To factor the expression [tex]\( 81 y^2 - 4 \)[/tex], follow these steps:
1. Recognize the Structure:
The given expression [tex]\( 81 y^2 - 4 \)[/tex] is a difference of squares. The difference of squares formula states that:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
2. Identify [tex]\(a^2\)[/tex] and [tex]\(b^2\)[/tex]:
In [tex]\( 81 y^2 - 4 \)[/tex], we can see that:
[tex]\[
a^2 = 81 y^2 \quad \text{and} \quad b^2 = 4
\][/tex]
3. Find [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
Take the square root of each term:
[tex]\[
a = \sqrt{81 y^2} = 9y
\][/tex]
and
[tex]\[
b = \sqrt{4} = 2
\][/tex]
4. Apply the Difference of Squares Formula:
Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the difference of squares formula:
[tex]\[
81 y^2 - 4 = (9y)^2 - 2^2 = (9y - 2)(9y + 2)
\][/tex]
Therefore, the factored form of the expression [tex]\( 81 y^2 - 4 \)[/tex] is:
[tex]\[
(9y - 2)(9y + 2)
\][/tex]
