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Factor [tex]$4b^2 - 9$[/tex].

A. [tex]$(2b + 3)(2b - 3)$[/tex]

B. [tex][tex]$(4b + 3)(b - 3)$[/tex][/tex]

C. [tex]$(2b - 4)(2b - 5)$[/tex]


Sagot :

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]
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