Given that \(\alpha\) and \(\beta\) are the roots of an equation such that \(\alpha + \beta = 3\) and \(\alpha \beta = 2\), find the equation.

Answer Details

We are given that \(\alpha\) and \(\beta\) are the roots of the equation, and we are also given two conditions based on these roots: 1. \(\alpha + \beta = 3\) 2. \(\alpha \beta = 2\) To find the equation, we can use the fact that if \(\alpha\) and \(\beta\) are roots of the equation, then the equation can be written in factored form as: $$(x - \alpha)(x - \beta) = 0$$ Expanding this equation, we get: $$x^2 - (\alpha + \beta)x + \alpha\beta = 0$$ Substituting the values of \(\alpha + \beta\) and \(\alpha \beta\) from the given conditions, we get: $$x^2 - 3x + 2 = 0$$ Therefore, the equation with roots \(\alpha\) and \(\beta\) is: $$x^2 - 3x + 2 = 0$$ Hence, the correct option is (a) \(x^{2} - 3x + 2 = 0\).