A square and a rectangle have equal areas. The side length of the square is x + 1 feet. The...

1. TRANSLATE the problem information

• Given information:
  • Square and rectangle have equal areas
  • Square side length: \(\mathrm{x + 1}\) feet
  • Rectangle length: \(\mathrm{x + 7}\) feet, width: \(\mathrm{3}\) feet
  • Find: value of x
• What this tells us: We need to set up an equation where both areas are equal

2. INFER the approach

• Since areas are equal, we'll write expressions for both areas and set them equal
• Square area uses the formula \(\mathrm{A = side^2}\)
• Rectangle area uses the formula \(\mathrm{A = length \times width}\)

3. TRANSLATE each area into mathematical expressions

• Square area = \(\mathrm{(x + 1)^2}\)
• Rectangle area = \(\mathrm{3(x + 7)}\)

4. Set up the equation and SIMPLIFY

• Equal areas means: \(\mathrm{(x + 1)^2 = 3(x + 7)}\)
• Expand the left side: \(\mathrm{x^2 + 2x + 1 = 3x + 21}\)
• Rearrange to standard form: \(\mathrm{x^2 + 2x + 1 - 3x - 21 = 0}\)
• Combine like terms: \(\mathrm{x^2 - x - 20 = 0}\)

5. SIMPLIFY by factoring the quadratic

• We need two numbers that multiply to \(\mathrm{-20}\) and add to \(\mathrm{-1}\)
• Those numbers are \(\mathrm{-5}\) and \(\mathrm{+4}\)
• Factor: \(\mathrm{(x - 5)(x + 4) = 0}\)
• Solutions: \(\mathrm{x = 5}\) or \(\mathrm{x = -4}\)

6. APPLY CONSTRAINTS to select the valid answer

• Check \(\mathrm{x = -4}\): Square side would be \(\mathrm{-4 + 1 = -3}\) feet (impossible - negative length)
• Check \(\mathrm{x = 5}\): Square side would be \(\mathrm{5 + 1 = 6}\) feet (valid)
• Therefore: \(\mathrm{x = 5}\)

7. Verify the answer

• When \(\mathrm{x = 5}\): Square area = \(\mathrm{6^2 = 36}\) square feet
• Rectangle area = \(\mathrm{3(12) = 36}\) square feet ✓

Answer: D) 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x + 1)^2}\) or when factoring the quadratic equation.

For example, they might expand \(\mathrm{(x + 1)^2}\) as \(\mathrm{x^2 + 1}\) (forgetting the middle term \(\mathrm{2x}\)), leading to the wrong equation \(\mathrm{x^2 + 1 = 3x + 21}\), which gives \(\mathrm{x^2 - 3x - 20 = 0}\). This factors differently and could lead them to select Choice A (2) or cause confusion and guessing.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students find both solutions \(\mathrm{x = 5}\) and \(\mathrm{x = -4}\) but fail to check which values make physical sense in the context.

They might randomly pick \(\mathrm{x = -4}\) without realizing this creates negative dimensions, or they might get confused about which solution to choose. Since \(\mathrm{-4}\) isn't among the answer choices, this leads to confusion and guessing.

The Bottom Line:

This problem challenges students to work systematically through a multi-step algebraic process while maintaining awareness that mathematical solutions must make sense in real-world contexts. Success requires both strong algebraic manipulation skills and the wisdom to validate answers against physical constraints.