\((\mathrm{x} + 2)(\mathrm{x} - 5)(\mathrm{x} + 9) = 0\)What is a positive solution to the given equation?

1. INFER the solution strategy

• Given: \((\mathrm{x} + 2)(\mathrm{x} - 5)(\mathrm{x} + 9) = 0\)
• Key insight: When a product equals zero, at least one factor must be zero
• Strategy: Apply the zero product property by setting each factor equal to zero

2. SIMPLIFY by creating separate equations

Set each factor equal to zero:

• \(\mathrm{x} + 2 = 0\)
• \(\mathrm{x} - 5 = 0\)
• \(\mathrm{x} + 9 = 0\)

3. SIMPLIFY each linear equation

Solve systematically:

• From \(\mathrm{x} + 2 = 0\): Subtract 2 from both sides → \(\mathrm{x} = -2\)
• From \(\mathrm{x} - 5 = 0\): Add 5 to both sides → \(\mathrm{x} = 5\)
• From \(\mathrm{x} + 9 = 0\): Subtract 9 from both sides → \(\mathrm{x} = -9\)

4. APPLY CONSTRAINTS to select final answer

• All solutions: \(\mathrm{x} = -2, 5, -9\)
• Question asks for positive solution
• Since only \(5 \gt 0\), the positive solution is \(\mathrm{x} = 5\)

Answer: C. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize they can use the zero product property with the factored form. Instead, they might try to expand \((\mathrm{x} + 2)(\mathrm{x} - 5)(\mathrm{x} + 9)\) first, creating a cubic equation that's much harder to solve. This leads to confusion and potentially guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students apply zero product property correctly but make arithmetic errors when solving the linear equations. For example, from \(\mathrm{x} + 2 = 0\), they might incorrectly get \(\mathrm{x} = 2\) instead of \(\mathrm{x} = -2\). This could lead them to think 2 is a positive solution, but since 2 isn't among the choices, this causes confusion and guessing.

The Bottom Line:

This problem tests whether students can recognize when to use the zero product property rather than expanding. The factored form is a gift - use it! The real challenge is strategic recognition, not complex calculations.