If exactely two integers lie between the roots of equatin `x ^(2) +ax-1=0.` Then integral value (s) of 'a' is/are :

Options

A. `-1`
B. `-2`
C. 1
D. 2

Answer

To solve the problem, we need to find the integral values of \( a \) such that there are exactly two integers lying between the roots of the quadratic equation \( x^2 + ax - 1 = 0 \).

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

### Step 2: Calculate the difference between the roots
Let the roots be \( r_1 \) and \( r_2 \):
\[ r_1 = \frac{-a + \sqrt{a^2 + 4}}{2}, \quad r_2 = \frac{-a - \sqrt{a^2 + 4}}{2} \]
The difference between the roots is:
\[ r_1 - r_2 = \sqrt{a^2 + 4} \]

### Step 3: Determine the condition for two integers between the roots
For there to be exactly two integers between the roots \( r_1 \) and \( r_2 \), the distance between the roots must be more than 2 but less than 4:
\[ 2 < r_1 - r_2 < 4 \]
Substituting the expression for the difference:
\[ 2 < \sqrt{a^2 + 4} < 4 \]

### Step 4: Solve the inequalities
1. **First Inequality**:
\[ \sqrt{a^2 + 4} > 2 \]
Squaring both sides:
\[ a^2 + 4 > 4 \implies a^2 > 0 \implies a \neq 0 \]

2. **Second Inequality**:
\[ \sqrt{a^2 + 4} < 4 \]
Squaring both sides:
\[ a^2 + 4 < 16 \implies a^2 < 12 \implies -\sqrt{12} < a < \sqrt{12} \]
Simplifying gives:
\[ -2\sqrt{3} < a < 2\sqrt{3} \]
Since \( \sqrt{3} \approx 1.732 \), we have:
\[ -3.464 < a < 3.464 \]

### Step 5: Combine the conditions
From the inequalities, we have:
1. \( a \neq 0 \)
2. \( -2\sqrt{3} < a < 2\sqrt{3} \)

### Step 6: Identify integral values of \( a \)
The integral values of \( a \) that satisfy \( -3.464 < a < 3.464 \) are:
\[ -3, -2, -1, 1, 2, 3 \]
However, since \( a \neq 0 \), we exclude 0.

### Final Answer
The integral values of \( a \) are:
\[ -3, -2, -1, 1, 2, 3 \]