Find the sum of all possible positive integer values of $b$ such that the quadratic equation $2x^2 + 5x + b = 0$ has rational roots.
To have rational roots, the discriminant of the quadratic equation must be a perfect square. The discriminant is $5^2 - 4(2)(b) = 25 - 8b$. So I need to find all positive integers $b$ such that $25 - 8b$ is a perfect square. I can try some small values of $b$ and see if they work. If $b = 1$, then $25 - 8b = 17$, which is not a perfect square. If $b = 2$, then $25 - 8b = 9$, which is a perfect square. If $b = 3$, then $25 - 8b = 1$, which is a perfect square. If $b = 4$, then $25 - 8b = -7$, which is negative and not a perfect square. If $b > 4$, then $25 - 8b$ will be negative and not a perfect square. So the only possible values of $b$ are $2$ and $3$. The sum of these values is $2 + 3 = 5$. 