Which of the following is a solution to the given equation? \((5\mathrm{x} + 4)(2\mathrm{x} - 5) = 0\)...

1. INFER the solution strategy

• Given: \((5\mathrm{x} + 4)(2\mathrm{x} - 5) = 0\)
• Key insight: When a product of factors equals zero, at least one factor must be zero
• Strategy: Set each factor equal to zero and solve separately

2. SIMPLIFY each factor equation

For the first factor: \(5\mathrm{x} + 4 = 0\)

• Subtract 4 from both sides: \(5\mathrm{x} = -4\)
• Divide by 5: \(\mathrm{x} = -\frac{4}{5}\)

For the second factor: \(2\mathrm{x} - 5 = 0\)

• Add 5 to both sides: \(2\mathrm{x} = 5\)
• Divide by 2: \(\mathrm{x} = \frac{5}{2}\)

3. Identify which solution appears in the answer choices

• Solutions found: \(\mathrm{x} = -\frac{4}{5}\) and \(\mathrm{x} = \frac{5}{2}\)
• Checking answer choices: \(-\frac{4}{5}\) matches Choice C

Answer: C. \(-\frac{4}{5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to expand \((5\mathrm{x} + 4)(2\mathrm{x} - 5)\) instead of recognizing the zero product property opportunity.

They multiply out to get \(10\mathrm{x}^2 - 25\mathrm{x} + 8\mathrm{x} - 20 = 0\), leading to \(10\mathrm{x}^2 - 17\mathrm{x} - 20 = 0\). This creates a much more complex quadratic that requires the quadratic formula or factoring, making the problem unnecessarily difficult. This leads to confusion and often guessing among the choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly apply zero product property but make sign errors when solving individual equations.

For example, when solving \(5\mathrm{x} + 4 = 0\), they might incorrectly get \(\mathrm{x} = \frac{4}{5}\) instead of \(\mathrm{x} = -\frac{4}{5}\), or when solving \(2\mathrm{x} - 5 = 0\), they might get \(\mathrm{x} = -\frac{5}{2}\) instead of \(\mathrm{x} = \frac{5}{2}\). This may lead them to select Choice A (\(-\frac{5}{2}\)) or choose incorrectly based on their arithmetic errors.

The Bottom Line:

This problem tests whether students recognize when to use the zero product property versus trying to solve by expansion. The key insight is seeing that the factored form is a gift - use it directly rather than making the problem harder.