Which expression is a difference of squares with a factor of [tex]$5 x-8$[/tex]?
A. [tex]$25 x^2-16$[/tex]
B. [tex]$25 x^2+16$[/tex]
C. [tex]$25 x^2-64$[/tex]
D. [tex]$25 x^2+64$[/tex]
Answer (1)
Check if the given expressions are a difference of squares.
Factor the expressions that are a difference of squares.
Identify the expression with a factor of 5 x − 8 .
The expression 25 x 2 − 64 is a difference of squares and has a factor of 5 x − 8 , so the answer is 25 x 2 − 64 .
Explanation
Problem Analysis We are given four expressions and need to identify the one that is a difference of squares and has a factor of 5 x − 8 . Let's analyze each option.
Checking Each Expression
25 x 2 − 16 : This is a difference of squares, since 25 x 2 = ( 5 x ) 2 and 16 = 4 2 . Factoring, we get ( 5 x − 4 ) ( 5 x + 4 ) . The factor 5 x − 8 is not present.
25 x 2 + 16 : This is a sum of squares, not a difference of squares, so we can eliminate it.
25 x 2 − 64 : This is a difference of squares, since 25 x 2 = ( 5 x ) 2 and 64 = 8 2 . Factoring, we get ( 5 x − 8 ) ( 5 x + 8 ) . The factor 5 x − 8 is present.
25 x 2 + 64 : This is a sum of squares, not a difference of squares, so we can eliminate it.
Conclusion The expression 25 x 2 − 64 is a difference of squares and has a factor of 5 x − 8 .
Examples
Difference of squares is a useful concept in algebra and can be applied in various real-life scenarios. For example, consider a rectangular garden whose area can be represented by the expression 25 x 2 − 64 . By factoring this expression into ( 5 x − 8 ) ( 5 x + 8 ) , we can determine the dimensions of the garden. This can help in planning the layout, calculating the amount of fencing needed, or optimizing the use of space. Understanding difference of squares allows for efficient problem-solving in geometry and design.
