Factor.
[tex]49 v^2-4[/tex]
Answer (1)
Recognize the expression as a difference of squares, identify the terms being squared, and apply the difference of squares factorization formula. The factored form of 49 v 2 − 4 is ( 7 v − 2 ) ( 7 v + 2 ) .
Explanation
Recognizing the Pattern We are asked to factor the expression 49 v 2 − 4 . This looks like a difference of squares, which has the form a 2 − b 2 .
Difference of Squares The difference of squares factorization is a 2 − b 2 = ( a − b ) ( a + b ) . We need to identify a and b in our expression.
Identifying a and b We have 49 v 2 − 4 . We can rewrite this as ( 7 v ) 2 − ( 2 ) 2 . So, a = 7 v and b = 2 .
Applying the Factorization Now we can apply the difference of squares factorization: ( a − b ) ( a + b ) = ( 7 v − 2 ) ( 7 v + 2 ) .
Examples
Factoring the difference of squares is a useful technique in many areas of mathematics and physics. For example, suppose you are designing a square garden with an area of 49 v 2 square feet and you want to reduce the area by 4 square feet. Factoring the expression 49 v 2 − 4 as ( 7 v − 2 ) ( 7 v + 2 ) helps you determine the new dimensions of the garden if it were to remain rectangular. This type of problem arises in various optimization and design scenarios.
