Total number of integral values of a such that `x^2 + ax + a + 1 = 0` has integral roots is equal to : (A) one 45. (B) two (C) three (D) four

Options

A. one
B. two
C. three
D. four

Explanation

To find the total number of integral values of \( a \) such that the quadratic equation \( x^2 + ax + (a + 1) = 0 \) has integral roots, we can follow these steps:

### Step 1: Identify the equation
The given quadratic equation is:
\[ x^2 + ax + (a + 1) = 0 \]

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

### Step 3: Calculate the discriminant
For the roots to be integral, the discriminant must be a perfect square. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac = a^2 - 4(1)(a + 1) = a^2 - 4a - 4 \]

### Step 4: Set the discriminant as a perfect square
Let \( D = k^2 \) for some integer \( k \):
\[ a^2 - 4a - 4 = k^2 \]
Rearranging gives:
\[ a^2 - 4a - k^2 - 4 = 0 \]

### Step 5: Analyze the quadratic in \( a \)
The above is a quadratic equation in \( a \). For \( a \) to have integral solutions, the discriminant of this equation must be a perfect square. The discriminant \( D_a \) of this quadratic in \( a \) is:
\[ D_a = (-4)^2 - 4(1)(-k^2 - 4) = 16 + 4k^2 + 16 = 4k^2 + 32 \]

### Step 6: Set the discriminant as a perfect square
We need \( 4k^2 + 32 \) to be a perfect square:
\[ 4k^2 + 32 = m^2 \quad \text{for some integer } m \]
Rearranging gives:
\[ m^2 - 4k^2 = 32 \]
This can be factored using the difference of squares:
\[ (m - 2k)(m + 2k) = 32 \]

### Step 7: Find pairs of factors of 32
The pairs of factors of 32 are:
1. \( (1, 32) \)
2. \( (2, 16) \)
3. \( (4, 8) \)
4. \( (-1, -32) \)
5. \( (-2, -16) \)
6. \( (-4, -8) \)

### Step 8: Solve for \( m \) and \( k \)
For each pair \( (p, q) \):
1. \( m - 2k = p \)
2. \( m + 2k = q \)

Adding these equations:
\[ 2m = p + q \implies m = \frac{p + q}{2} \]
Subtracting the first from the second:
\[ 4k = q - p \implies k = \frac{q - p}{4} \]

### Step 9: Check integer solutions
We need \( m \) and \( k \) to be integers. We can check each factor pair:
1. For \( (1, 32) \): \( m = \frac{33}{2} \) (not an integer)
2. For \( (2, 16) \): \( m = 9, k = 3.5 \) (not an integer)
3. For \( (4, 8) \): \( m = 6, k = 1 \) (valid)
4. For \( (-1, -32) \): \( m = -16.5 \) (not an integer)
5. For \( (-2, -16) \): \( m = -9, k = -3.5 \) (not an integer)
6. For \( (-4, -8) \): \( m = -6, k = -1 \) (valid)

### Step 10: Calculate values of \( a \)
From valid \( k \) values, we can find corresponding \( a \) values:
1. For \( k = 1 \): \( a^2 - 4a - 5 = 0 \) gives \( a = 5, -1 \).
2. For \( k = -1 \): \( a^2 - 4a - 3 = 0 \) gives \( a = 3, 1 \).

### Conclusion
The integral values of \( a \) are \( -1, 1, 3, 5 \). Thus, the total number of integral values of \( a \) is 4.

### Final Answer
The total number of integral values of \( a \) is **4** (Option D).

---