If p is a prime and both roots of `x^(2)-px-444p=0` are integers, then p is equal to

Options

A. 2
B. 3
C. 37
D. none of these

Answer

To solve the problem, we need to determine the prime number \( p \) such that both roots of the quadratic equation \( x^2 - px - 444p = 0 \) are integers.

### Step-by-Step Solution:

1. **Identify the quadratic equation**:
The given quadratic equation is:
\[ x^2 - px - 444p = 0 \]

2. **Use the quadratic formula**:
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
where \( D = b^2 - 4ac \) is the discriminant.

3. **Calculate the discriminant**:
Here, \( a = 1 \), \( b = -p \), and \( c = -444p \). Thus, the discriminant \( D \) is:
\[ D = (-p)^2 - 4 \cdot 1 \cdot (-444p) = p^2 + 1776p \]

4. **Set the discriminant as a perfect square**:
For the roots to be integers, \( D \) must be a perfect square. Let’s set:
\[ D = k^2 \quad \text{for some integer } k \]
Therefore, we have:
\[ p^2 + 1776p = k^2 \]

5. **Rearranging the equation**:
Rearranging gives us:
\[ k^2 - p^2 - 1776p = 0 \]
This is a quadratic in \( p \):
\[ p^2 + 1776p - k^2 = 0 \]

6. **Finding integer roots**:
For \( p \) to be an integer, the discriminant of this new quadratic must also be a perfect square:
\[ D' = 1776^2 + 4k^2 \]

7. **Testing prime candidates**:
Now we will test the prime candidates given in the options: 2, 3, and 37.

- **For \( p = 2 \)**:
\[ D = 2^2 + 1776 \cdot 2 = 4 + 3552 = 3556 \quad (\text{not a perfect square}) \]

- **For \( p = 3 \)**:
\[ D = 3^2 + 1776 \cdot 3 = 9 + 5328 = 5337 \quad (\text{not a perfect square}) \]

- **For \( p = 37 \)**:
\[ D = 37^2 + 1776 \cdot 37 = 1369 + 65712 = 67081 \]
Now check if \( 67081 \) is a perfect square:
\[ \sqrt{67081} = 259 \quad (\text{which is an integer}) \]

8. **Conclusion**:
Since \( D \) is a perfect square when \( p = 37 \), and \( 37 \) is a prime number, the solution to the problem is:
\[ p = 37 \]