smc

The left-hand side of the equation can be factored as $(x-b)(x-a+x-c)=(x-b)(2x-(a+c))$, from which it follows that the roots of the equation are $x=b$, and $x=\frac{a+c}{2}$. The sum of the roots is therefore $b+\frac{a+c}{2}$, and the maximum is achieved by choosing $b=9$, and $\{a,c\}=\{7,8\}$. Therefore the answer is $9+\frac{7+8}{2}=9+7.5=\boxed{16.5}.$