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 = \(\sqrt{2\mathrm{x}}\) units
  • 7 times the diagonal length = 56

• What we need to find: The value of x

2. TRANSLATE the key relationship into an equation

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

• This gives us our working equation: \(7\sqrt{2\mathrm{x}} = 56\)

3. SIMPLIFY to solve for x

• First, divide both sides by 7:
  \(\sqrt{2\mathrm{x}} = 56 \div 7 = 8\)

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

• Finally, divide by 2:
  \(\mathrm{x} = 32\)

4. Verify the answer

• Check: 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 struggle to convert "7 times the length of the diagonal is 56" into the correct equation setup.

Some students might write equations like "\(7\mathrm{x} = 56\)" (confusing the diagonal length with the area x) or "\(7\sqrt{2\mathrm{x}} + \text{something} = 56\)" (adding unnecessary terms). This fundamental translation error prevents them from setting up the correct starting equation, leading to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic mistakes during the solution process.

Common calculation errors include: forgetting to square both sides properly (thinking \(\sqrt{2\mathrm{x}} = 8\) means \(2\mathrm{x} = 8\)), arithmetic mistakes when dividing 56 by 7, or errors when squaring 8. These calculation slip-ups often lead them to select Choice A (8) or Choice B (16) instead of the correct answer.

The Bottom Line:

This problem tests both translation skills (converting English to math) and algebraic manipulation under the pressure of working with square roots. Success requires careful attention to both the initial setup and the step-by-step solution process.