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) \(x^{2} - 3x + 2 = 0\).