To factor the given quadratic expression [tex]\( 9x^2 - 25 \)[/tex] completely, we will follow a step-by-step approach:
1. Recognize the form of the expression: Notice that the given expression, [tex]\( 9x^2 - 25 \)[/tex], is a difference of squares. The difference of squares can be factored using the identity:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
2. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the expression [tex]\( 9x^2 - 25 \)[/tex]:
- In our case, [tex]\( 9x^2 \)[/tex] is a perfect square, and so is [tex]\( 25 \)[/tex].
- We can write [tex]\( 9x^2 \)[/tex] as [tex]\( (3x)^2 \)[/tex] and [tex]\( 25 \)[/tex] as [tex]\( 5^2 \)[/tex].
3. Apply the difference of squares identity:
Given:
[tex]\[
9x^2 - 25
\][/tex]
Write it as:
[tex]\[
(3x)^2 - 5^2
\][/tex]
Now, apply the difference of squares formula:
[tex]\[
(3x)^2 - 5^2 = (3x - 5)(3x + 5)
\][/tex]
4. Verify the factors given in the multiple-choice options:
- Option 1: [tex]\( (3x + 5)(3x - 5) \)[/tex]
- Option 2: [tex]\( (3x - 5)(3x - 5) \)[/tex]
- Option 3: [tex]\( (9x + 5)(x - 5) \)[/tex]
- Option 4: [tex]\( (9x - 5)(x + 5) \)[/tex]
Comparing these with our factorization [tex]\( (3x - 5)(3x + 5) \)[/tex], we see that the correct factorization corresponds to Option 1:
Thus, the completely factored form of [tex]\( 9x^2 - 25 \)[/tex] is:
[tex]\[
(3x - 5)(3x + 5)
\][/tex]
