solve for factoring perfect square trinomial: n^2+16n+64

To factor the perfect square trinomial \( n^2 + 16n + 64 \), follow these steps:

1. **Identify the trinomial:** \( n^2 + 16n + 64 \).

2. **Check if it fits the perfect square trinomial form:** A perfect square trinomial takes the form \( a^2 + 2ab + b^2 \), which can be factored as \( (a + b)^2 \).

   - Here, \( a^2 = n^2 \), so \( a = n \).
   - \( b^2 = 64 \), so \( b = 8 \).
   - \( 2ab = 2 \times n \times 8 = 16n \), which matches the middle term.

3. **Factor the trinomial:** Since the trinomial matches the form of a perfect square, it can be factored as:
   \[
   n^2 + 16n + 64 = (n + 8)^2
   \]

So, the factored form of \( n^2 + 16n + 64 \) is \( (n + 8)^2 \).

Would you like more details on this process, or do you have any questions?

Here are some related questions you might consider:

1. How would you factor \( n^2 - 16n + 64 \)?
2. What is the factored form of \( x^2 + 10x + 25 \)?
3. How do you expand \( (n + 8)^2 \) back into a trinomial?
4. What is a general approach to recognizing perfect square trinomials?
5. How can you factor \( n^2 + 20n + 100 \)?

**Tip:** Always verify your factorization by expanding the binomial back into its original trinomial form to ensure accuracy.

Learn how to factor the perfect square trinomial n^2 + 16n + 64 with a step-by-step solution. Understand the concept of perfect square trinomials and how to apply the formula (a + b)^2 = a^2 + 2ab + b^2. Suitable for students in Grades 9-10 studying quadratic equations.

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