Answer :

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

1. Identify Common Factors:
- First, we look for the greatest common factor (GCF) in the expression. The GCF of 27 and 75 is 3.
- Factor out the GCF from the expression:
[tex]\[
27x^2 - 75y^2 = 3(9x^2 - 25y^2)
\][/tex]

2. Recognize the Difference of Squares:
- The expression inside the parentheses, [tex]\( 9x^2 - 25y^2 \)[/tex], is a difference of squares.
- Remember the identity for factoring a difference of squares: [tex]\( a^2 - b^2 = (a-b)(a+b) \)[/tex].

3. Apply the Difference of Squares Formula:
- In [tex]\( 9x^2 - 25y^2 \)[/tex], [tex]\( a^2 = (3x)^2 \)[/tex] and [tex]\( b^2 = (5y)^2 \)[/tex].
- Use the difference of squares identity to factor:
[tex]\[
9x^2 - 25y^2 = (3x - 5y)(3x + 5y)
\][/tex]

4. Combine All Components:
- Substitute back into the expression after factoring out the GCF:
[tex]\[
27x^2 - 75y^2 = 3(9x^2 - 25y^2) = 3(3x - 5y)(3x + 5y)
\][/tex]

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

Other Questions