Certainly! Let's find the factored form of the given expression [tex]\(4x^2 - 64\)[/tex] and then verify which of the provided options are equivalent to it.
Step 1: Understand the given expression.
We start with:
[tex]\[ 4x^2 - 64 \][/tex]
Step 2: Factor out the common term.
We notice that 4 is a common factor:
[tex]\[ 4(x^2 - 16) \][/tex]
Step 3: Recognize the difference of squares.
The expression inside the parentheses, [tex]\(x^2 - 16\)[/tex], is a difference of squares. This can be factored as:
[tex]\[ x^2 - 16 = (x - 4)(x + 4) \][/tex]
Step 4: Combine the factors.
Putting it all together, we get:
[tex]\[ 4(x^2 - 16) = 4(x - 4)(x + 4) \][/tex]
So, the factored form of the given expression is:
[tex]\[ 4(x - 4)(x + 4) \][/tex]
Step 5: Verify the provided options to see if they are equivalent to [tex]\(4(x - 4)(x + 4)\)[/tex]:
1. [tex]\( 4(x + 4)(x - 4) \)[/tex]
- This matches our factored form exactly.
- Therefore, this is equivalent.
2. [tex]\( 4(x + 8)(x - 8) \)[/tex]
- If we expand this expression:
[tex]\[ 4(x + 8)(x - 8) = 4(x^2 - 64) = 4x^2 - 256 \][/tex]
- This is not equivalent to [tex]\(4x^2 - 64\)[/tex].
3. [tex]\( 2(x + 4)(x - 4) \)[/tex]
- If we expand this expression:
[tex]\[ 2(x + 4)(x - 4) = 2(x^2 - 16) = 2x^2 - 32 \][/tex]
- This is not equivalent to [tex]\(4x^2 - 64\)[/tex].
4. [tex]\( 2(2x + 4)(x - 8) \)[/tex]
- If we expand this expression:
[tex]\[ 2 (2x + 4)(x - 8) = 2(2x^2 - 16x + 4x - 32) = 4x^2 - 24x - 64 \][/tex]
- This is not equivalent to [tex]\(4x^2 - 64\)[/tex].
5. [tex]\( (2x + 8)(2x - 8) \)[/tex]
- If we expand this expression:
[tex]\[ (2x + 8)(2x - 8) = 4x^2 - 64 \][/tex]
- This is equivalent to [tex]\(4x^2 - 64\)[/tex].
Step 6: Summary of equivalent expressions.
The equivalent expressions to [tex]\(4x^2 - 64\)[/tex] are:
- [tex]\(4(x + 4)(x - 4)\)[/tex]
- [tex]\((2x + 8)(2x - 8)\)[/tex]
Thus, the correct selections are:
- [tex]\(4(x+4)(x-4)\)[/tex]
- [tex]\((2x+8)(2x-8)\)[/tex]
