The area of a rectangular banner is 2,661 square inches. The banner's length x, in inches, is 24 inches longer...

1. TRANSLATE the problem information

• Given information:
  • Area of rectangle = 2,661 square inches
  • Length = x inches
  • Length is 24 inches longer than width
• What this tells us: Width = \(\mathrm{(x - 24)}\) inches

2. INFER the approach

• We need to use the rectangle area formula to create an equation
• Since we want to match the given answer choices, we'll need to rearrange our equation to standard form (everything on one side equal to zero)

3. Set up the area equation

• \(\mathrm{Area = length \times width}\)
• \(\mathrm{2,661 = x(x - 24)}\)

4. SIMPLIFY by expanding

• \(\mathrm{2,661 = x^2 - 24x}\)

5. INFER the final step

• The answer choices all have zero on the left side, so we need to rearrange
• Subtract 2,661 from both sides: \(\mathrm{0 = x^2 - 24x - 2,661}\)

Answer: A. \(\mathrm{0 = x^2 - 24x - 2,661}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "length is 24 inches longer than width" and incorrectly set up width as \(\mathrm{(x + 24)}\) instead of \(\mathrm{(x - 24)}\).

Their reasoning: "If length is 24 inches longer, then width must be 24 inches more too." This shows confusion about which variable represents which dimension. With width = \(\mathrm{(x + 24)}\), their area equation becomes \(\mathrm{x(x + 24) = 2,661}\), which expands to \(\mathrm{x^2 + 24x = 2,661}\), leading to \(\mathrm{0 = x^2 + 24x - 2,661}\).

This may lead them to select Choice C (\(\mathrm{0 = x^2 + 24x - 2,661}\)).

The Bottom Line:

The key challenge is correctly translating the relationship between length and width. Students must carefully track which dimension is the reference point and which is being described in terms of the other.