Factor [tex]25 x^2-64[/tex].
[tex](5 x+8)(5 x-8)[/tex]
[tex](-5 x+8)(5 x-8)[/tex]
[tex](5 x+8)(-5 x-8)[/tex]
[tex](5 x-8)^2[/tex]
Answer (1)
The expression 25 x 2 − 64 is a difference of squares. We identify a = 5 x and b = 8 . Applying the difference of squares factorization, we get ( 5 x + 8 ) ( 5 x − 8 ) . Therefore, the factored form of the expression is ( 5 x + 8 ) ( 5 x − 8 ) .
Explanation
Recognizing the Pattern We are asked to factor the expression 25 x 2 − 64 . This looks like a difference of squares, which has a specific factoring pattern.
Identifying a and b The difference of squares pattern is a 2 − b 2 = ( a + b ) ( a − b ) . We need to identify what 'a' and 'b' are in our expression. We can rewrite 25 x 2 − 64 as ( 5 x ) 2 − ( 8 ) 2 . So, a = 5 x and b = 8 .
Applying the Pattern Now we can apply the difference of squares pattern: ( 5 x ) 2 − ( 8 ) 2 = ( 5 x + 8 ) ( 5 x − 8 ) .
Selecting the Correct Option Comparing our factored expression ( 5 x + 8 ) ( 5 x − 8 ) with the given options, we see that it matches the first option.
Examples
Factoring the difference of squares is a common technique in algebra and can be used in various real-world applications. For example, consider a rectangular garden whose area is given by 25 x 2 − 64 . By factoring this expression into ( 5 x + 8 ) ( 5 x − 8 ) , we can determine the dimensions of the garden in terms of x . This can help in planning the layout, calculating the amount of fencing needed, or determining the amount of fertilizer to apply. Understanding factoring allows us to solve practical problems involving areas, volumes, and other geometric quantities.
