(Easy) question about second degree equations

Question

Let $x^2-ax+b=0$ be a second degree equation, where $x, a$ and $b$ are all positive integers greater than 0. Then, for a given $a$,

1. Can we calculate how many different values of $b$ are there?
2. Can we calculate the divisors of $b$? And its primality?
3. Is there any other "important" characteristic about $b$ that we could obtain?
4. (Would it be possible to know the exact values of $b$?)

Answer 1

if x, a and b are all integers. Then the roots of the polynomial are integers.

$(x+r_1)(x+r_2) = x^2 + ax + b = 0$

In order for $a$ and $b$ to both be greater than $0,$ then $r_1, r_2$ must be greater than 0.

But that would indicate that $x<0$

There is no polynomial that meets the required conditions.