The roots of the quadratic equation \(2x^{2} - 5x + m = 0\) are \(\alpha\) and \(\beta\), where m is a constant. Find \(\alpha^{2} + \beta^{2}\) in terms of...

Answer Details

To find \(\alpha^{2} + \beta^{2}\), we need to use the following identities: \[\alpha + \beta = \frac{-b}{a}\quad\text{and}\quad \alpha\beta = \frac{c}{a}\] where a, b, and c are the coefficients of the quadratic equation \(ax^{2} + bx + c = 0\). In this case, the quadratic equation is \(2x^{2} - 5x + m = 0\), so a = 2, b = -5, and c = m. Using the identity \(\alpha + \beta = \frac{-b}{a}\), we have: \[\alpha + \beta = \frac{-(-5)}{2} = \frac{5}{2}\] Using the identity \(\alpha\beta = \frac{c}{a}\), we have: \[\alpha\beta = \frac{m}{2}\] Now, we can use the identity: \[\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta\] Substituting the values we found earlier, we get: \[\alpha^{2} + \beta^{2} = \left(\frac{5}{2}\right)^{2} - 2\left(\frac{m}{2}\right)\] Simplifying, we get: \[\alpha^{2} + \beta^{2} = \frac{25}{4} - m\] Therefore, the answer is \(\frac{25}{4} - m\), which is.