To solve the problem of factoring the expression [tex]\(x^2 - 81\)[/tex], let's follow a detailed, step-by-step approach.
### Step-by-Step Solution:
1. Identify the expression:
The given expression is [tex]\(x^2 - 81\)[/tex].
2. Recognize the form:
Notice that [tex]\(x^2 - 81\)[/tex] is a classic example of the difference of squares. The difference of squares formula states that:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
3. Rewrite the expression in terms of the difference of squares:
In this expression, [tex]\(a^2\)[/tex] is [tex]\(x^2\)[/tex] and [tex]\(b^2\)[/tex] is [tex]\(81\)[/tex]. We know that [tex]\(81\)[/tex] can be written as [tex]\(9^2\)[/tex]. Thus, we can rewrite the expression as:
[tex]\[
x^2 - 81 = x^2 - 9^2
\][/tex]
4. Apply the difference of squares formula:
Using the formula [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex], where [tex]\(a = x\)[/tex] and [tex]\(b = 9\)[/tex], we can factor the expression:
[tex]\[
x^2 - 9^2 = (x - 9)(x + 9)
\][/tex]
5. Identify the value of [tex]\(b\)[/tex]:
In the factored form [tex]\((x - 9)(x + 9)\)[/tex], it's clear that [tex]\(b = 9\)[/tex], since the difference of squares utilizes the value of 9 squared (which is 81).
### Conclusion:
The factored form of the expression [tex]\(x^2 - 81\)[/tex] is [tex]\((x - 9)(x + 9)\)[/tex].
Therefore, the value of [tex]\(b\)[/tex] is 9.
