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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].
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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