Find the sum of all possible positive integer values of b such that the quadratic equation \(2x^2 + 5x + b = 0\) has rational roots.
+26388 Find the sum of all possible positive integer values of b
\(\begin{array}{|rcll|} \hline b & < & \frac{25}{8} \\ b & < & 3.125 \\\\ &\text{positive integer:}& b = 1 \qquad b = 2 \qquad b = 3\\\\ &\text{rational roots:}& b = 2 \qquad b = 3\\\\ && 2+3 =5\\ \hline \end{array}\)
+26388 Find the sum of all possible positive integer values of b such that the quadratic equation
\(2x^2 + 5x + b = 0\)
has rational roots.
\(\begin{array}{|rcll|} \hline && 2x^2 + 5x + b = 0 \\\\ x& =& \frac{-5\pm\sqrt{5^2-4\cdot 2\cdot b}}{2\cdot 2} \\\\ x &=& \frac{-5\pm\sqrt{25-8\cdot b}}{4} \\\\ 25-8\cdot b &\ge& 0 \\ 25 &\ge& 8\cdot b \\ 8\cdot b &\mathbf{<}& 25\\ \mathbf{b} &\mathbf{<}& \mathbf{\frac{25}{8}} \\ \hline \end{array}\)
+33653 So the positive integers that satisfy the conditions are: b = 2 [x = -1/2 and -2] and b = 3 [x = -1 and -3/4]
Hence the requested sum is 2 + 3 → 5
+26388 Find the sum of all possible positive integer values of b
\(\begin{array}{|rcll|} \hline b & < & \frac{25}{8} \\ b & < & 3.125 \\\\ &\text{positive integer:}& b = 1 \qquad b = 2 \qquad b = 3\\\\ &\text{rational roots:}& b = 2 \qquad b = 3\\\\ && 2+3 =5\\ \hline \end{array}\)
