Which of the following is a solution to the equation 2x^2 - 4 = x^2?

1. SIMPLIFY the equation by collecting like terms

• Given: \(\mathrm{2x^2 - 4 = x^2}\)
• Subtract \(\mathrm{x^2}\) from both sides: \(\mathrm{2x^2 - x^2 - 4 = 0}\)
• This gives us: \(\mathrm{x^2 - 4 = 0}\)

2. SIMPLIFY further to isolate x²

• Add 4 to both sides: \(\mathrm{x^2 - 4 + 4 = 0 + 4}\)
• This gives us: \(\mathrm{x^2 = 4}\)

3. CONSIDER ALL CASES when taking the square root

• Take the square root of both sides: \(\mathrm{x = ±\sqrt{4}}\)
• Since \(\mathrm{\sqrt{4} = 2}\), we have: \(\mathrm{x = ±2}\)
• This means \(\mathrm{x = 2}\) or \(\mathrm{x = -2}\)

4. Match with answer choices

• Looking at the choices: A. 1, B. 2, C. 3, D. 4
• We found \(\mathrm{x = 2}\), which is choice B

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make arithmetic mistakes when combining like terms, such as incorrectly calculating \(\mathrm{2x^2 - x^2}\) or making sign errors when moving terms across the equals sign.

For example, they might incorrectly get \(\mathrm{x^2 + 4 = 0}\) instead of \(\mathrm{x^2 - 4 = 0}\), leading them to try to find the square root of a negative number and becoming confused. This leads to guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students rush through the algebraic steps without careful attention to operations, such as forgetting to apply the same operation to both sides of the equation consistently.

They might correctly identify that they need to isolate \(\mathrm{x^2}\) but make computational errors in the process, potentially arriving at incorrect values like \(\mathrm{x^2 = 1}\) or \(\mathrm{x^2 = 9}\), which would lead them to select Choice A (1) or Choice C (3) respectively.

The Bottom Line:

This problem requires systematic algebraic manipulation through multiple steps. Students who try to solve it mentally or skip steps often make computational errors that derail their entire solution path.