Example 5. When $b \neq c$, if the parabolas $y=x^{2}+b x+c$ and $y=-x^{2}+c x+b$ are tangent to each other, then $b, c$ satisfy the relation $\qquad$
(1992, Taiyuan City Junior High School Mathematics Competition)
Solving for the tangency of two parabolas, we get the system of equations
$$
\left\{\begin{array}{l}
y=x^{2}+b x+c, \\
y=-x^{2}+c x+b .
\end{array}\right.
$$

Eliminating \( y \) and simplifying, we get
$$
2 x^{2}+(b-c) x+c-b=0 .
$$

By the property of tangency, this equation has two identical real roots, so
$$
\Delta=(b-c)^{2}-4 \times 2(c-b)=0 .
$$

Since \( b \neq c \), we have \( b=c+8 \).