If the equation `x^(2) + kx+ 1 = 0 " has the roots "alpha and beta` , then what is the value of `(alpha + beta) xx ( alpha^(-1) +beta ^(-1))`

Options

A. `k^(2)`
B. ` 1/k^(2)`
C. ` 2 k^(2)`
D. ` 1/((2k^(2)))`

Explanation

To solve the problem, we need to find the value of \((\alpha + \beta) \times (\alpha^{-1} + \beta^{-1})\) given that the roots of the quadratic equation \(x^2 + kx + 1 = 0\) are \(\alpha\) and \(\beta\).

### Step 1: Identify the sum and product of the roots
From Vieta's formulas, for the quadratic equation \(x^2 + kx + 1 = 0\):
- The sum of the roots \(\alpha + \beta = -\frac{k}{1} = -k\).
- The product of the roots \(\alpha \beta = \frac{1}{1} = 1\).

### Step 2: Calculate \(\alpha^{-1} + \beta^{-1}\)
The expression for \(\alpha^{-1} + \beta^{-1}\) can be rewritten using the product of the roots:
\[ \alpha^{-1} + \beta^{-1} = \frac{1}{\alpha} + \frac{1}{\beta} = \frac{\beta + \alpha}{\alpha \beta} = \frac{\alpha + \beta}{\alpha \beta} \]
Substituting the values we found:
\[ \alpha^{-1} + \beta^{-1} = \frac{-k}{1} = -k \]

### Step 3: Substitute into the original expression
Now we substitute \(\alpha + \beta\) and \(\alpha^{-1} + \beta^{-1}\) into the expression:
\[ (\alpha + \beta) \times (\alpha^{-1} + \beta^{-1}) = (-k) \times (-k) = k^2 \]

### Final Answer
Thus, the value of \((\alpha + \beta) \times (\alpha^{-1} + \beta^{-1})\) is:
\[ \boxed{k^2} \]