If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} + 5x + n = 0\), such that \(\alpha\beta = 2\), find the value of n.

If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} + 5x + n = 0\), such that \(\alpha\beta = 2\), find the value of n.

Answer Details

If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} + 5x + n = 0\), then by the quadratic formula, we have: \[\alpha, \beta = \frac{-5 \pm \sqrt{5^{2} - 4(2)(n)}}{2(2)}\] Simplifying, we get: \[\alpha, \beta = \frac{-5 \pm \sqrt{25 - 8n}}{4}\] We are also given that \(\alpha\beta = 2\). Therefore, we have: \[\alpha\beta = \frac{-5 + \sqrt{25 - 8n}}{4} \cdot \frac{-5 - \sqrt{25 - 8n}}{4} = 2\] Expanding the left-hand side, we get: \[\frac{25 - (25 - 8n)}{16} = 2\] Simplifying, we get: \[\frac{8n}{16} = 2\] \[n = 4\] Therefore, the answer is the fourth option, 4.