Solve the equation `x^(2)-6x+25=0` by factorization method.

Options

A. `4pm4i`
B. `3pm3i`
C. `3pm4i`
D. `2pm4i`

Explanation

To solve the equation \( x^2 - 6x + 25 = 0 \) using the factorization method, we will follow these steps:

### Step 1: Identify the equation
We start with the quadratic equation:
\[ x^2 - 6x + 25 = 0 \]

### Step 2: Rewrite the equation
We want to express the quadratic in a form that can be factored. We can rewrite the constant term \( 25 \) as \( 9 + 16 \):
\[ x^2 - 6x + (9 + 16) = 0 \]
This can be rearranged as:
\[ x^2 - 6x + 9 + 16 = 0 \]

### Step 3: Complete the square
Next, we can complete the square for the first three terms:
\[ (x - 3)^2 + 16 = 0 \]
This is because \( (x - 3)^2 = x^2 - 6x + 9 \).

### Step 4: Isolate the constant
Now, we can isolate the constant term:
\[ (x - 3)^2 = -16 \]

### Step 5: Take the square root
Taking the square root of both sides gives us:
\[ x - 3 = \pm 4i \]
where \( i \) is the imaginary unit.

### Step 6: Solve for \( x \)
Now we can solve for \( x \):
1. \( x - 3 = 4i \) leads to \( x = 3 + 4i \)
2. \( x - 3 = -4i \) leads to \( x = 3 - 4i \)

### Step 7: Write the final solutions
Thus, the solutions to the equation \( x^2 - 6x + 25 = 0 \) are:
\[ x = 3 + 4i \quad \text{and} \quad x = 3 - 4i \]

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