Sure, let's go through the process of factoring the given expression step-by-step.
Given the expression:
[tex]\[ 4 - 36v^2 \][/tex]
1. Identify the expression as a difference of squares:
We notice that both terms [tex]\(4\)[/tex] and [tex]\(36v^2\)[/tex] are perfect squares. Recall that a difference of squares takes the form [tex]\(a^2 - b^2\)[/tex].
- [tex]\(4\)[/tex] can be written as [tex]\((2)^2\)[/tex]
- [tex]\(36v^2\)[/tex] can be written as [tex]\((6v)^2\)[/tex]
This fits the form [tex]\(a^2 - b^2\)[/tex], where [tex]\(a = 2\)[/tex] and [tex]\(b = 6v\)[/tex].
2. Apply the difference of squares factoring formula:
The formula is:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
Here, replacing [tex]\(a\)[/tex] and [tex]\(b\)[/tex] with our identified terms:
[tex]\[ 4 - 36v^2 = (2)^2 - (6v)^2 \][/tex]
So, we have:
[tex]\[ = (2 - 6v)(2 + 6v) \][/tex]
3. Combine the terms into the final factored form:
The fully factored form of the expression [tex]\(4 - 36v^2\)[/tex] is:
[tex]\[ (2 - 6v)(2 + 6v) \][/tex]
So, the completely factored form of [tex]\(4 - 36v^2\)[/tex] is:
[tex]\[ \boxed{(2 - 6v)(2 + 6v)} \][/tex]
