If `x^(2) - px + q = 0` has equal integral roots, then

Options

A. p and q are even integers
B. p and q are odd integers
C. p an even integer and q is a perfect square of a positive integer
D. none of these

Explanation

To solve the problem, we need to analyze the quadratic equation given:

\[ x^2 - px + q = 0 \]

We know that this equation has equal integral roots. Let's denote the equal roots as \( \alpha \).

### Step 1: Understanding the roots
Since the roots are equal, we can use the property of quadratic equations that states:

- The sum of the roots \( \alpha + \alpha = 2\alpha \) is equal to \( p \).
- The product of the roots \( \alpha \cdot \alpha = \alpha^2 \) is equal to \( q \).

Thus, we can express \( p \) and \( q \) in terms of \( \alpha \):

\[ p = 2\alpha \quad \text{and} \quad q = \alpha^2 \]

### Step 2: Analyzing \( p \)
Since \( p = 2\alpha \), we can conclude that \( p \) is always an even integer because it is two times an integer (\( \alpha \)).

### Step 3: Analyzing \( q \)
Next, since \( q = \alpha^2 \), we know that \( q \) is the square of an integer. The square of any integer is always a perfect square.

### Conclusion
From our analysis, we have determined that:
- \( p \) is an even integer.
- \( q \) is a perfect square.

Thus, the correct option from the given choices is:

**Option C: p is an even integer and q is a perfect square.**