What is the solution to the equation below?(6x^2 - 24)/(x + 2) = 12

1. TRANSLATE the problem information

• Given equation: \(\frac{6\mathrm{x}^2 - 24}{\mathrm{x} + 2} = 12\)
• We need to find the value of x that makes this equation true

2. INFER the solution strategy

• The numerator looks like it might factor nicely
• If we can factor and find common terms with the denominator, we can simplify
• Strategy: Factor the numerator, then cancel common factors

3. SIMPLIFY by factoring the numerator

• Factor out the common factor of 6: \(6\mathrm{x}^2 - 24 = 6(\mathrm{x}^2 - 4)\)
• Recognize difference of squares: \(\mathrm{x}^2 - 4 = (\mathrm{x} - 2)(\mathrm{x} + 2)\)
• Complete factoring: \(6\mathrm{x}^2 - 24 = 6(\mathrm{x} - 2)(\mathrm{x} + 2)\)

4. APPLY CONSTRAINTS to identify domain restrictions

• The denominator \((\mathrm{x} + 2)\) cannot equal zero
• Therefore: \(\mathrm{x} \neq -2\)

5. SIMPLIFY by canceling common factors

• Rewrite equation: \(\frac{6(\mathrm{x} - 2)(\mathrm{x} + 2)}{\mathrm{x} + 2} = 12\)
• For \(\mathrm{x} \neq -2\), cancel \((\mathrm{x} + 2)\): \(6(\mathrm{x} - 2) = 12\)

6. SIMPLIFY to solve the linear equation

• Divide both sides by 6: \(\mathrm{x} - 2 = 2\)
• Add 2 to both sides: \(\mathrm{x} = 4\)

7. APPLY CONSTRAINTS to verify the solution

• Check: Does \(\mathrm{x} = 4\) violate our restriction \(\mathrm{x} \neq -2\)? No!
• Therefore \(\mathrm{x} = 4\) is our valid solution

Answer: C) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students attempt to solve without factoring the numerator first, trying to multiply both sides by \((\mathrm{x} + 2)\) immediately. This creates a more complex equation: \(6\mathrm{x}^2 - 24 = 12(\mathrm{x} + 2) = 12\mathrm{x} + 24\), leading to \(6\mathrm{x}^2 - 12\mathrm{x} - 48 = 0\). While this is solvable, it's unnecessarily complicated and more error-prone. This approach often leads to calculation errors or students getting overwhelmed by the quadratic formula, causing them to guess or select Choice A (-2) if they confuse the domain restriction with the solution.

Second Most Common Error:

Missing APPLY CONSTRAINTS reasoning: Students correctly factor and cancel to get \(\mathrm{x} = 4\), but they don't consider whether this solution is valid given the original equation's domain. However, since \(\mathrm{x} = 4\) doesn't create any issues and matches one of the answer choices, this error path typically doesn't lead to wrong answer selection in this particular problem.

The Bottom Line:

The key insight is recognizing that factoring first makes this problem much simpler than it initially appears. Students who jump straight into algebraic manipulation without strategic thinking often create unnecessary complexity for themselves.