If α and β are the roots of the equation 3x\(^2\) + 5x - 2 = 0, find the value of 1/α + 1/β
The correct answer is D. 5/2
Given the quadratic equation \(3x^2 + 5x - 2 = 0\), let's first find the roots (\(\alpha\) and \(\beta\)) using the quadratic formula:
The quadratic formula states that for an equation of the form \(ax^2 + bx + c = 0\), the solutions are given by:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
In this case, \(a = 3\), \(b = 5\), and \(c = -2\). Plugging in these values:
\[\alpha = \frac{-5 + \sqrt{5^2 - 4 \cdot 3 \cdot (-2)}}{2 \cdot 3} = \frac{-5 + \sqrt{49}}{6} = \frac{-5 + 7}{6} = \frac{2}{6} = \frac{1}{3}\]
\[\beta = \frac{-5 - \sqrt{5^2 - 4 \cdot 3 \cdot (-2)}}{2 \cdot 3} = \frac{-5 - \sqrt{49}}{6} = \frac{-5 - 7}{6} = \frac{-12}{6} = -2\]
Now, we need to find the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\):
\[\frac{1}{\alpha} + \frac{1}{\beta} = \frac{1}{\frac{1}{3}} + \frac{1}{-2} = 3 - \frac{1}{2} = \frac{5}{2}\]
Therefore, the value of \(\frac{1}{\alpha} + \frac{1}{\beta}\) is \(\frac{5}{2}\).
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