5. Let $b$ take even numbers from 1 to 11, and $c$ take any positive integer. Then the number of quadratic equations $x^{2}-b x+c=0$ that can be formed with two distinct real roots is ( ).
(A) 50
(C) 55
(C) 57
(D) 58
5.A.

When $b=2$, from $b^{2}-4 c=4-4 c>0$, we get $c<1$, so, $c=0$.
When $b=4$, from $b^{2}-4 c=16-4 c>0$, we get $c<4$, so, $c=0,1,2,3$.
When $b=6$, from $b^{2}-4 c=36-4 c>0$, we get $c<9$, so, $c=0,1, \cdots, 8$.
When $b=8$, from $b^{2}-4 c=64-4 c>0$, we get $c<16$, so, $c=0,1, \cdots, 15$.
When $b=10$, from $b^{2}-4 c=100-4 c>0$, we get $c<25$, so, $c=1,2, \cdots, 24$.
Therefore, the number of equations that satisfy the conditions is
$$
3+8+15+24=50 \text { (equations). }
$$