The roots of the equation \(2x^{2} + kx + 5 = 0\) are \(\alpha\) and \(\beta\), where k is a constant. If \(\alpha^{2} + \beta^{2} = -1\), find the values o...

The sum of the roots of the quadratic equation \(ax^{2} + bx + c = 0\) is given by the formula \(-\frac{b}{a}\), and the product of the roots is given by the formula \(\frac{c}{a}\). Therefore, for the quadratic equation \(2x^{2} + kx + 5 = 0\), we have: \[\alpha + \beta = -\frac{k}{2}\] \[\alpha \beta = \frac{5}{2}\] Squaring the first equation, we get: \[(\alpha + \beta)^{2} = \alpha^{2} + 2\alpha\beta + \beta^{2} = \frac{k^{2}}{4}\] Substituting the given value of \(\alpha^{2} + \beta^{2} = -1\), we have: \[-1 + 2\alpha\beta = \frac{k^{2}}{4}\] Substituting the value of \(\alpha \beta = \frac{5}{2}\), we have: \[-1 + 2\left(\frac{5}{2}\right) = \frac{k^{2}}{4}\] Simplifying the left-hand side, we get: \[4 = \frac{k^{2}}{4}\] Multiplying both sides by 4, we get: \[16 = k^{2}\] Taking the square root of both sides, we get: \[k = \pm 4\] Therefore, the values of k are \(\pm 4\).