Solve the equation `x^(2)-(2sqrt(2)+3i)x+6isqrt(2)=0` by factorization method.

Explanation

To solve the equation \( x^2 - (2\sqrt{2} + 3i)x + 6i\sqrt{2} = 0 \) using the factorization method, we will follow these steps:

### Step 1: Write the equation in standard form
The equation is already in standard quadratic form:
\[ x^2 - (2\sqrt{2} + 3i)x + 6i\sqrt{2} = 0 \]

### Step 2: Identify coefficients
From the equation, we can identify:
- \( a = 1 \) (coefficient of \( x^2 \))
- \( b = -(2\sqrt{2} + 3i) \) (coefficient of \( x \))
- \( c = 6i\sqrt{2} \) (constant term)

### Step 3: Factor the quadratic
To factor the quadratic, we need to find two numbers that multiply to \( ac \) (which is \( 6i\sqrt{2} \)) and add up to \( b \) (which is \( -(2\sqrt{2} + 3i) \)).

We can rewrite the equation:
\[ x^2 - (2\sqrt{2} + 3i)x + 6i\sqrt{2} = 0 \]

We will look for two numbers \( p \) and \( q \) such that:
- \( p + q = -(2\sqrt{2} + 3i) \)
- \( pq = 6i\sqrt{2} \)

### Step 4: Guess and check for factors
Let's try \( p = -2\sqrt{2} \) and \( q = -3i \):
- Check the sum:
\[ -2\sqrt{2} - 3i = -(2\sqrt{2} + 3i) \quad \text{(correct)} \]
- Check the product:
\[ (-2\sqrt{2})(-3i) = 6i\sqrt{2} \quad \text{(correct)} \]

### Step 5: Write the factors
Thus, we can factor the quadratic as:
\[ (x - (-2\sqrt{2}))(x - (-3i)) = (x + 2\sqrt{2})(x + 3i) = 0 \]

### Step 6: Set each factor to zero
Now, we set each factor to zero:
1. \( x + 2\sqrt{2} = 0 \)
2. \( x + 3i = 0 \)

### Step 7: Solve for \( x \)
From the first equation:
\[ x = -2\sqrt{2} \]
From the second equation:
\[ x = -3i \]

### Final Solution
The solutions to the equation \( x^2 - (2\sqrt{2} + 3i)x + 6i\sqrt{2} = 0 \) are:
\[ x = -2\sqrt{2} \quad \text{and} \quad x = -3i \]

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