To factor the quadratic expression [tex]\(4b^2 - 9\)[/tex], you can use the difference of squares method.
Here's the step-by-step solution:
1. Identify the pattern:
The expression [tex]\(4b^2 - 9\)[/tex] is in the form of [tex]\(a^2 - c^2\)[/tex], which is a difference of squares. The difference of squares can be factored using the formula:
[tex]\[
a^2 - c^2 = (a - c)(a + c)
\][/tex]
2. Rewrite the terms:
Notice that [tex]\(4b^2\)[/tex] is a perfect square and so is [tex]\(9\)[/tex]. Specifically:
[tex]\[
4b^2 = (2b)^2 \quad \text{and} \quad 9 = 3^2
\][/tex]
So, we can rewrite the expression [tex]\(4b^2 - 9\)[/tex] as:
[tex]\[
(2b)^2 - 3^2
\][/tex]
3. Apply the difference of squares formula:
Using the formula [tex]\(a^2 - c^2 = (a - c)(a + c)\)[/tex], we get:
[tex]\[
(2b - 3)(2b + 3)
\][/tex]
Therefore, the factored form of [tex]\(4b^2 - 9\)[/tex] is:
[tex]\[
(2b - 3)(2b + 3)
\][/tex]
The given answer options include [tex]\((2b + 3)(2b - 3)\)[/tex], which is equivalent to our result [tex]\( (2b - 3)(2b + 3) \)[/tex] due to the commutative property of multiplication.
Hence, the correct factored form of the expression [tex]\(4b^2 - 9\)[/tex] is:
[tex]\[
(2b - 3)(2b + 3)
\][/tex]
