Question

a group of students is designing a large banner for their school project. the length of the banner is $3x - 2$ meters, and the width is $x - 4$ meters. the area of the banner is 32 square meters. they want to determine the dimensions of the banner.
determine the steps they need to take to find the dimensions of the banner.
drag one answer to each box to complete the sentences.

1. they can first

write the equation $(3x - 2)(x - 4) = 32$
to set up the equation based on the given information.

1. next, they

factor the equation $(3x - 4)(x + 6) = 0$
to simplify the equation and set it to standard form.

1. then, they

Explanation:

Step1: Analyze the first step

The area of a rectangle is length times width. Given length \( 3x - 2 \) and width \( x - 4 \), and area 32, so the equation \( (3x - 2)(x - 4)=32 \) is correct.

Step2: Analyze the second step

First, expand \( (3x - 2)(x - 4)=32 \):

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Now, factor \( 3x^{2}-14x - 24 \). We need two numbers \( a \) and \( b \) such that \( a\times b=3\times(-24)= - 72 \) and \( a + b=-14 \). The numbers are -18 and 4.

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So the correct factoring step should be to get the equation in standard form \( 3x^{2}-14x - 24 = 0 \) first (by expanding and simplifying the initial equation) and then factor it. The given factoring \( (3x - 4)(x + 6)=0 \) is wrong because \( (3x - 4)(x + 6)=3x^{2}+18x - 4x - 24=3x^{2}+14x - 24
eq3x^{2}-14x - 24 \). So the second step should be to expand and simplify the equation \( (3x - 2)(x - 4)=32 \) to standard quadratic form \( 3x^{2}-14x - 24 = 0 \).

Step3: Analyze the third step (if needed)

After factoring (correctly) to \( (3x + 4)(x - 6)=0 \), we set each factor to zero: \( 3x + 4 = 0 \) or \( x - 6 = 0 \). Solving \( 3x+4 = 0 \) gives \( x=-\frac{4}{3} \) (discarded as dimension can't be negative), and \( x - 6 = 0 \) gives \( x = 6 \). Then length \( 3(6)-2 = 16 \), width \( 6 - 4 = 2 \). But for the second step, the error is in the factoring, the correct action is to expand the initial equation to standard quadratic form.

Answer:

For the second step, the correct action is to expand and simplify the equation \( (3x - 2)(x - 4)=32 \) to get the quadratic equation in standard form \( 3x^{2}-14x - 24 = 0 \) (instead of the incorrect factoring given).