Number of possible value(s) of integer 'a' for which the quadratic equation `x^(2) + ax + 16 = 0` has integral roots, is

Options

A. 4
B. 6
C. 2
D. none of these

Explanation

To find the number of possible integer values of 'a' for which the quadratic equation \( x^2 + ax + 16 = 0 \) has integral roots, we can follow these steps:

### Step 1: Understand the condition for integral roots
For the quadratic equation \( ax^2 + bx + c = 0 \) to have integral roots, the discriminant \( D \) must be a perfect square. The discriminant \( D \) for our equation is given by:
\[ D = b^2 - 4ac \]
In our case, \( a = a \), \( b = a \), and \( c = 16 \). Therefore, the discriminant becomes:
\[ D = a^2 - 4 \cdot 1 \cdot 16 = a^2 - 64 \]

### Step 2: Set the discriminant as a perfect square
We set the discriminant equal to a perfect square:
\[ a^2 - 64 = k^2 \]
where \( k \) is an integer. Rearranging gives:
\[ a^2 - k^2 = 64 \]
This can be factored using the difference of squares:
\[ (a - k)(a + k) = 64 \]

### Step 3: Find the factor pairs of 64
Next, we need to find the integer factor pairs of 64. The pairs are:
1. \( (1, 64) \)
2. \( (2, 32) \)
3. \( (4, 16) \)
4. \( (8, 8) \)
5. \( (-1, -64) \)
6. \( (-2, -32) \)
7. \( (-4, -16) \)
8. \( (-8, -8) \)

### Step 4: Solve for 'a' from each factor pair
For each factor pair \( (m, n) \), we have:
\[ a - k = m \quad \text{and} \quad a + k = n \]
Adding these two equations gives:
\[ 2a = m + n \implies a = \frac{m + n}{2} \]
Subtracting the first equation from the second gives:
\[ 2k = n - m \implies k = \frac{n - m}{2} \]

Now we will calculate 'a' for each factor pair:

1. For \( (1, 64) \):
\[ a = \frac{1 + 64}{2} = \frac{65}{2} \quad \text{(not an integer)} \]

2. For \( (2, 32) \):
\[ a = \frac{2 + 32}{2} = 17 \quad \text{(integer)} \]

3. For \( (4, 16) \):
\[ a = \frac{4 + 16}{2} = 10 \quad \text{(integer)} \]

4. For \( (8, 8) \):
\[ a = \frac{8 + 8}{2} = 8 \quad \text{(integer)} \]

5. For \( (-1, -64) \):
\[ a = \frac{-1 - 64}{2} = -32.5 \quad \text{(not an integer)} \]

6. For \( (-2, -32) \):
\[ a = \frac{-2 - 32}{2} = -17 \quad \text{(integer)} \]

7. For \( (-4, -16) \):
\[ a = \frac{-4 - 16}{2} = -10 \quad \text{(integer)} \]

8. For \( (-8, -8) \):
\[ a = \frac{-8 - 8}{2} = -8 \quad \text{(integer)} \]

### Step 5: List all possible integer values of 'a'
The possible integer values of 'a' are:
- \( 17 \)
- \( 10 \)
- \( 8 \)
- \( -17 \)
- \( -10 \)
- \( -8 \)

### Conclusion
Thus, the number of possible integer values of 'a' is \( 6 \).