If \(\alpha\) and \(\beta\) are the roots of the equation \(2x^{2} - 6x + 5 = 0\), evaluate \(\frac{\beta}{\alpha} + \frac{\alpha}{\beta}\).
A. \(\frac{24}{5}\)
B. \(\frac{8}{5}\)
C. \(\frac{5}{8}\)
D. \(\frac{5}{24}\)
Correct Answer: B
Explanation
\(2x^{2} - 6x + 5 = 0 \implies a = 2, b = -6, c = 5\)
\(\alpha + \beta = \frac{-b}{a} = \frac{-(-6)}{2} = 3\)
\(\alpha\beta = \frac{c}{a} = \frac{5}{2} \)
\(\frac{\beta}{\alpha} + \frac{\alpha}{\beta} = \frac{\beta^{2} + \alpha^{2}}{\alpha\beta}\)
\(\frac{(\alpha + \beta)^{2} - 2\alpha\beta}{\alpha\beta} = \frac{3^{2} - 2(\frac{5}{2})}{\frac{5}{2}}\)
= \(\frac{4}{\frac{5}{2}} = \frac{8}{5}\)
