\(38\mathrm{x}^2 = 38(9)\) What is the negative solution to the given equation?...

1. SIMPLIFY the equation by eliminating coefficients

• Given: \(38\mathrm{x}^2 = 38(9)\)
• Divide both sides by 38 to isolate \(\mathrm{x}^2\):
  \(\frac{38\mathrm{x}^2}{38} = \frac{38(9)}{38}\)
  \(\mathrm{x}^2 = 9\)

2. SIMPLIFY further by taking square roots

• Take the square root of both sides:
  \(\sqrt{\mathrm{x}^2} = \sqrt{9}\)
  \(\mathrm{x} = ±3\)

3. CONSIDER ALL CASES to identify all solutions

• The equation \(\mathrm{x}^2 = 9\) gives us two solutions:
  • \(\mathrm{x} = +3\) (positive solution)
  • \(\mathrm{x} = -3\) (negative solution)

4. Select the requested solution

• The question asks specifically for the negative solution
• Therefore: \(\mathrm{x} = -3\)

Answer: -3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors during the division step or forget to consider both positive and negative square roots.

Some students might divide incorrectly and get \(\mathrm{x}^2 = 38×9\) instead of \(\mathrm{x}^2 = 9\), leading them to work with \(\mathrm{x}^2 = 342\) and get confused with non-integer solutions. Others might take the square root but only consider the positive solution \(\mathrm{x} = 3\), missing that square roots have two values.

This leads to confusion and potentially selecting the wrong solution or getting stuck entirely.

The Bottom Line:

This problem tests students' understanding that quadratic equations typically have two solutions, and their ability to systematically work through algebraic simplification while keeping track of both positive and negative square roots.