Answer :
To factor the expression [tex]\(27x^2 - 48y^2\)[/tex] completely, follow these steps:
1. Identify the Form:
The given expression is of the form [tex]\(a^2 - b^2\)[/tex], which is a difference of squares. In general, a difference of squares can be factored using the identity:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
2. Rewrite the Terms:
Express the given terms in a form that resembles the difference of squares:
[tex]\[
27x^2 - 48y^2 = (3x)^2 - (4\sqrt{3}y)^2
\][/tex]
Here, [tex]\(a = 3x\)[/tex] and [tex]\(b = 4\sqrt{3}y\)[/tex].
3. Apply the Difference of Squares Formula:
Using the difference of squares formula:
[tex]\[
(3x)^2 - (4\sqrt{3}y)^2 = (3x - 4\sqrt{3}y)(3x + 4\sqrt{3}y)
\][/tex]
4. Factor Completely:
Notice that both factors involve a common numerical coefficient [tex]\(3\)[/tex] that can be factored out from each term:
- Factor the number 3 out of each term:
[tex]\[
27x^2 - 48y^2 = 3(9x^2 - 16y^2)
\][/tex]
5. Recognize Further Difference of Squares:
The expression inside the parentheses, [tex]\(9x^2 - 16y^2\)[/tex], is itself a difference of squares, which can be factored as:
[tex]\[
9x^2 - 16y^2 = (3x)^2 - (4y)^2 = (3x - 4y)(3x + 4y)
\][/tex]
6. Combine the Factors:
Plug these factors back with the previously factored 3:
[tex]\[
3(9x^2 - 16y^2) = 3(3x - 4y)(3x + 4y)
\][/tex]
Therefore, the expression [tex]\(27x^2 - 48y^2\)[/tex] factors completely to:
[tex]\[
3(3x - 4y)(3x + 4y)
\][/tex]
1. Identify the Form:
The given expression is of the form [tex]\(a^2 - b^2\)[/tex], which is a difference of squares. In general, a difference of squares can be factored using the identity:
[tex]\[
a^2 - b^2 = (a - b)(a + b)
\][/tex]
2. Rewrite the Terms:
Express the given terms in a form that resembles the difference of squares:
[tex]\[
27x^2 - 48y^2 = (3x)^2 - (4\sqrt{3}y)^2
\][/tex]
Here, [tex]\(a = 3x\)[/tex] and [tex]\(b = 4\sqrt{3}y\)[/tex].
3. Apply the Difference of Squares Formula:
Using the difference of squares formula:
[tex]\[
(3x)^2 - (4\sqrt{3}y)^2 = (3x - 4\sqrt{3}y)(3x + 4\sqrt{3}y)
\][/tex]
4. Factor Completely:
Notice that both factors involve a common numerical coefficient [tex]\(3\)[/tex] that can be factored out from each term:
- Factor the number 3 out of each term:
[tex]\[
27x^2 - 48y^2 = 3(9x^2 - 16y^2)
\][/tex]
5. Recognize Further Difference of Squares:
The expression inside the parentheses, [tex]\(9x^2 - 16y^2\)[/tex], is itself a difference of squares, which can be factored as:
[tex]\[
9x^2 - 16y^2 = (3x)^2 - (4y)^2 = (3x - 4y)(3x + 4y)
\][/tex]
6. Combine the Factors:
Plug these factors back with the previously factored 3:
[tex]\[
3(9x^2 - 16y^2) = 3(3x - 4y)(3x + 4y)
\][/tex]
Therefore, the expression [tex]\(27x^2 - 48y^2\)[/tex] factors completely to:
[tex]\[
3(3x - 4y)(3x + 4y)
\][/tex]
