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 will use the quadratic formula. The quadratic formula states that for any equation of the form \( ax^2 + bx + c = 0 \), the roots can be found using:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 1: Identify coefficients
In our equation \( x^2 + 6ix - 9 = 0 \), we can identify the coefficients as follows:
- \( a = 1 \)
- \( b = 6i \)
- \( c = -9 \)

### Step 2: Calculate the discriminant
Next, we need to calculate the discriminant \( D = b^2 - 4ac \):

\[ D = (6i)^2 - 4 \cdot 1 \cdot (-9) \]

Calculating \( (6i)^2 \):

\[ (6i)^2 = 36i^2 = 36(-1) = -36 \]

Now substituting back into the discriminant:

\[ D = -36 - 4 \cdot 1 \cdot (-9) = -36 + 36 = 0 \]

### Step 3: Calculate the roots
Since the discriminant \( D = 0 \), there is one repeated root. We can now substitute \( D \) back into the quadratic formula:

\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6i \pm \sqrt{0}}{2 \cdot 1} \]

This simplifies to:

\[ x = \frac{-6i}{2} = -3i \]

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

\[ x = -3i \]