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 $\qquad$.
5. 12 Detailed Explanation: From the problem, we know that the two roots of the equation are integers, one positive and one 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. In total, there are 12.