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Answer :
To solve the expression [tex]\(27x^2 - 48\)[/tex], we will simplify it by factoring.
### Step-by-Step Solution:
1. Identify the Common Factor:
- Look at the coefficients in the expression: 27 and 48.
- Both numbers can be divided by 3, which is the greatest common factor.
2. Factor Out the Greatest Common Factor:
- Factor out 3 from each term in the expression:
[tex]\[
27x^2 - 48 = 3(9x^2) - 3(16)
\][/tex]
3. Rewrite the Expression:
- After factoring out 3, rewrite the expression:
[tex]\[
27x^2 - 48 = 3(9x^2 - 16)
\][/tex]
4. Recognize a Difference of Squares:
- The expression inside the parenthesis, [tex]\(9x^2 - 16\)[/tex], is a difference of squares.
- It can be written as [tex]\( (3x)^2 - 4^2 \)[/tex].
5. Factor the Difference of Squares:
- Use the identity [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
- Factor [tex]\(9x^2 - 16\)[/tex] as [tex]\((3x - 4)(3x + 4)\)[/tex].
6. Combine Everything:
- Write the entire factored expression:
[tex]\[
27x^2 - 48 = 3(3x - 4)(3x + 4)
\][/tex]
The expression [tex]\(27x^2 - 48\)[/tex] is simplified and factored completely as [tex]\(3(3x - 4)(3x + 4)\)[/tex].
### Step-by-Step Solution:
1. Identify the Common Factor:
- Look at the coefficients in the expression: 27 and 48.
- Both numbers can be divided by 3, which is the greatest common factor.
2. Factor Out the Greatest Common Factor:
- Factor out 3 from each term in the expression:
[tex]\[
27x^2 - 48 = 3(9x^2) - 3(16)
\][/tex]
3. Rewrite the Expression:
- After factoring out 3, rewrite the expression:
[tex]\[
27x^2 - 48 = 3(9x^2 - 16)
\][/tex]
4. Recognize a Difference of Squares:
- The expression inside the parenthesis, [tex]\(9x^2 - 16\)[/tex], is a difference of squares.
- It can be written as [tex]\( (3x)^2 - 4^2 \)[/tex].
5. Factor the Difference of Squares:
- Use the identity [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex].
- Factor [tex]\(9x^2 - 16\)[/tex] as [tex]\((3x - 4)(3x + 4)\)[/tex].
6. Combine Everything:
- Write the entire factored expression:
[tex]\[
27x^2 - 48 = 3(3x - 4)(3x + 4)
\][/tex]
The expression [tex]\(27x^2 - 48\)[/tex] is simplified and factored completely as [tex]\(3(3x - 4)(3x + 4)\)[/tex].
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