Find the sum of all possible positive integer values of b such that the quadratic equation 2x^2 + 5x + b = 0 has rational roots.

Find the sum of all possible positive integer values of b

$\begin{array}{|rcll|} \hline b & < & \frac{25}{8} \\ b & < & 3.125 \\\\ &\text{positive integer:}& b = 1 \qquad b = 2 \qquad b = 3\\\\ &\text{rational roots:}& b = 2 \qquad b = 3\\\\ && 2+3 =5\\ \hline \end{array}$

Find the sum of all possible positive integer values of b such that the quadratic equation

$2x^2 + 5x + b = 0$

has rational roots.

$\begin{array}{|rcll|} \hline && 2x^2 + 5x + b = 0 \\\\ x& =& \frac{-5\pm\sqrt{5^2-4\cdot 2\cdot b}}{2\cdot 2} \\\\ x &=& \frac{-5\pm\sqrt{25-8\cdot b}}{4} \\\\ 25-8\cdot b &\ge& 0 \\ 25 &\ge& 8\cdot b \\ 8\cdot b &\mathbf{<}& 25\\ \mathbf{b} &\mathbf{<}& \mathbf{\frac{25}{8}} \\ \hline \end{array}$