\((\mathrm{x} - 4)^2 = (\mathrm{x} + 2)^2 - 12\)Which of the following is the solution to the given equation?

1. INFER the best approach

• We have an equation with squared binomials on both sides
• Rather than trying to take square roots immediately, expanding both sides will lead to a simpler linear equation
• This approach avoids dealing with ± solutions early on

2. SIMPLIFY by expanding the left side

• \((x - 4)^2 = x^2 - 8x + 16\)
• Remember: \((a - b)^2 = a^2 - 2ab + b^2\)

3. SIMPLIFY by expanding the right side

• \((x + 2)^2 - 12 = x^2 + 4x + 4 - 12 = x^2 + 4x - 8\)
• Expand first: \((x + 2)^2 = x^2 + 4x + 4\)
• Then subtract 12: \(x^2 + 4x + 4 - 12 = x^2 + 4x - 8\)

4. SIMPLIFY by setting expressions equal and solving

• \(x^2 - 8x + 16 = x^2 + 4x - 8\)
• Subtract \(x^2\) from both sides: \(-8x + 16 = 4x - 8\)
• Add \(8x\) to both sides: \(16 = 12x - 8\)
• Add 8 to both sides: \(24 = 12x\)
• Divide by 12: \(x = 2\)

Answer: C (2)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making sign errors during binomial expansion, especially with \((x - 4)^2\)

Students often write \((x - 4)^2 = x^2 - 16\) or \(x^2 + 8x + 16\), forgetting that \((a - b)^2 = a^2 - 2ab + b^2\). This creates incorrect coefficients that lead to wrong solutions.

This may lead them to select Choice A (-2) or get confused and guess.

Second Most Common Error:

Poor SIMPLIFY reasoning: Making arithmetic errors while combining like terms or solving the linear equation

After correctly expanding, students might make mistakes like forgetting to change signs when moving terms across the equals sign, or incorrectly combining \(-8x\) and \(4x\).

This leads to confusion and incorrect final answers, possibly leading them to select Choice B (0) or Choice D (4).

The Bottom Line:

This problem tests systematic algebraic manipulation skills. The key insight is that expanding both sides converts a seemingly complex equation with squares into a straightforward linear equation, but success depends on careful attention to signs and arithmetic throughout multiple steps.