Answer :

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

1. Look for a Greatest Common Factor (GCF):

The terms in the expression are [tex]\(27x^2\)[/tex] and [tex]\(-75y^2\)[/tex]. Both terms are divisible by 3. So, the GCF is 3.

2. Factor out the GCF:

Factor out the GCF of 3 from the expression:

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

3. Recognize the Difference of Squares:

Within the parentheses, we have [tex]\(9x^2 - 25y^2\)[/tex]. This is a difference of squares, which has the form [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].

4. Identify [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:

Here, [tex]\(9x^2\)[/tex] is [tex]\((3x)^2\)[/tex] and [tex]\(25y^2\)[/tex] is [tex]\((5y)^2\)[/tex]. Therefore, [tex]\(a = 3x\)[/tex] and [tex]\(b = 5y\)[/tex].

5. Apply the Difference of Squares Formula:

Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the formula:

[tex]\[
9x^2 - 25y^2 = (3x - 5y)(3x + 5y)
\][/tex]

6. Combine with the GCF:

Put it all together:

[tex]\[
27x^2 - 75y^2 = 3(3x - 5y)(3x + 5y)
\][/tex]

So, the expression [tex]\(27x^2 - 75y^2\)[/tex] is factored completely as [tex]\(3(3x - 5y)(3x + 5y)\)[/tex].

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