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

Explanation

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

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

### Step 2: Identify the constant term
We notice that the constant term is 25. We can try to express 25 in a way that helps us factor the equation. We can break down 25 as \( 9 + 16 \):
\[ x^2 - 6x + 9 + 16 = 0 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (x^2 - 6x + 9) + 16 = 0 \]
The expression \( x^2 - 6x + 9 \) can be factored as:
\[ (x - 3)^2 \]
So, we rewrite the equation:
\[ (x - 3)^2 + 16 = 0 \]

### Step 4: Rewrite the equation in terms of squares
Next, we can express 16 as \( (4i)^2 \) (since \( i^2 = -1 \)):
\[ (x - 3)^2 - (4i)^2 = 0 \]

### Step 5: Apply the difference of squares formula
Now we can use the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \):
\[ (x - 3 - 4i)(x - 3 + 4i) = 0 \]

### Step 6: Set each factor to zero
Now we can set each factor to zero:
1. \( x - 3 - 4i = 0 \)
2. \( x - 3 + 4i = 0 \)

### Step 7: Solve for \( x \)
From the first equation:
\[ x - 3 = 4i \implies x = 3 + 4i \]
From the second equation:
\[ x - 3 = -4i \implies x = 3 - 4i \]

### Final Solution
Thus, the roots of the equation \( x^2 - 6x + 25 = 0 \) are:
\[ x = 3 + 4i \quad \text{and} \quad x = 3 - 4i \]

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