {x}_{0}$, ${x}_{0} \ne 0$ is a root of $f \left(x\right)$. By the given property, then, we have

$0 = f \left(7 + {x}_{0}\right) = f \left(7 - {x}_{0}\right)$.

Then, as ${x}_{0} \ne 0$, we have a second distinct root as $7 - {x}_{0}$.

We can clearly see that any root of the form $7 - x$ with $x \ne 0$ will result in a second root (the reflection of the first root about the line $x = 7$). Thus, to have a single third root, it must be at $7$.

Thus, our three roots are $7 , 7 + {x}_{0}$, and $7 - {x}_{0}$. Adding them, we get

$7 + \left(7 + {x}_{0}\right) + \left(7 - {x}_{0}\right) = 21$