To factor the given expression [tex]\( 36 c^2 - 121 d^2 \)[/tex] completely, we begin by recognizing that this expression is a difference of squares. A difference of squares can always be factored using the following formula:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
First, we identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] such that the expression fits the form [tex]\(a^2 - b^2\)[/tex].
In our given expression, [tex]\( 36 c^2 - 121 d^2 \)[/tex], we can see that:
[tex]\[
a^2 = 36 c^2 \quad \text{and} \quad b^2 = 121 d^2
\][/tex]
We need to determine [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
a = 6c \quad \text{because} \quad (6c)^2 = 36c^2
\][/tex]
[tex]\[
b = 11d \quad \text{because} \quad (11d)^2 = 121d^2
\][/tex]
Now that we have identified [tex]\(a = 6c\)[/tex] and [tex]\(b = 11d\)[/tex], we can apply the difference of squares formula:
[tex]\[
36 c^2 - 121 d^2 = (6c)^2 - (11d)^2 = (6c - 11d)(6c + 11d)
\][/tex]
Thus, the completely factored form of the expression [tex]\(36 c^2 - 121 d^2\)[/tex] is:
[tex]\[
(6c - 11d)(6c + 11d)
\][/tex]
