A square has area x square units. The length of its diagonal, in units, is sqrt(2x). If 7 times the...

1. TRANSLATE the problem information

• Given information:
  • Square has area \(\mathrm{x}\) square units
  • Diagonal length is \(\sqrt{2\mathrm{x}}\) units
  • 7 times the diagonal length equals 56

• What this tells us: We need to find the value of \(\mathrm{x}\) using the constraint about the diagonal.

2. TRANSLATE the key relationship into an equation

• The phrase "7 times the length of the diagonal is 56" becomes:
  \(7 \times \sqrt{2\mathrm{x}} = 56\)

• This gives us our working equation to solve for \(\mathrm{x}\).

3. SIMPLIFY to isolate the square root term

• Divide both sides by 7:
  \(\sqrt{2\mathrm{x}} = 8\)

• Now we have the square root expression equal to a simple number.

4. SIMPLIFY by eliminating the square root

• Square both sides to eliminate the square root:
  \((\sqrt{2\mathrm{x}})^2 = 8^2\)
  \(2\mathrm{x} = 64\)

5. SIMPLIFY to find the final answer

• Divide both sides by 2:
  \(\mathrm{x} = 32\)

• Check our work: If \(\mathrm{x} = 32\), then diagonal = \(\sqrt{2 \times 32} = \sqrt{64} = 8\), and \(7 \times 8 = 56\) ✓

Answer: C (32)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students might misinterpret "7 times the length of the diagonal is 56" and set up incorrect equations like \(7\mathrm{x} = 56\) or \(7\sqrt{\mathrm{x}} = 56\), missing the factor of 2 inside the square root.

Working from \(7\mathrm{x} = 56\) leads to \(\mathrm{x} = 8\), while \(7\sqrt{\mathrm{x}} = 56\) leads to \(\sqrt{\mathrm{x}} = 8\), so \(\mathrm{x} = 64\). This may lead them to select Choice A (8) or Choice D (64).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(7\sqrt{2\mathrm{x}} = 56\) but make algebraic mistakes. A common error is incorrectly squaring the equation before isolating the square root term, leading to computational mistakes.

This leads to confusion and often results in selecting an incorrect answer choice through faulty calculations.

The Bottom Line:

This problem tests students' ability to carefully translate word relationships into mathematical equations and then execute multi-step algebraic simplification without computational errors. The key insight is recognizing that the constraint gives us exactly the information needed to solve for \(\mathrm{x}\) directly.