If the difference between the roots of the equation `x^(2)+ax+1=0` is less than `sqrt(5)`, then the set of possible values of a is (i) `(3,oo)` (ii) `(-oo,-3)` (iii) `(-3,3)` (iv) `(-3,oo)`

Options

A. `(3,oo)`
B. `(-oo,-3)`
C. `(-3,3)`
D. `(-3,oo)`

Answer

To solve the problem, we need to find the set of possible values of \( a \) such that the difference between the roots of the quadratic equation \( x^2 + ax + 1 = 0 \) is less than \( \sqrt{5} \).

### Step-by-Step Solution:

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

2. **Find the difference between the roots**:
The difference between the roots \( \alpha \) and \( \beta \) is given by:
\[ |\alpha - \beta| = \left| \frac{\sqrt{a^2 - 4}}{2} - \left(-\frac{\sqrt{a^2 - 4}}{2}\right) \right| = \frac{\sqrt{a^2 - 4}}{1} \]
Thus, we have:
\[ |\alpha - \beta| = \sqrt{a^2 - 4} \]

3. **Set up the inequality**:
According to the problem, the difference between the roots must be less than \( \sqrt{5} \):
\[ \sqrt{a^2 - 4} < \sqrt{5} \]

4. **Square both sides**:
To eliminate the square root, we square both sides of the inequality:
\[ a^2 - 4 < 5 \]

5. **Rearrange the inequality**:
Adding 4 to both sides gives:
\[ a^2 < 9 \]

6. **Take the square root**:
Taking the square root of both sides, we find:
\[ -3 < a < 3 \]

7. **Conclusion**:
The set of possible values of \( a \) is:
\[ a \in (-3, 3) \]

### Final Answer:
Thus, the correct option is:
(iii) \((-3, 3)\)