Factorise the following `(x^(2)-6xy+9y^(2))-25`

Options

A. `(x-5)(x+5)`
B. `(x-y-5)(x-y+5)`
C. `(x+3y+5)(x-3y-5)`
D. `(x-3y-5)(x-3y+5)`

Answer

To factorize the expression \( (x^2 - 6xy + 9y^2) - 25 \), we can follow these steps:

### Step 1: Recognize the structure of the expression
The expression \( x^2 - 6xy + 9y^2 \) can be recognized as a perfect square trinomial.

### Step 2: Rewrite the perfect square trinomial
The expression \( x^2 - 6xy + 9y^2 \) can be rewritten as:
\[ (x - 3y)^2 \]
This is because:
\[ (x - 3y)(x - 3y) = x^2 - 3yx - 3yx + 9y^2 = x^2 - 6xy + 9y^2 \]

### Step 3: Substitute back into the original expression
Now, we can substitute this back into the original expression:
\[ (x - 3y)^2 - 25 \]

### Step 4: Recognize this as a difference of squares
The expression \( (x - 3y)^2 - 25 \) can be recognized as a difference of squares. Recall that \( a^2 - b^2 = (a + b)(a - b) \).

### Step 5: Identify \( a \) and \( b \)
Here, let:
- \( a = (x - 3y) \)
- \( b = 5 \) (since \( 25 = 5^2 \))

### Step 6: Apply the difference of squares formula
Using the difference of squares formula:
\[ (x - 3y)^2 - 5^2 = (x - 3y + 5)(x - 3y - 5) \]

### Final Factorized Form
Thus, the factorized form of the expression \( (x^2 - 6xy + 9y^2) - 25 \) is:
\[ (x - 3y + 5)(x - 3y - 5) \]