What is the product of $(2 x-5)(2 x+5)$?
A. $4 x^2-25$
B. $4 x^2+20 x-10$
C. $4 x^2+20 x-25$
D. $4 x^2-10
Answer (2)
Recognize the expression as a difference of squares: ( a − b ) ( a + b ) = a 2 − b 2 .
Identify a = 2 x and b = 5 .
Apply the formula: ( 2 x − 5 ) ( 2 x + 5 ) = ( 2 x ) 2 − ( 5 ) 2 .
Simplify to get the final answer: 4 x 2 − 25 .
Explanation
Recognizing the Pattern We are asked to find the product of two binomials: ( 2 x − 5 ) and ( 2 x + 5 ) . This looks like a special product, specifically the difference of squares.
Identifying a and b The expression ( 2 x − 5 ) ( 2 x + 5 ) is in the form of ( a − b ) ( a + b ) , which equals a 2 − b 2 . In this case, a = 2 x and b = 5 .
Applying the Formula Now, we apply the difference of squares formula: ( 2 x − 5 ) ( 2 x + 5 ) = ( 2 x ) 2 − ( 5 ) 2
Simplifying the Expression Next, we simplify the expression: ( 2 x ) 2 = 4 x 2 and ( 5 ) 2 = 25 Therefore, ( 2 x ) 2 − ( 5 ) 2 = 4 x 2 − 25
Final Answer So, the product of ( 2 x − 5 ) ( 2 x + 5 ) is 4 x 2 − 25 . Looking at the multiple-choice options, we see that the correct answer is A.
Examples
The difference of squares pattern is useful in various applications, such as simplifying algebraic expressions, solving equations, and even in engineering and physics. For example, if you are designing a square garden with side length 2 x and want to remove a square section of side length 5 , the remaining area can be expressed as 4 x 2 − 25 , which factors into ( 2 x − 5 ) ( 2 x + 5 ) . This factorization can help you determine the dimensions of a rectangular garden with the same area.
The product of ( 2 x − 5 ) ( 2 x + 5 ) is 4 x 2 − 25 using the difference of squares formula. The answer is option A. This is derived by recognizing that the expression follows the pattern a 2 − b 2 .
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