A rectangle has a length that is 6 units greater than its width. The area of the rectangle is 16...

1. TRANSLATE the problem information

• Given information:
  • Width = \(\mathrm{x}\) units
  • Length = \(\mathrm{x + 6}\) units (6 units greater than width)
  • Area = \(\mathrm{16}\) square units

2. INFER the solution approach

• We need to use the area formula for rectangles
• Set up equation: \(\mathrm{Area = length \times width}\)
• This gives us: \(\mathrm{16 = x(x + 6)}\)

3. SIMPLIFY through algebraic steps

• Expand: \(\mathrm{x(x + 6) = x^2 + 6x}\)
• Set equal to area: \(\mathrm{x^2 + 6x = 16}\)
• Rearrange to standard form: \(\mathrm{x^2 + 6x - 16 = 0}\)

4. SIMPLIFY by factoring the quadratic

• Find two numbers that multiply to -16 and add to 6
• Those numbers are 8 and -2: \(\mathrm{(8)(-2) = -16}\) and \(\mathrm{8 + (-2) = 6}\)
• Factor: \(\mathrm{(x + 8)(x - 2) = 0}\)
• Solutions: \(\mathrm{x = -8}\) or \(\mathrm{x = 2}\)

5. APPLY CONSTRAINTS to select final answer

• Since width must be positive in real-world context: \(\mathrm{x = 2}\)
• Verify: \(\mathrm{width = 2, length = 8, area = 2 \times 8 = 16}\) ✓

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students struggle to set up the relationship between width and length correctly. They might write the length as just "6" instead of "\(\mathrm{x + 6}\)", or confuse which dimension is greater.

This leads to incorrect equations like \(\mathrm{x \times 6 = 16}\), giving \(\mathrm{x = 8/3}\), which doesn't match the integer requirement and causes confusion and guessing.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students get the correct quadratic \(\mathrm{x^2 + 6x - 16 = 0}\) but struggle with factoring. They might attempt to use the quadratic formula incorrectly or make arithmetic errors in factoring.

This often leads to getting stuck partway through and randomly guessing an answer.

The Bottom Line:

This problem combines word problem translation with quadratic solving skills. Students need to carefully parse the language to set up the correct equation, then execute multi-step algebra accurately. The constraint application (positive width) is the final crucial step many forget.