Factor $4x^2 - 9$.
A. $(2x+3)(2x-3)$
B. $(2x-3)^2$
C. $(2x+3)(-2x-3)$
D. $(-2x+3)(2x-3)$
Answer (1)
The expression 4 x 2 − 9 is a difference of squares. We identify a = 2 x and b = 3 . Applying the difference of squares factorization, we get ( 2 x + 3 ) ( 2 x − 3 ) . The factored form of the expression is ( 2 x + 3 ) ( 2 x − 3 ) .
Explanation
Recognizing the Pattern We are asked to factor the expression 4 x 2 − 9 . This looks like a difference of squares, which has the form a 2 − b 2 . We can factor this as ( a + b ) ( a − b ) .
Identifying a and b In our expression, 4 x 2 − 9 , we can identify a 2 = 4 x 2 and b 2 = 9 . Taking the square root of both sides, we get a = 2 x and b = 3 .
Applying the Factorization Now we can substitute these values into the difference of squares factorization: ( a + b ) ( a − b ) = ( 2 x + 3 ) ( 2 x − 3 ) .
Examples
Factoring the difference of squares is a useful technique in many areas of mathematics and physics. For example, it can be used to simplify expressions in algebra, solve equations, and analyze the motion of objects. Imagine you are designing a rectangular garden where the area is represented by 4 x 2 − 9 . By factoring this expression into ( 2 x + 3 ) ( 2 x − 3 ) , you determine the dimensions of the garden, ensuring it fits perfectly within your available space. This skill is crucial for optimizing space and resources in various practical scenarios.
