Jacob helps paint a square mural in his classroom. Then he helps paint a mural in the hallway whose length is 6 feet longer and whose width is 2 feet shorter. Let [tex]$x$[/tex] represent the side length of the mural in Jacob's classroom. Write an expression that represents the two binomials you would multiply to find the area of the hallway mural. Use that expression to find the area of the hallway mural if each side of the classroom mural is 8 feet long.
A. [tex]$(x-6)(x+2) ; 20$[/tex] square feet
B. [tex]$(x+6)(x+2) ; 140$[/tex] square feet
C. [tex]$(x-6)(x-2) ; 12$[/tex] square feet
D. [tex]$(x+6)(x-2) ; 84$[/tex] square feet
Answer (1)
The area of the hallway mural is represented by the expression ( x + 6 ) ( x − 2 ) .
Substitute x = 8 into the expression: ( 8 + 6 ) ( 8 − 2 ) .
Calculate the values: ( 14 ) ( 6 ) .
The area of the hallway mural is 84 square feet.
Explanation
Problem Analysis Let's analyze the problem. We have a square mural in the classroom with side length x . The hallway mural has a length that is 6 feet longer than the classroom mural, so its length is x + 6 . The hallway mural's width is 2 feet shorter than the classroom mural, so its width is x − 2 . We need to find the area of the hallway mural, which is given by the product of its length and width. Then, we need to find the area when x = 8 feet.
Area Expression The area of the hallway mural is given by the expression ( x + 6 ) ( x − 2 ) . This represents the product of the two binomials that define the length and width of the hallway mural.
Substitution Now, we substitute x = 8 into the expression to find the area of the hallway mural when the side length of the classroom mural is 8 feet. So we have ( 8 + 6 ) ( 8 − 2 ) .
Calculation We calculate the values inside the parentheses: 8 + 6 = 14 and 8 − 2 = 6 . Then, we multiply these values: 14 × 6 = 84 .
Final Answer Therefore, the area of the hallway mural is 84 square feet. Comparing this to the given options, we see that option D, ( x + 6 ) ( x − 2 ) ; 84 square feet, is the correct answer.
Examples
Imagine you're designing a rectangular garden where the length is 6 feet more than a square flower bed and the width is 2 feet less. Knowing how to express the garden's area as ( x + 6 ) ( x − 2 ) and calculate it for a specific flower bed size (like x = 8 feet) helps you determine the garden's total area. This is useful for planning how much space you need, how many plants to buy, and how to arrange them efficiently. Understanding binomial multiplication and area calculations is essential for practical design and resource management.
