5. Let $A=\{1,2, \cdots, 10\}$. If the equation $x^{2}-b x- c=0$ satisfies $b, c \in A$, and the equation has at least one root $a \in A$, then the equation is called a "beautiful equation". The number of beautiful equations is ( ).
(A) 8
(B) 10
(C) 12
(D) 14
5.C.

From the problem, we know that the two roots of the equation are both integers and one is positive while the other is negative. When one root is -1, there are 9 beautiful equations that meet the requirements; when one root is -2, there are 3 beautiful equations that meet the requirements. Therefore, there are a total of 12.