imc

Split the equation into two cases, where the value inside the absolute value is positive and nonpositive. Case 1: The equation yields $x^2-12x+34=2$, which is equal to $(x-4)(x-8)=0$. Therefore, the two values for the positive case is $4$ and $8$. Case 2: Similarly, taking the nonpositive case for the value inside the absolute value notation yields $x^2-12x+34=-2$. Factoring and simplifying gives $(x-6{)}^2=0$, so the only value for this case is $6$. Summing all the values results in $4+8+6=\boxed{18}$.