Solve the following equations by factorization method : `x^(2)+6ix-9=0`

Explanation

To solve the equation \( x^2 + 6ix - 9 = 0 \) using the factorization method, we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ x^2 + 6ix - 9 = 0 \]

### Step 2: Identify the coefficients
In this quadratic equation, we can identify:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = 6i \) (coefficient of \( x \))
- \( c = -9 \) (constant term)

### Step 3: Factor the quadratic expression
We want to express the quadratic in the form of a perfect square. We can rewrite the equation as:
\[ x^2 + 6ix + (3i)^2 - 9 = 0 \]
This is because \( (3i)^2 = 9(-1) = -9 \).

### Step 4: Complete the square
Now we can rewrite the equation as:
\[ (x + 3i)^2 - 9 = 0 \]
This can be rearranged to:
\[ (x + 3i)^2 = 9 \]

### Step 5: Take the square root of both sides
Taking the square root of both sides gives us:
\[ x + 3i = \pm 3 \]

### Step 6: Solve for \( x \)
Now we can solve for \( x \):
1. \( x + 3i = 3 \) leads to:
\[ x = 3 - 3i \]
2. \( x + 3i = -3 \) leads to:
\[ x = -3 - 3i \]

### Final Solutions
Thus, the solutions to the equation \( x^2 + 6ix - 9 = 0 \) are:
\[ x = 3 - 3i \quad \text{and} \quad x = -3 - 3i \]

---