Let the roots be $\alpha, \beta$.
$$
\begin{array}{l}
\alpha+\beta=c, \alpha \beta=b=30, \alpha-\beta=1 \\
(\alpha-\beta)^{2}=1 \\
\Rightarrow(\alpha+\beta)^{2}-4 \alpha \beta=1 \\
c^{2}-4 \times 30=1 \\
\Rightarrow c=11 \text { or }-11
\end{array}
$$

The greatest possible value of $c=11$
I1.3 If the equation $x^{2}-c x+b=0$ has two distinct real roots and their difference is 1, determine the greatest possible value of the sum of the roots, $c$.