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.
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Correct Answer: Option D
Explanation:
An equation can be written as \(x^{2} - (\alpha + \beta)x + (\alpha\beta) = 0\)
Making the coefficient of \(x^{2}\) = 1 in the given equation, we have
\(x^{2} + \frac{5}{2}x + \frac{n}{2} = 0\)
Comparing, we have \(\alpha\beta = \frac{n}{2} = 2 \implies n = 4\)
An equation can be written as \(x^{2} - (\alpha + \beta)x + (\alpha\beta) = 0\)
Making the coefficient of \(x^{2}\) = 1 in the given equation, we have
\(x^{2} + \frac{5}{2}x + \frac{n}{2} = 0\)
Comparing, we have \(\alpha\beta = \frac{n}{2} = 2 \implies n = 4\)