If `2+sqrt(3)i` is a root of the equation `x^2+p x+q=0` , then write the values of `p\ a n d\ q` .

Explanation

To find the values of \( p \) and \( q \) in the quadratic equation \( x^2 + px + q = 0 \) given that \( 2 + \sqrt{3}i \) is a root, we can follow these steps:

### Step 1: Identify the conjugate root
Since the coefficients \( p \) and \( q \) are real numbers, the complex roots must occur in conjugate pairs. Therefore, if \( 2 + \sqrt{3}i \) is a root, then its conjugate \( 2 - \sqrt{3}i \) is also a root.

### Step 2: Use Vieta's formulas
According to Vieta's formulas, for a quadratic equation \( x^2 + px + q = 0 \):
- The sum of the roots \( r_1 + r_2 = -p \)
- The product of the roots \( r_1 \cdot r_2 = q \)

Let \( r_1 = 2 + \sqrt{3}i \) and \( r_2 = 2 - \sqrt{3}i \).

### Step 3: Calculate the sum of the roots
\[ r_1 + r_2 = (2 + \sqrt{3}i) + (2 - \sqrt{3}i) = 2 + 2 = 4 \]
Thus, according to Vieta's, we have:
\[ -p = 4 \implies p = -4 \]

### Step 4: Calculate the product of the roots
\[ r_1 \cdot r_2 = (2 + \sqrt{3}i)(2 - \sqrt{3}i) \]
Using the difference of squares:
\[ = 2^2 - (\sqrt{3}i)^2 = 4 - (-3) = 4 + 3 = 7 \]
Thus, we have:
\[ q = 7 \]

### Final Values
The values of \( p \) and \( q \) are:
\[ p = -4, \quad q = 7 \]