Question

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`?

Answer 1

`a+b+c = 21`

Explanation:

Suppose `7+x_0`, `x_0!=0` is a root of `f(x)`. By the given property, then, we have

`0 = f(7+x_0) = f(7-x_0)`.

Then, as `x_0!=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!=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+(7+x_0)+(7-x_0) = 21`