Solve
(i) `x^(2) -(3sqrt2-2i)x-6sqrt2i=0`

Answer

To solve the quadratic equation \( x^2 - (3\sqrt{2} - 2i)x - 6\sqrt{2}i = 0 \), we can use the factorization method. Here’s a step-by-step solution:

### Step 1: Rewrite the equation
The given equation is:
\[ x^2 - (3\sqrt{2} - 2i)x - 6\sqrt{2}i = 0 \]
We can rewrite it as:
\[ x^2 - (3\sqrt{2})x + (2i)x - 6\sqrt{2}i = 0 \]

### Step 2: Group the terms
Now, we can group the terms:
\[ x^2 - (3\sqrt{2})x + (2i)x - 6\sqrt{2}i = 0 \]
This can be rearranged to:
\[ x^2 - (3\sqrt{2}x - 2ix) - 6\sqrt{2}i = 0 \]

### Step 3: Factor by grouping
We can factor by grouping:
\[ x(x - 3\sqrt{2}) + 2i(x - 3\sqrt{2}) = 0 \]
Now, we can factor out the common term \((x - 3\sqrt{2})\):
\[ (x - 3\sqrt{2})(x + 2i) = 0 \]

### Step 4: Set each factor to zero
Now, we can set each factor to zero:
1. \( x - 3\sqrt{2} = 0 \)
2. \( x + 2i = 0 \)

### Step 5: Solve for \( x \)
From the first equation:
\[ x = 3\sqrt{2} \]
From the second equation:
\[ x = -2i \]

### Final Solution
Thus, the solutions to the quadratic equation are:
\[ x = 3\sqrt{2} \quad \text{and} \quad x = -2i \]

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