The roots of the equation `x^(2) +6ix-9=0` are :

Options

A. `pmi`
B. `pm2i`
C. `pm3i`
D. None of these

Answer

To find the roots of the equation \( x^2 + 6ix - 9 = 0 \), we can follow these steps:

### Step 1: Identify the coefficients
The given equation is in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = 6i \)
- \( c = -9 \)

### Step 2: Apply the quadratic formula
The roots of a quadratic equation can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ x = \frac{-6i \pm \sqrt{(6i)^2 - 4 \cdot 1 \cdot (-9)}}{2 \cdot 1} \]

### Step 3: Calculate \( b^2 - 4ac \)
First, calculate \( b^2 \):
\[ (6i)^2 = 36i^2 = 36(-1) = -36 \]
Next, calculate \( -4ac \):
\[ -4 \cdot 1 \cdot (-9) = 36 \]
Now, combine these results:
\[ b^2 - 4ac = -36 + 36 = 0 \]

### Step 4: Substitute back into the quadratic formula
Since \( b^2 - 4ac = 0 \), we have:
\[ x = \frac{-6i \pm \sqrt{0}}{2} \]
This simplifies to:
\[ x = \frac{-6i}{2} = -3i \]

### Step 5: Determine the roots
Since the discriminant is zero, there is one repeated root:
\[ x = -3i \]

### Final Answer
The roots of the equation \( x^2 + 6ix - 9 = 0 \) are:
\[ x = -3i \quad \text{(repeated root)} \]