\((\mathrm{x}^2 - 3)(2\mathrm{x} + 8) = 0\) What is the product of the solutions to the given equation?...

1. INFER the solution strategy

• Given: \((\mathrm{x}^2 - 3)(2\mathrm{x} + 8) = 0\)
• Key insight: When a product of factors equals zero, at least one factor must be zero
• Strategy: Apply the Zero Product Property by setting each factor equal to zero

2. SIMPLIFY the first factor

• Set \(2\mathrm{x} + 8 = 0\)
• Subtract 8: \(2\mathrm{x} = -8\)
• Divide by 2: \(\mathrm{x} = -4\)

3. SIMPLIFY the second factor and CONSIDER ALL CASES

• Set \(\mathrm{x}^2 - 3 = 0\)
• Add 3: \(\mathrm{x}^2 = 3\)
• Take square root of both sides: \(\mathrm{x} = \pm\sqrt{3}\)
• This gives us two solutions: \(\mathrm{x} = \sqrt{3}\) and \(\mathrm{x} = -\sqrt{3}\)

4. SIMPLIFY the product calculation

• All three solutions: \(-4\), \(\sqrt{3}\), and \(-\sqrt{3}\)
• Product = \((-4) \times (\sqrt{3}) \times (-\sqrt{3})\)
• Since \(\sqrt{3} \times (-\sqrt{3}) = -3\)
• Product = \((-4) \times (-3) = 12\)

Answer: D. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve \(\mathrm{x}^2 = 3\) but only use the positive square root, finding \(\mathrm{x} = \sqrt{3}\) while missing \(\mathrm{x} = -\sqrt{3}\).

They calculate the product as \((-4) \times (\sqrt{3}) = -4\sqrt{3}\), which doesn't match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students make an arithmetic error when calculating \(\sqrt{3} \times (-\sqrt{3})\), thinking it equals \(+3\) instead of \(-3\).

This leads them to calculate: \((-4) \times (+3) = -12\), causing them to select Choice A (-12).

The Bottom Line:

This problem tests whether students can systematically find all solutions from a factored equation and correctly handle radical multiplication. The key challenge is remembering that square root equations always have two solutions and being careful with signs during multiplication.