Square the first term minus square the second term.
[tex](3 x)^2=9 x^2(4)^2=16[/tex]
[tex](3 x+4)(3 x-4)=9 x^2-16[/tex]
1. [tex](x+2)(x-2) \quad x^2-4[/tex]
2. [tex](x+6)(x-6) \quad x^2-36[/tex]
3. [tex](5 x+3)(5 x-3) 25 x^2-9[/tex]
4. [tex](3 x+7)(3 x-7)-9 x^2-49[/tex]
5. [tex](2 x+4)(2 x-4) 4 x^2-16[/tex]
Answer (2)
The question involves verifying the application of the difference of squares formula. Each example provided correctly demonstrates the formula, confirming the answers given. Thus, all solutions align with a 2 − b 2 calculation principles.
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Apply the difference of squares formula ( a + b ) ( a − b ) = a 2 − b 2 .
Verify each given solution by identifying 'a' and 'b', calculating a 2 and b 2 , and confirming the result.
All the given answers are correct.
The solutions are: 1. x 2 − 4 , 2. x 2 − 36 , 3. 25 x 2 − 9 , 4. 9 x 2 − 49 , 5. 4 x 2 − 16 .
Explanation
Understanding the Problem The problem involves applying the difference of squares formula, which states that ( a + b ) ( a − b ) = a 2 − b 2 . We need to verify if the given solutions are correct.
Applying the Difference of Squares Formula
( x + 2 ) ( x − 2 ) : Here, a = x and b = 2 . So, a 2 = x 2 and b 2 = 2 2 = 4 . Therefore, ( x + 2 ) ( x − 2 ) = x 2 − 4 . This matches the given answer.
( x + 6 ) ( x − 6 ) : Here, a = x and b = 6 . So, a 2 = x 2 and b 2 = 6 2 = 36 . Therefore, ( x + 6 ) ( x − 6 ) = x 2 − 36 . This matches the given answer.
( 5 x + 3 ) ( 5 x − 3 ) : Here, a = 5 x and b = 3 . So, a 2 = ( 5 x ) 2 = 25 x 2 and b 2 = 3 2 = 9 . Therefore, ( 5 x + 3 ) ( 5 x − 3 ) = 25 x 2 − 9 . This matches the given answer.
( 3 x + 7 ) ( 3 x − 7 ) : Here, a = 3 x and b = 7 . So, a 2 = ( 3 x ) 2 = 9 x 2 and b 2 = 7 2 = 49 . Therefore, ( 3 x + 7 ) ( 3 x − 7 ) = 9 x 2 − 49 . This matches the given answer.
( 2 x + 4 ) ( 2 x − 4 ) : Here, a = 2 x and b = 4 . So, a 2 = ( 2 x ) 2 = 4 x 2 and b 2 = 4 2 = 16 . Therefore, ( 2 x + 4 ) ( 2 x − 4 ) = 4 x 2 − 16 . This matches the given answer.
Conclusion All the given answers are correct applications of the difference of squares formula.
Examples
The difference of squares formula is useful in various fields, such as engineering, physics, and computer science. For example, it can be used to simplify calculations in structural analysis, signal processing, and cryptography. In real life, this formula helps in simplifying complex problems, making them easier to solve.
