To factor the quadratic expression \(9w^2 - 48w + 64\), we will follow these steps:

1. **Check if the quadratic expression can be factored as a perfect square trinomial:**

   A quadratic expression \(ax^2 + bx + c\) is a perfect square trinomial if it can be factored as \((px + q)^2\) where:
   \[
   a = p^2, \quad b = 2pq, \quad \text{and} \quad c = q^2
   \]
   For the expression \(9w^2 - 48w + 64\):
   - \(a = 9\) (which is \(3^2\))
   - \(c = 64\) (which is \(8^2\))
   - \(b = -48\) (which should be \(2pq\))

2. **Verify if \(b = 2pq\) holds true:**
   \[
   2pq = 2(3)(-8) = -48
   \]
   This matches our \(b\) term.

Since the conditions are satisfied, the expression is indeed a perfect square trinomial.

3. **Factor the quadratic expression:**

   \[
   9w^2 - 48w + 64 = (3w - 8)^2
   \]

So, the factored form of \(9w^2 - 48w + 64\) is \((3w - 8)^2\).

Would you like further details or have any questions?

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Here are 5 related questions to expand your understanding:

1. How would you factor the expression if it was not a perfect square trinomial?
2. What is the significance of the discriminant in factoring quadratics?
3. How do you solve \(9w^2 - 48w + 64 = 0\) for \(w\)?
4. How can you verify that \((3w - 8)^2\) is the correct factorization?
5. What are some methods to factor a quadratic expression when the leading coefficient is not 1?

**Tip:** Always check if a quadratic can be factored as a perfect square trinomial before using other methods, as it simplifies the process significantly.

Learn how to factor the quadratic expression 9w^2 - 48w + 64 using the perfect square trinomial method. This problem involves algebraic concepts such as quadratic equations, factoring, and recognizing perfect square trinomials. Suitable for students in Grades 9-10.

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