Answer :

To factor the expression [tex]\(75x^2 - 27y^2\)[/tex] completely, follow these steps:

1. Identify the Greatest Common Factor (GCF):

First, we need to find the greatest common factor of the terms in the expression. The terms are [tex]\(75x^2\)[/tex] and [tex]\(-27y^2\)[/tex].

- The GCF of the coefficients 75 and 27 is 3.

Therefore, we factor out 3 from the expression:

[tex]\[
75x^2 - 27y^2 = 3(25x^2 - 9y^2)
\][/tex]

2. Recognize the Difference of Squares:

After factoring out the GCF, we have the expression [tex]\(25x^2 - 9y^2\)[/tex] left to factor. This expression is a difference of squares, which follows the formula:

[tex]\[
a^2 - b^2 = (a + b)(a - b)
\][/tex]

Here, we identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex] as follows:

- [tex]\(a^2 = 25x^2\)[/tex], so [tex]\(a = 5x\)[/tex]
- [tex]\(b^2 = 9y^2\)[/tex], so [tex]\(b = 3y\)[/tex]

3. Apply the Difference of Squares Formula:

Using the difference of squares formula, we can factor [tex]\(25x^2 - 9y^2\)[/tex] as:

[tex]\[
(5x + 3y)(5x - 3y)
\][/tex]

4. Combine Both Steps:

Therefore, the completely factored form of the original expression [tex]\(75x^2 - 27y^2\)[/tex] is:

[tex]\[
3(5x + 3y)(5x - 3y)
\][/tex]

This is the complete factorization of the expression [tex]\(75x^2 - 27y^2\)[/tex].

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