Is `x^2 - 6xy + 36y^2` a perfect square trinomial ?

Answer

To determine if the expression \( x^2 - 6xy + 36y^2 \) is a perfect square trinomial, we can follow these steps:

### Step 1: Identify the terms
The given expression is:
\[ x^2 - 6xy + 36y^2 \]

### Step 2: Rewrite the last term
Notice that \( 36y^2 \) can be rewritten as \( (6y)^2 \). Thus, we can express the trinomial as:
\[ x^2 - 6xy + (6y)^2 \]

### Step 3: Check the middle term
For a trinomial to be a perfect square, the middle term must be twice the product of the square roots of the first and last terms. The square root of \( x^2 \) is \( x \) and the square root of \( (6y)^2 \) is \( 6y \). Therefore, we check:
\[ 2 \cdot x \cdot 6y = 12xy \]
However, the middle term in our expression is \( -6xy \).

### Step 4: Compare the middle term
Since \( -6xy \) does not equal \( 12xy \), we can conclude that:
\[ -6xy \neq 12xy \]

### Step 5: Conclusion
Since the middle term does not satisfy the condition for a perfect square trinomial, we conclude that:
\[ x^2 - 6xy + 36y^2 \text{ is not a perfect square trinomial.} \]

### Final Answer
Thus, the expression \( x^2 - 6xy + 36y^2 \) is **not** a perfect square trinomial.

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