Liz just returned from a trip to visit some of the national parks with her family, and she wants to print her favorite picture from the trip to hang on her wall. She is not quite sure what size print she wants to get, although she does know that she wants the print to be square. She also wants to add a border that is $3$ inches wide around the sides of the photo. $\newline$ The total area of the photo and border in square inches can be modeled by the expression $(x + 6)(x + 6)$ , where $x$ is the width of the photo in inches. This expression can also be written in the form $x^2 + 12x + 36$ . What does the quantity $12x + 36$ represent in the expression? $\newline$ Choices: $\newline$ (A)the length of the photo and border together in inches $\newline$ (B)the area of the photo in square inches $\newline$ (C)the area of the border in square inches $\newline$ (D)the perimeter around the outside of the border in inches

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Algebra 1

Interpret parts of quadratic expressions: word problems

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Q. Liz just returned from a trip to visit some of the national parks with her family, and she wants to print her favorite picture from the trip to hang on her wall. She is not quite sure what size print she wants to get, although she does know that she wants the print to be square. She also wants to add a border that is

3

inches wide around the sides of the photo.

\newline

The total area of the photo and border in square inches can be modeled by the expression

(x + 6)(x + 6)

, where

x

is the width of the photo in inches. This expression can also be written in the form

x^2 + 12x + 36

. What does the quantity

12x + 36

represent in the expression?

\newline

Choices:

\newline

(A)the length of the photo and border together in inches

\newline

(B)the area of the photo in square inches

\newline

(C)the area of the border in square inches

\newline

(D)the perimeter around the outside of the border in inches

Total Area Expression: The expression for the total area is $x + 6)(x + 6)\, which is equivalent to \$x^2 + 12x + 36$ .
Area of Photo: $x$ represents the width of the photo, so $x^2$ is the area of the photo itself.
Area of Border Strips: The term $12x$ comes from expanding the expression $(x + 6)(x + 6)$ , where we get $2 \times 6x$ , which is the area of the four border strips that are $x$ inches by $3$ inches.
Area of Corner Squares: The constant $36$ comes from squaring the border width, which is $6$ inches, so $6 \times 6$ gives us the area of the four corner squares of the border.
Total Border Area: Therefore, $12x$ represents the combined area of the four border strips, and $36$ represents the combined area of the four corner squares of the border.
Total Border Area: Therefore, $12x$ represents the combined area of the four border strips, and $36$ represents the combined area of the four corner squares of the border.So, $12x + 36$ represents the total area of the border around the photo.