jmc

By Vieta's, $a$, $b$, and $c$ are the solutions to the cubic equation $$x^3-2x^2-7x+14=0.$$ We group and factor as follows: $$\begin{array}{rl}{x}^3-2x^2-7x+14=0& =(x^3-7x)-(2x^2-14)\\ & =x(x^2-7)-2(x^2-7)\\ & =(x-2)(x^2-7).\end{array}$$ Thus, the three solutions are $x=2$, $x=\sqrt{7}$, and $x=-\sqrt{7}$. The largest of these numbers is $\boxed{\sqrt{7}}$.