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 m.
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Correct Answer: Option A
Explanation:
\(2x^{2} - 5x + m = 0\)
\(a = 2, b = -5, c = m\)
\(\alpha + \beta = \frac{-b}{a} = \frac{5}{2}\)
\(\alpha \beta = \frac{c}{a} = \frac{m}{2}\)
\(\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta\)
= \((\frac{5}{2})^{2} - 2(\frac{m}{2}) \)
= \(\frac{25}{4} - m\)
\(2x^{2} - 5x + m = 0\)
\(a = 2, b = -5, c = m\)
\(\alpha + \beta = \frac{-b}{a} = \frac{5}{2}\)
\(\alpha \beta = \frac{c}{a} = \frac{m}{2}\)
\(\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta\)
= \((\frac{5}{2})^{2} - 2(\frac{m}{2}) \)
= \(\frac{25}{4} - m\)