Answer :
To rewrite the expression [tex]\( x^2 - 7 \)[/tex] as a Difference of Squares, follow these steps:
1. Identify the given expression: The expression we have is [tex]\( x^2 - 7 \)[/tex].
2. Recognize the form of a Difference of Squares: The formula for a Difference of Squares is [tex]\( a^2 - b^2 = (a + b)(a - b) \)[/tex].
3. Determine if the given expression fits the form:
- We already have [tex]\( x^2 \)[/tex] which is a perfect square, but we need to rewrite 7 in a way that also fits the form of a square.
- Notice that [tex]\( 7 \)[/tex] can be written as [tex]\( (\sqrt{7})^2 \)[/tex].
4. Rewrite the expression accordingly:
- Rewrite [tex]\( x^2 - 7 \)[/tex] as [tex]\( x^2 - (\sqrt{7})^2 \)[/tex].
5. Apply the Difference of Squares formula:
- Here, [tex]\( a = x \)[/tex] and [tex]\( b = \sqrt{7} \)[/tex].
- Substituting into the formula [tex]\( a^2 - b^2 = (a + b)(a - b) \)[/tex]:
[tex]\[ x^2 - (\sqrt{7})^2 = (x + \sqrt{7})(x - \sqrt{7}) \][/tex]
6. Review options to find the correct one:
- Option A: [tex]\( (x + \sqrt{7})(x - \sqrt{7}) \)[/tex]
- Option B: [tex]\( (x - 7)(x + 7) \)[/tex] does not match.
- Option C: [tex]\( (x + 49)(x - 49) \)[/tex] involves incorrect square terms.
- Option D: [tex]\( (x + 7)^2 \)[/tex] is not in the form of the Difference of Squares.
The correct answer is:
A. [tex]\( (x + \sqrt{7})(x - \sqrt{7}) \)[/tex]
1. Identify the given expression: The expression we have is [tex]\( x^2 - 7 \)[/tex].
2. Recognize the form of a Difference of Squares: The formula for a Difference of Squares is [tex]\( a^2 - b^2 = (a + b)(a - b) \)[/tex].
3. Determine if the given expression fits the form:
- We already have [tex]\( x^2 \)[/tex] which is a perfect square, but we need to rewrite 7 in a way that also fits the form of a square.
- Notice that [tex]\( 7 \)[/tex] can be written as [tex]\( (\sqrt{7})^2 \)[/tex].
4. Rewrite the expression accordingly:
- Rewrite [tex]\( x^2 - 7 \)[/tex] as [tex]\( x^2 - (\sqrt{7})^2 \)[/tex].
5. Apply the Difference of Squares formula:
- Here, [tex]\( a = x \)[/tex] and [tex]\( b = \sqrt{7} \)[/tex].
- Substituting into the formula [tex]\( a^2 - b^2 = (a + b)(a - b) \)[/tex]:
[tex]\[ x^2 - (\sqrt{7})^2 = (x + \sqrt{7})(x - \sqrt{7}) \][/tex]
6. Review options to find the correct one:
- Option A: [tex]\( (x + \sqrt{7})(x - \sqrt{7}) \)[/tex]
- Option B: [tex]\( (x - 7)(x + 7) \)[/tex] does not match.
- Option C: [tex]\( (x + 49)(x - 49) \)[/tex] involves incorrect square terms.
- Option D: [tex]\( (x + 7)^2 \)[/tex] is not in the form of the Difference of Squares.
The correct answer is:
A. [tex]\( (x + \sqrt{7})(x - \sqrt{7}) \)[/tex]
