If the difference between the roots of equation `x^(2) + kx + 1 = 0` is striclly less than `sqrt(5)`. Where |k| `ge 2`. Then k can be any element of the interval.

Options

A. `(-3 ,-2) cap (2,3)`
B. `(-3,3)`
C. `(-3,-2) cap (2,3)`
D. None of these

Answer

To solve the problem, we need to analyze the quadratic equation \( x^2 + kx + 1 = 0 \) and determine the conditions under which the difference between its roots is strictly less than \( \sqrt{5} \) while also considering the constraint \( |k| \geq 2 \).

### Step-by-Step Solution:

1. **Identify the Roots**:
The roots of the quadratic equation \( x^2 + kx + 1 = 0 \) can be found using the quadratic formula:
\[ \alpha, \beta = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = k \), and \( c = 1 \). Thus, the roots are:
\[ \alpha, \beta = \frac{-k \pm \sqrt{k^2 - 4}}{2} \]

2. **Calculate the Difference Between the Roots**:
The difference between the roots \( \alpha - \beta \) is given by:
\[ \alpha - \beta = \frac{\sqrt{k^2 - 4}}{1} = \sqrt{k^2 - 4} \]

3. **Set Up the Inequality**:
According to the problem, we need the difference to be strictly less than \( \sqrt{5} \):
\[ \sqrt{k^2 - 4} < \sqrt{5} \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides of the inequality:
\[ k^2 - 4 < 5 \]
This simplifies to:
\[ k^2 < 9 \]

5. **Solve the Inequality**:
Taking the square root of both sides gives:
\[ -3 < k < 3 \]

6. **Combine with the Given Condition**:
We also have the condition \( |k| \geq 2 \), which translates to:
\[ k \leq -2 \quad \text{or} \quad k \geq 2 \]

7. **Find the Intersection of Intervals**:
Now, we need to find the intersection of the intervals:
- From \( -3 < k < 3 \), we have two subintervals: \( -3 < k < -2 \) and \( 2 < k < 3 \).
- From \( |k| \geq 2 \), we have \( k \leq -2 \) or \( k \geq 2 \).

Thus, the valid intervals for \( k \) are:
- \( -3 < k < -2 \)
- \( 2 < k < 3 \)

8. **Final Answer**:
Therefore, the values of \( k \) can be any element of the intervals:
\[ (-3, -2) \cup (2, 3) \]