If 3+4i is a root of equation `x^(2)+px+q=0` where `p, q in R` then

Options

A. p=6, q=25
B. p=6, q=-25
C. p= -6, q= -7
D. p= -6, q=25

Answer

To solve the problem, we need to find the values of \( p \) and \( q \) in the quadratic equation \( x^2 + px + q = 0 \) given that \( 3 + 4i \) is a root. Since the coefficients \( p \) and \( q \) are real numbers, the complex conjugate \( 3 - 4i \) must also be a root.

### Step-by-Step Solution:

1. **Identify the Roots**:
Let \( \alpha = 3 + 4i \) and \( \beta = 3 - 4i \).

2. **Find the Sum of the Roots**:
The sum of the roots \( \alpha + \beta \) is calculated as follows:
\[ \alpha + \beta = (3 + 4i) + (3 - 4i) = 3 + 4i + 3 - 4i = 6 \]

3. **Calculate \( p \)**:
According to Vieta's formulas, \( p \) is the negative of the sum of the roots:
\[ p = -(\alpha + \beta) = -6 \]

4. **Find the Product of the Roots**:
The product of the roots \( \alpha \beta \) is calculated using:
\[ \alpha \beta = (3 + 4i)(3 - 4i) = 3^2 - (4i)^2 = 9 - (-16) = 9 + 16 = 25 \]

5. **Calculate \( q \)**:
According to Vieta's formulas, \( q \) is equal to the product of the roots:
\[ q = \alpha \beta = 25 \]

6. **Final Values**:
Thus, we have found:
\[ p = -6 \quad \text{and} \quad q = 25 \]

### Conclusion:
The values of \( p \) and \( q \) are \( p = -6 \) and \( q = 25 \).