If the equation `x^(2)+px+q=0` has integral solutions for prime number p and q, then `p+q=`

Options

A. 3
B. 5
C. 4
D. 10

Explanation

To solve the equation \(x^2 + px + q = 0\) for integral solutions where \(p\) and \(q\) are prime numbers, we can follow these steps:

### Step 1: Understand the nature of the roots
For the quadratic equation \(x^2 + px + q = 0\) to have integral solutions, the discriminant must be a perfect square. The discriminant \(D\) is given by:
\[ D = p^2 - 4q \]
For \(D\) to be a perfect square, we can set \(D = k^2\) for some integer \(k\).

### Step 2: Set up the equation
We need to find prime numbers \(p\) and \(q\) such that:
\[ p^2 - 4q = k^2 \]
This can be rearranged to:
\[ p^2 - k^2 = 4q \]
Factoring the left side gives:
\[ (p - k)(p + k) = 4q \]

### Step 3: Analyze the factors
Since \(q\) is a prime number, \(4q\) has limited factorizations. The factors of \(4q\) can be \(1 \times 4q\), \(2 \times 2q\), or \(4 \times q\).

### Step 4: Test small prime numbers
Let's test small prime numbers for \(p\) and \(q\):
- Let \(p = 3\) (the smallest odd prime) and \(q = 2\) (the smallest prime).

Substituting these values into the equation gives:
\[ x^2 + 3x + 2 = 0 \]

### Step 5: Factor the quadratic
Factoring the quadratic:
\[ x^2 + 3x + 2 = (x + 1)(x + 2) = 0 \]
This gives the roots:
\[ x = -1 \quad \text{and} \quad x = -2 \]
Both are integral solutions.

### Step 6: Calculate \(p + q\)
Now, we can calculate:
\[ p + q = 3 + 2 = 5 \]

### Conclusion
Thus, the value of \(p + q\) is:
\[ \boxed{5} \]