Solve the equation `x^(2)+9=0` by factorization method.

Explanation

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

### Step 1: Rewrite the equation
Start with the original equation:
\[ x^2 + 9 = 0 \]

### Step 2: Move 9 to the other side
Rearranging the equation gives:
\[ x^2 = -9 \]

### Step 3: Recognize the square of a complex number
We can express \(-9\) as \(- (3^2)\), which allows us to use the property of complex numbers:
\[ x^2 = - (3^2) = -1 \cdot 3^2 \]

### Step 4: Introduce the imaginary unit \(i\)
Since \(i^2 = -1\), we can rewrite the equation as:
\[ x^2 = (3i)^2 \]

### Step 5: Factor the equation
Now we can factor the equation using the difference of squares:
\[ x^2 - (3i)^2 = 0 \]
This can be factored as:
\[ (x - 3i)(x + 3i) = 0 \]

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

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

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

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