What is one of the solutions to the equation 2x^3 - 18x = 0?

1. INFER the solution strategy

• Given equation: \(2\mathrm{x}^3 - 18\mathrm{x} = 0\)
• Key insight: This is a cubic equation that can be solved by factoring rather than using complex cubic formulas
• Strategy: Look for common factors first, then continue factoring

2. SIMPLIFY by factoring out the greatest common factor

• Both terms \(2\mathrm{x}^3\) and \(-18\mathrm{x}\) contain the factor \(2\mathrm{x}\)
• Factor out \(2\mathrm{x}\): \(2\mathrm{x}(\mathrm{x}^2 - 9) = 0\)

3. SIMPLIFY further by recognizing patterns

• The expression \(\mathrm{x}^2 - 9\) is a difference of squares \((\mathrm{x}^2 - 3^2)\)
• Apply the difference of squares pattern: \(\mathrm{x}^2 - 9 = (\mathrm{x} - 3)(\mathrm{x} + 3)\)
• Complete factorization: \(2\mathrm{x}(\mathrm{x} - 3)(\mathrm{x} + 3) = 0\)

4. INFER how to find solutions using zero product property

• If a product of factors equals zero, at least one factor must be zero
• Set each factor equal to zero:
  • \(2\mathrm{x} = 0 \rightarrow \mathrm{x} = 0\)
  • \(\mathrm{x} - 3 = 0 \rightarrow \mathrm{x} = 3\)
  • \(\mathrm{x} + 3 = 0 \rightarrow \mathrm{x} = -3\)

5. TRANSLATE solutions to answer the question

• All solutions: \(\mathrm{x} = -3, 0, \text{ or } 3\)
• The question asks for "one of the solutions"
• Check which solution appears in the answer choices: 3 is choice (C)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Students struggle with the systematic factoring process, particularly missing the difference of squares pattern. They might factor out \(2\mathrm{x}\) correctly to get \(2\mathrm{x}(\mathrm{x}^2 - 9) = 0\), but then fail to recognize that \(\mathrm{x}^2 - 9\) can be factored further. This leaves them with an incomplete factorization, leading to either missing solutions or making calculation errors when trying to solve \(\mathrm{x}^2 - 9 = 0\) directly.

This confusion often leads to guessing among the answer choices.

Second Most Common Error:

Weak INFER reasoning: Students don't recognize factoring as the appropriate strategy and instead attempt to solve the cubic equation through expansion or guessing. Without the strategic insight to factor first, they become overwhelmed by the complexity of the cubic equation and abandon systematic solution approaches.

This leads to random answer selection or attempting to substitute answer choices back into the original equation.

The Bottom Line:

The key challenge is recognizing that systematic factoring (GCF first, then special patterns) transforms a seemingly complex cubic equation into simple linear equations through the zero product property.