`3-2i` and `3+2i` are roots to which of the following quadratic equations ?

Options

A. `x^(2)+6x+13=0`
B. `x^(2)-6x+13=0`
C. `x^(2)-6x-13=0`
D. `x^(2)+6x+7=0`

Explanation

To find the quadratic equation for which the roots are \(3 - 2i\) and \(3 + 2i\), we can use the relationship between the roots and the coefficients of the quadratic equation. The general form of a quadratic equation with roots \(p\) and \(q\) is given by:

\[ x^2 - (p + q)x + (pq) = 0 \]

### Step 1: Identify the roots
The given roots are:
- \(p = 3 - 2i\)
- \(q = 3 + 2i\)

### Step 2: Calculate the sum of the roots
Now, we calculate the sum of the roots \(p + q\):

\[ p + q = (3 - 2i) + (3 + 2i) = 3 + 3 + (-2i + 2i) = 6 \]

### Step 3: Calculate the product of the roots
Next, we calculate the product of the roots \(pq\):

\[ pq = (3 - 2i)(3 + 2i) \]

Using the formula for the product of a complex number and its conjugate:

\[ pq = 3^2 - (2i)^2 = 9 - (-4) = 9 + 4 = 13 \]

### Step 4: Write the quadratic equation
Now, substituting the sum and product of the roots into the quadratic equation form:

\[ x^2 - (p + q)x + (pq) = 0 \]

Substituting the values we found:

\[ x^2 - 6x + 13 = 0 \]

### Conclusion
Thus, the quadratic equation for which \(3 - 2i\) and \(3 + 2i\) are roots is:

\[ x^2 - 6x + 13 = 0 \]

### Answer
The correct option is **b. \(x^2 - 6x + 13 = 0\)**.

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