If \((2\mathrm{x} - 3)^2 = \mathrm{x}(4\mathrm{x} - 6) - 9\), what is the value of 6x?

1. TRANSLATE the problem setup

• Given equation: \((2\mathrm{x} - 3)^2 = \mathrm{x}(4\mathrm{x} - 6) - 9\)
• Find: The value of \(6\mathrm{x}\) (not \(\mathrm{x}\) itself)

2. SIMPLIFY both sides by expanding

• Left side: \((2\mathrm{x} - 3)^2 = (2\mathrm{x})^2 - 2(2\mathrm{x})(3) + 3^2 = 4\mathrm{x}^2 - 12\mathrm{x} + 9\)
• Right side: \(\mathrm{x}(4\mathrm{x} - 6) - 9 = 4\mathrm{x}^2 - 6\mathrm{x} - 9\)

The equation becomes: \(4\mathrm{x}^2 - 12\mathrm{x} + 9 = 4\mathrm{x}^2 - 6\mathrm{x} - 9\)

3. SIMPLIFY by eliminating common terms

• Notice that \(4\mathrm{x}^2\) appears on both sides, so subtract it from both sides
• This leaves: \(-12\mathrm{x} + 9 = -6\mathrm{x} - 9\)

4. INFER the most efficient path to the answer

• Rather than solving for \(\mathrm{x}\) and then multiplying by 6, we can isolate \(6\mathrm{x}\) directly
• Add \(12\mathrm{x}\) to both sides: \(9 = 6\mathrm{x} - 9\)
• Add 9 to both sides: \(18 = 6\mathrm{x}\)

Answer: 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make sign errors when expanding \((2\mathrm{x} - 3)^2\), writing it as \(4\mathrm{x}^2 - 6\mathrm{x} + 9\) instead of \(4\mathrm{x}^2 - 12\mathrm{x} + 9\). They forget that the middle term should be \(-2(2\mathrm{x})(3) = -12\mathrm{x}\), not just \(-6\mathrm{x}\).

This error leads to the wrong equation \(4\mathrm{x}^2 - 6\mathrm{x} + 9 = 4\mathrm{x}^2 - 6\mathrm{x} - 9\), which simplifies to \(9 = -9\), creating confusion about whether there's no solution or leading to random guessing.

Second Most Common Error:

Incomplete INFER reasoning: Students solve correctly until they reach \(18 = 6\mathrm{x}\), but then unnecessarily divide both sides by 6 to get \(\mathrm{x} = 3\), and then multiply by 6 again to get \(6\mathrm{x} = 18\). While this gives the right answer, it shows they didn't recognize they already had what the problem asked for.

This inefficient approach can introduce calculation errors and wastes time, though it doesn't necessarily lead to a wrong answer.

The Bottom Line:

This problem tests careful algebraic manipulation combined with strategic thinking about what the question actually asks for. The key insight is recognizing when you've already found what you're looking for.