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Factorize completely the expression [tex]$75x^2 - 27y^2$[/tex].

Answer :

Sure! Let's factorize the expression [tex]\(75x^2 - 27y^2\)[/tex] step-by-step:

1. Identify the Greatest Common Factor (GCF):

First, look for the greatest common factor in the expression.

[tex]\[
\text{GCF of } 75 \text{ and } 27 \text{ is } 3.
\][/tex]

So, factor out the GCF (3) from the entire expression:

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

2. Recognize the Difference of Squares:

Inside the parenthesis, we have [tex]\(25x^2 - 9y^2\)[/tex]. This is a difference of squares because each term is a perfect square:

[tex]\[
25x^2 = (5x)^2 \quad \text{and} \quad 9y^2 = (3y)^2.
\][/tex]

3. Apply the Difference of Squares Formula:

Recall that the difference of squares formula is:

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

So, apply this to [tex]\(25x^2 - 9y^2\)[/tex]:

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

4. Combine with the GCF:

Now, bring back the factor that was factored out earlier (the GCF of 3):

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

Thus, the expression [tex]\(75x^2 - 27y^2\)[/tex] factorizes completely to:

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

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