Which of the following is NOT equivalent to $(3 x-12)(x+4)$?
A. $3(x^2-8 x+16)$
B. $3(x^2-16)$
C. $3 x^2-48$
D. $3 x(x+4)-12(x+4)$
Answer (1)
Expand the original expression: ( 3 x − 12 ) ( x + 4 ) = 3 x 2 − 48 .
Expand each of the given expressions.
Compare the expanded forms to the original expression.
Identify the non-equivalent expression: 3 ( x 2 − 8 x + 16 ) .
Explanation
Understanding the Problem We are given the expression ( 3 x − 12 ) ( x + 4 ) and four other expressions. We need to find the expression that is NOT equivalent to the given expression.
Expanding the Original Expression First, let's expand the given expression ( 3 x − 12 ) ( x + 4 ) : ( 3 x − 12 ) ( x + 4 ) = 3 x ( x + 4 ) − 12 ( x + 4 ) = 3 x 2 + 12 x − 12 x − 48 = 3 x 2 − 48
Expanding the Other Expressions Now, let's expand and simplify each of the four given expressions:
3 ( x 2 − 8 x + 16 ) = 3 x 2 − 24 x + 48
3 ( x 2 − 16 ) = 3 x 2 − 48
3 x 2 − 48 = 3 x 2 − 48
3 x ( x + 4 ) − 12 ( x + 4 ) = 3 x 2 + 12 x − 12 x − 48 = 3 x 2 − 48
Identifying the Non-Equivalent Expression Comparing the simplified forms, we see that:
( 3 x − 12 ) ( x + 4 ) = 3 x 2 − 48
3 ( x 2 − 8 x + 16 ) = 3 x 2 − 24 x + 48
3 ( x 2 − 16 ) = 3 x 2 − 48
3 x 2 − 48 = 3 x 2 − 48
3 x ( x + 4 ) − 12 ( x + 4 ) = 3 x 2 − 48
The expression 3 ( x 2 − 8 x + 16 ) is not equivalent to the original expression.
Final Answer Therefore, the expression that is NOT equivalent to ( 3 x − 12 ) ( x + 4 ) is 3 ( x 2 − 8 x + 16 ) .
Examples
Understanding equivalent expressions is crucial in algebra, especially when simplifying equations or solving for variables. For instance, if you're designing a rectangular garden where the area is represented by ( 3 x − 12 ) ( x + 4 ) , knowing that this is equivalent to 3 x 2 − 48 allows you to easily calculate the area for different values of x . However, using a non-equivalent expression like 3 ( x 2 − 8 x + 16 ) would lead to incorrect area calculations, affecting your garden design and resource planning.
