Answer :
To factor the expression [tex]\(27 - 48x^2\)[/tex] completely, we can follow these steps:
1. Look for a common factor: Check if there is a number that divides all terms in the expression. We can see that both 27 and 48 have a common factor of 3. Factoring out the 3, we get:
[tex]\[
27 - 48x^2 = 3(9 - 16x^2)
\][/tex]
2. Recognize the difference of squares: The expression inside the parentheses, [tex]\(9 - 16x^2\)[/tex], is a difference of squares. A difference of squares has the form [tex]\(a^2 - b^2\)[/tex] and can be factored as [tex]\((a - b)(a + b)\)[/tex].
In our case, [tex]\(9\)[/tex] is [tex]\(3^2\)[/tex] and [tex]\(16x^2\)[/tex] is [tex]\((4x)^2\)[/tex]. So, we can apply the difference of squares formula:
[tex]\[
9 - 16x^2 = (3 - 4x)(3 + 4x)
\][/tex]
3. Combine the factored parts: Now that we have factored [tex]\(9 - 16x^2\)[/tex] into [tex]\((3 - 4x)(3 + 4x)\)[/tex], we multiply this by the common factor we initially factored out (3):
[tex]\[
27 - 48x^2 = 3(3 - 4x)(3 + 4x)
\][/tex]
To put it in standard order, we can write the factors of the quadratic terms first:
[tex]\[
27 - 48x^2 = -3(4x - 3)(4x + 3)
\][/tex]
Thus, the factorization of [tex]\(27 - 48x^2\)[/tex] is [tex]\(-3(4x - 3)(4x + 3)\)[/tex].
1. Look for a common factor: Check if there is a number that divides all terms in the expression. We can see that both 27 and 48 have a common factor of 3. Factoring out the 3, we get:
[tex]\[
27 - 48x^2 = 3(9 - 16x^2)
\][/tex]
2. Recognize the difference of squares: The expression inside the parentheses, [tex]\(9 - 16x^2\)[/tex], is a difference of squares. A difference of squares has the form [tex]\(a^2 - b^2\)[/tex] and can be factored as [tex]\((a - b)(a + b)\)[/tex].
In our case, [tex]\(9\)[/tex] is [tex]\(3^2\)[/tex] and [tex]\(16x^2\)[/tex] is [tex]\((4x)^2\)[/tex]. So, we can apply the difference of squares formula:
[tex]\[
9 - 16x^2 = (3 - 4x)(3 + 4x)
\][/tex]
3. Combine the factored parts: Now that we have factored [tex]\(9 - 16x^2\)[/tex] into [tex]\((3 - 4x)(3 + 4x)\)[/tex], we multiply this by the common factor we initially factored out (3):
[tex]\[
27 - 48x^2 = 3(3 - 4x)(3 + 4x)
\][/tex]
To put it in standard order, we can write the factors of the quadratic terms first:
[tex]\[
27 - 48x^2 = -3(4x - 3)(4x + 3)
\][/tex]
Thus, the factorization of [tex]\(27 - 48x^2\)[/tex] is [tex]\(-3(4x - 3)(4x + 3)\)[/tex].
