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

Options

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

Answer

To solve the equation \( x^2 + ax + (a + 1) = 0 \) and find the integral values of \( a \) such that exactly two integers lie between the roots, we can follow these steps:

### Step 1: Understand the condition for the roots
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) are given by the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
where \( D = b^2 - 4ac \) is the discriminant. For our equation, we have:
- \( a = 1 \)
- \( b = a \)
- \( c = a + 1 \)

### Step 2: Calculate the discriminant
The discriminant \( D \) is:
\[ D = a^2 - 4(1)(a + 1) = a^2 - 4a - 4 \]
For the roots to be real and distinct, we need \( D > 0 \):
\[ a^2 - 4a - 4 > 0 \]

### Step 3: Solve the inequality
To solve the inequality \( a^2 - 4a - 4 > 0 \), we first find the roots of the equation \( a^2 - 4a - 4 = 0 \) using the quadratic formula:
\[ a = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-4)}}{2(1)} = \frac{4 \pm \sqrt{16 + 16}}{2} = \frac{4 \pm \sqrt{32}}{2} = \frac{4 \pm 4\sqrt{2}}{2} = 2 \pm 2\sqrt{2} \]
The roots are \( 2 + 2\sqrt{2} \) and \( 2 - 2\sqrt{2} \).

### Step 4: Determine the intervals
The quadratic \( a^2 - 4a - 4 \) opens upwards (since the coefficient of \( a^2 \) is positive). Thus, the inequality \( a^2 - 4a - 4 > 0 \) holds outside the roots:
\[ a < 2 - 2\sqrt{2} \quad \text{or} \quad a > 2 + 2\sqrt{2} \]

Calculating the approximate values:
- \( 2 - 2\sqrt{2} \approx -0.828 \)
- \( 2 + 2\sqrt{2} \approx 4.828 \)

### Step 5: Find integral values of \( a \)
The integral values of \( a \) that satisfy the inequality are:
- For \( a < 2 - 2\sqrt{2} \): The only integer is \( -1, -2, -3, \ldots \)
- For \( a > 2 + 2\sqrt{2} \): The integers are \( 5, 6, 7, \ldots \)

### Step 6: Check the condition for two integers between the roots
To ensure exactly two integers lie between the roots, the distance between the roots must be greater than 2. The distance between the roots is given by:
\[ \text{Distance} = \frac{\sqrt{D}}{a} = \frac{\sqrt{a^2 - 4a - 4}}{2} \]
We need:
\[ \frac{\sqrt{a^2 - 4a - 4}}{2} > 2 \implies \sqrt{a^2 - 4a - 4} > 4 \implies a^2 - 4a - 4 > 16 \implies a^2 - 4a - 20 > 0 \]
Finding the roots of \( a^2 - 4a - 20 = 0 \):
\[ a = \frac{4 \pm \sqrt{16 + 80}}{2} = \frac{4 \pm \sqrt{96}}{2} = \frac{4 \pm 4\sqrt{6}}{2} = 2 \pm 2\sqrt{6} \]
Calculating approximate values:
- \( 2 - 2\sqrt{6} \approx -2.898 \)
- \( 2 + 2\sqrt{6} \approx 6.898 \)

### Final Integral Values of \( a \)
The integral values of \( a \) that satisfy both conditions are:
- From \( a < 2 - 2\sqrt{6} \): \( -3, -2, -1, 0, 1, 2 \)
- From \( a > 2 + 2\sqrt{6} \): \( 7, 8, 9, \ldots \)

Thus, the integral values of \( a \) are \( -1, -2, 1, 2 \).

### Summary of Integral Values
The integral values of \( a \) that satisfy the condition are:
\[ \text{Values of } a: -1, -2, 1, 2 \]