2x^2/(x^2 - 4) - x/(x + 2) = 4/(x - 2) What value of x satisfies the equation above? -{2}...

1. TRANSLATE the problem information

• Given equation: \(\frac{2\mathrm{x}^2}{\mathrm{x}^2 - 4} - \frac{\mathrm{x}}{\mathrm{x} + 2} = \frac{4}{\mathrm{x} - 2}\)
• Need to find the value of x that satisfies this equation

2. INFER the solution strategy

• The key insight is recognizing that \(\mathrm{x}^2 - 4\) can be factored as \((\mathrm{x} + 2)(\mathrm{x} - 2)\)
• This reveals that all denominators are related, suggesting we need a common denominator approach
• The strategy: factor first, then find common denominators to combine terms

3. SIMPLIFY by factoring and rewriting

• Factor: \(\mathrm{x}^2 - 4 = (\mathrm{x} + 2)(\mathrm{x} - 2)\)
• Equation becomes: \(\frac{2\mathrm{x}^2}{(\mathrm{x} + 2)(\mathrm{x} - 2)} - \frac{\mathrm{x}}{\mathrm{x} + 2} = \frac{4}{\mathrm{x} - 2}\)
• The common denominator should be \((\mathrm{x} + 2)(\mathrm{x} - 2)\)

4. SIMPLIFY by creating equivalent fractions

• Multiply the middle term by \(\frac{\mathrm{x} - 2}{\mathrm{x} - 2}\):
• \(\frac{2\mathrm{x}^2}{(\mathrm{x} + 2)(\mathrm{x} - 2)} - \frac{\mathrm{x}(\mathrm{x} - 2)}{(\mathrm{x} + 2)(\mathrm{x} - 2)} = \frac{4}{\mathrm{x} - 2}\)

5. SIMPLIFY by combining the left side

• Combine: \(\frac{2\mathrm{x}^2 - \mathrm{x}(\mathrm{x} - 2)}{(\mathrm{x} + 2)(\mathrm{x} - 2)} = \frac{4}{\mathrm{x} - 2}\)
• Expand: \(\frac{2\mathrm{x}^2 - \mathrm{x}^2 + 2\mathrm{x}}{(\mathrm{x} + 2)(\mathrm{x} - 2)} = \frac{4}{\mathrm{x} - 2}\)
• Simplify: \(\frac{\mathrm{x}^2 + 2\mathrm{x}}{(\mathrm{x} + 2)(\mathrm{x} - 2)} = \frac{4}{\mathrm{x} - 2}\)

6. SIMPLIFY by clearing denominators

• Multiply both sides by \((\mathrm{x} + 2)(\mathrm{x} - 2)\):
• \(\mathrm{x}^2 + 2\mathrm{x} = 4(\mathrm{x} + 2)\)
• \(\mathrm{x}^2 + 2\mathrm{x} = 4\mathrm{x} + 8\)
• \(\mathrm{x}^2 - 2\mathrm{x} - 8 = 0\)

7. SIMPLIFY by solving the quadratic

• Factor: \((\mathrm{x} - 4)(\mathrm{x} + 2) = 0\)
• Solutions: \(\mathrm{x} = 4\) or \(\mathrm{x} = -2\)

8. APPLY CONSTRAINTS to check validity

• Check \(\mathrm{x} = -2\): This makes \((\mathrm{x} + 2) = 0\), creating division by zero in the original equation
• Therefore \(\mathrm{x} = -2\) is extraneous and must be rejected
• Check \(\mathrm{x} = 4\): All denominators are non-zero, so this is valid

9. SIMPLIFY by verification

• Substitute \(\mathrm{x} = 4\): \(\frac{2(16)}{12} - \frac{4}{6} = \frac{4}{2}\)
• Calculate: \(\frac{32}{12} - \frac{4}{6} = 2\)
• Simplify: \(\frac{8}{3} - \frac{2}{3} = 2\)
• Result: \(\frac{6}{3} = 2\) ✓

Answer: C (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students solve the quadratic correctly to get \(\mathrm{x} = 4\) or \(\mathrm{x} = -2\), but fail to check which solutions are valid in the original equation. They see \(\mathrm{x} = -2\) as a solution and may select Choice A (-2) without realizing it creates division by zero in the original rational equation.

Second Most Common Error:

Poor INFER strategy: Students struggle to recognize that \(\mathrm{x}^2 - 4\) should be factored, or they don't see the connection between the denominators. This leads to attempting to clear denominators incorrectly or getting overwhelmed by the complex fractions. This causes them to get stuck and abandon systematic solution, leading to guessing.

The Bottom Line:

Rational equations require both algebraic manipulation skills and careful attention to domain restrictions. The key insight is recognizing structural patterns in denominators and remembering that not all algebraic solutions are valid in the original equation.