To factor the expression [tex]\(49 x^2 - 81\)[/tex] completely, we can recognize that this is a difference of squares. The difference of squares is a common algebraic pattern that can be factored using the formula:
[tex]\[ a^2 - b^2 = (a - b)(a + b) \][/tex]
In this case, we need to identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that [tex]\(49 x^2\)[/tex] and [tex]\(81\)[/tex] are perfect squares.
1. Notice that [tex]\(49 x^2\)[/tex] is a perfect square:
[tex]\[
49 x^2 = (7x)^2
\][/tex]
So [tex]\(a = 7x\)[/tex].
2. Notice that [tex]\(81\)[/tex] is also a perfect square:
[tex]\[
81 = 9^2
\][/tex]
So [tex]\(b = 9\)[/tex].
Now we apply the difference of squares formula:
[tex]\[
49 x^2 - 81 = (7x)^2 - 9^2 = (7x - 9)(7x + 9)
\][/tex]
Thus, the factorization of [tex]\(49 x^2 - 81\)[/tex] is:
[tex]\[
(7x - 9)(7x + 9)
\][/tex]
So, the completely factored form of [tex]\(49 x^2 - 81\)[/tex] is:
[tex]\[
(7x - 9)(7x + 9)
\][/tex]
