If `x^(2)+px+q=0` has roots 2i + 3, 2i–3 then the discriminant of the equation is

Options

A. 36
B. 32
C. -36
D. -16

Answer

To find the discriminant of the quadratic equation \(x^2 + px + q = 0\) with roots \(2i + 3\) and \(2i - 3\), we will follow these steps:

### Step 1: Identify the roots
The roots of the quadratic equation are given as:
- \(\alpha = 2i + 3\)
- \(\beta = 2i - 3\)

### Step 2: Calculate the sum of the roots
Using the property of roots, we know that:
\[ p = -(\alpha + \beta) \]
Calculating \(\alpha + \beta\):
\[ \alpha + \beta = (2i + 3) + (2i - 3) = 4i \]
Thus,
\[ p = -4i \]

### Step 3: Calculate the product of the roots
Using the property of roots again, we have:
\[ q = \alpha \cdot \beta \]
Calculating \(\alpha \cdot \beta\):
\[ \alpha \cdot \beta = (2i + 3)(2i - 3) = (2i)^2 - 3^2 = 4i^2 - 9 \]
Since \(i^2 = -1\):
\[ \alpha \cdot \beta = 4(-1) - 9 = -4 - 9 = -13 \]
Thus,
\[ q = -13 \]

### Step 4: Write the quadratic equation
Now, substituting \(p\) and \(q\) into the quadratic equation:
\[ x^2 - 4ix - 13 = 0 \]

### Step 5: Calculate the discriminant
The discriminant \(D\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by:
\[ D = b^2 - 4ac \]
In our case, \(a = 1\), \(b = -4i\), and \(c = -13\). Thus, we calculate:
\[ D = (-4i)^2 - 4 \cdot 1 \cdot (-13) \]
Calculating \((-4i)^2\):
\[ (-4i)^2 = 16i^2 = 16(-1) = -16 \]
Now substituting this back into the discriminant formula:
\[ D = -16 - 4 \cdot 1 \cdot (-13) = -16 + 52 = 36 \]

### Conclusion
The discriminant of the equation is:
\[ \boxed{36} \]