Absolutely, let's go through the step-by-step factorization of the expression [tex]\(9 - 25w^2\)[/tex].
### Step 1: Recognize the form of the expression
Notice that the given expression [tex]\(9 - 25w^2\)[/tex] is a difference of squares. The difference of squares formula is:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
### Step 2: Identify the squares
To use this formula, we need to identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in our expression. Observe that:
[tex]\[
9 = 3^2 \quad \text{and} \quad 25w^2 = (5w)^2
\][/tex]
So we can write:
[tex]\[
9 - 25w^2 = 3^2 - (5w)^2
\][/tex]
### Step 3: Apply the difference of squares formula
Now, using the difference of squares formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], where [tex]\(a = 3\)[/tex] and [tex]\(b = 5w\)[/tex], we get:
[tex]\[
3^2 - (5w)^2 = (3 - 5w)(3 + 5w)
\][/tex]
### Step 4: Account for the negative sign (optional for clarity)
To align precisely with the structure of our result, we recognize:
[tex]\[
-(5w - 3)(5w + 3) = -(5w+3)(5w-3)
\][/tex]
Therefore, the factorized form of [tex]\(9 - 25w^2\)[/tex] is:
[tex]\[
-(5w - 3)(5w + 3)
\][/tex]
This is the fully factorized form of the expression [tex]\(9 - 25w^2\)[/tex].
