4x^2/(x^2 - 9) - 2x/(x + 3) = 1/(x - 3) What value of x satisfies the equation above?...

1. INFER the solution strategy

• Given: \(\frac{4x^2}{x^2 - 9} - \frac{2x}{x + 3} = \frac{1}{x - 3}\)
• Key insight: Notice that \(x^2 - 9 = (x + 3)(x - 3)\), so we can use \((x^2 - 9)\) as our common denominator
• Strategy: Convert all fractions to this common denominator, then solve

2. SIMPLIFY each fraction to the common denominator

• First fraction: \(\frac{4x^2}{x^2 - 9}\) already has the common denominator
• Second fraction: \(\frac{2x}{x + 3} \times \frac{x - 3}{x - 3} = \frac{2x^2 - 6x}{x^2 - 9}\)
• Third fraction: \(\frac{1}{x - 3} \times \frac{x + 3}{x + 3} = \frac{x + 3}{x^2 - 9}\)

3. SIMPLIFY the equation with common denominators

• Equation becomes: \(\frac{4x^2}{x^2 - 9} - \frac{2x^2 - 6x}{x^2 - 9} = \frac{x + 3}{x^2 - 9}\)
• Multiply everything by \((x^2 - 9)\): \(4x^2 - (2x^2 - 6x) = x + 3\)
• Distribute: \(4x^2 - 2x^2 + 6x = x + 3\)
• Combine: \(2x^2 + 6x = x + 3\)

4. SIMPLIFY to standard quadratic form

• Move all terms to one side: \(2x^2 + 6x - x - 3 = 0\)
• Combine like terms: \(2x^2 + 5x - 3 = 0\)
• Factor: \((2x - 1)(x + 3) = 0\)
• Solutions: \(x = \frac{1}{2}\) or \(x = -3\)

5. APPLY CONSTRAINTS to eliminate extraneous solutions

• Check \(x = -3\): In the original equation, this makes the denominator \((x + 3) = 0\)
• Since division by zero is undefined, \(x = -3\) is extraneous
• Valid solution: \(x = \frac{1}{2}\)

Answer: C. 1/2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that \(x^2 - 9\) can be factored as \((x + 3)(x - 3)\), so they struggle to find an efficient common denominator. Instead, they might try to use \((x^2 - 9)(x + 3)(x - 3)\) as the common denominator, leading to much more complex algebra. This creates calculation errors and confusion, causing them to get stuck and guess randomly.

Second Most Common Error:

Missing APPLY CONSTRAINTS: Students correctly solve the quadratic to get \(x = \frac{1}{2}\) and \(x = -3\), but they don't check whether both solutions work in the original equation. They see \(x = -3\) as one of the answer choices and select it without realizing it's extraneous. This may lead them to select Choice A (-3).

The Bottom Line:

This problem tests whether students can strategically work with rational expressions and remember to check their solutions. The key insight is recognizing the factored form of \(x^2 - 9\), and the critical final step is validating that solutions don't create division by zero.