Expand $(2 x-y)(2 x+y)$
Answer (1)
Recognize the expression as a difference of squares: ( a − b ) ( a + b ) = a 2 − b 2 .
Identify a = 2 x and b = y .
Substitute into the formula: ( 2 x ) 2 − ( y ) 2 .
Simplify to obtain the final answer: 4 x 2 − y 2 .
Explanation
Understanding the Problem We are asked to expand the expression ( 2 x − y ) ( 2 x + y ) . This is a product of two binomials. The binomials are of the form ( a − b ) ( a + b ) where a = 2 x and b = y . We can use the difference of squares formula to expand this expression.
Stating the Objective The objective is to expand the expression ( 2 x − y ) ( 2 x + y ) .
Applying the Difference of Squares Formula We will use the difference of squares formula, which states that ( a − b ) ( a + b ) = a 2 − b 2 . We identify a = 2 x and b = y . Substituting these into the formula, we get ( 2 x ) 2 − ( y ) 2 .
Simplifying the Expression Now, we simplify the expression: ( 2 x ) 2 − ( y ) 2 = 4 x 2 − y 2 .
Final Answer Therefore, the expanded form of ( 2 x − y ) ( 2 x + y ) is 4 x 2 − y 2 .
Examples
The difference of squares is a useful concept in algebra and can be applied in various real-life scenarios. For instance, consider a rectangular garden whose area is to be expanded while keeping the total amount of fencing the same. If the original garden has sides 2 x + y and 2 x − y , the area is ( 2 x + y ) ( 2 x − y ) . If we want to change the dimensions such that the area is easily calculated, we can use the expanded form 4 x 2 − y 2 . This concept is also used in engineering to simplify calculations involving areas and volumes.
