Answer :
Let's solve the expression [tex]\(27x^2 - 48y^2\)[/tex] by simplifying or factoring it, if possible.
This expression can be recognized as a difference of squares. The difference of squares formula is given by:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
However, before applying the difference of squares formula, let's see if we can factor out any common factors from the expression first.
1. Identify the Greatest Common Factor (GCF):
The GCF of [tex]\(27x^2\)[/tex] and [tex]\(48y^2\)[/tex] is 3. Let's factor out 3 from the expression:
[tex]\[ 27x^2 - 48y^2 = 3(9x^2 - 16y^2) \][/tex]
2. Factor the Expression Inside the Parentheses:
Now, within the parentheses, [tex]\(9x^2 - 16y^2\)[/tex] is a difference of squares because:
- [tex]\(9x^2\)[/tex] is [tex]\((3x)^2\)[/tex]
- [tex]\(16y^2\)[/tex] is [tex]\((4y)^2\)[/tex]
We can apply the difference of squares formula:
[tex]\[ 9x^2 - 16y^2 = (3x + 4y)(3x - 4y) \][/tex]
3. Combine Everything:
Now plug it back into the expression we factored the GCF out of:
[tex]\[ 27x^2 - 48y^2 = 3(3x + 4y)(3x - 4y) \][/tex]
So, the fully factored form of the expression [tex]\(27x^2 - 48y^2\)[/tex] is:
[tex]\[ 3(3x + 4y)(3x - 4y) \][/tex]
This solution involves recognizing the structure of the expression, identifying the difference of squares, and applying the appropriate factoring steps.
This expression can be recognized as a difference of squares. The difference of squares formula is given by:
[tex]\[ a^2 - b^2 = (a + b)(a - b) \][/tex]
However, before applying the difference of squares formula, let's see if we can factor out any common factors from the expression first.
1. Identify the Greatest Common Factor (GCF):
The GCF of [tex]\(27x^2\)[/tex] and [tex]\(48y^2\)[/tex] is 3. Let's factor out 3 from the expression:
[tex]\[ 27x^2 - 48y^2 = 3(9x^2 - 16y^2) \][/tex]
2. Factor the Expression Inside the Parentheses:
Now, within the parentheses, [tex]\(9x^2 - 16y^2\)[/tex] is a difference of squares because:
- [tex]\(9x^2\)[/tex] is [tex]\((3x)^2\)[/tex]
- [tex]\(16y^2\)[/tex] is [tex]\((4y)^2\)[/tex]
We can apply the difference of squares formula:
[tex]\[ 9x^2 - 16y^2 = (3x + 4y)(3x - 4y) \][/tex]
3. Combine Everything:
Now plug it back into the expression we factored the GCF out of:
[tex]\[ 27x^2 - 48y^2 = 3(3x + 4y)(3x - 4y) \][/tex]
So, the fully factored form of the expression [tex]\(27x^2 - 48y^2\)[/tex] is:
[tex]\[ 3(3x + 4y)(3x - 4y) \][/tex]
This solution involves recognizing the structure of the expression, identifying the difference of squares, and applying the appropriate factoring steps.
