\(\sqrt{(\mathrm{x} - 2)^2} = \sqrt{3\mathrm{x} + 34}\) What is the smallest solution to the given equation?...

1. TRANSLATE the problem information

• Given equation: \(\sqrt{(\mathrm{x} - 2)^2} = \sqrt{3\mathrm{x} + 34}\)
• Need to find: smallest solution

2. INFER the solving strategy

• Both sides contain square roots, making direct solving difficult
• Key insight: Since both sides are non-negative, we can square both sides to eliminate the radicals
• This will create a quadratic equation we can solve

3. SIMPLIFY by squaring both sides

• Square both sides: \((\sqrt{(\mathrm{x} - 2)^2})^2 = (\sqrt{3\mathrm{x} + 34})^2\)
• Left side: \((\mathrm{x} - 2)^2\)
• Right side: \(3\mathrm{x} + 34\)
• Result: \((\mathrm{x} - 2)^2 = 3\mathrm{x} + 34\)

4. SIMPLIFY the quadratic equation

• Expand: \(\mathrm{x}^2 - 4\mathrm{x} + 4 = 3\mathrm{x} + 34\)
• Move all terms to one side: \(\mathrm{x}^2 - 4\mathrm{x} + 4 - 3\mathrm{x} - 34 = 0\)
• Combine like terms: \(\mathrm{x}^2 - 7\mathrm{x} - 30 = 0\)

5. SIMPLIFY by factoring

• Look for two numbers that multiply to \(-30\) and add to \(-7\)
• Those numbers are \(-10\) and \(3\): \((-10)(3) = -30\), \((-10) + 3 = -7\)
• Factor: \((\mathrm{x} - 10)(\mathrm{x} + 3) = 0\)

6. CONSIDER ALL CASES using zero product property

• If \((\mathrm{x} - 10)(\mathrm{x} + 3) = 0\), then either:
  • \(\mathrm{x} - 10 = 0\), so \(\mathrm{x} = 10\)
  • \(\mathrm{x} + 3 = 0\), so \(\mathrm{x} = -3\)

7. APPLY CONSTRAINTS by checking solutions

• Check \(\mathrm{x} = 10\): \(\sqrt{(10-2)^2} = \sqrt{64} = 8\), \(\sqrt{3(10)+34} = \sqrt{64} = 8\) ✓
• Check \(\mathrm{x} = -3\): \(\sqrt{(-3-2)^2} = \sqrt{25} = 5\), \(\sqrt{3(-3)+34} = \sqrt{25} = 5\) ✓
• Both solutions are valid

8. INFER the final answer

• We have two valid solutions: \(10\) and \(-3\)
• The smallest solution is \(-3\)

Answer: -3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Making algebraic errors when expanding \((\mathrm{x} - 2)^2\) or when rearranging terms

Students might expand incorrectly as \(\mathrm{x}^2 - 4\mathrm{x} + 2\) instead of \(\mathrm{x}^2 - 4\mathrm{x} + 4\), or make sign errors when moving terms. These mistakes lead to an incorrect quadratic equation, producing wrong solutions entirely. This leads to confusion and guessing among the available answer choices.

Second Most Common Error:

Inadequate CONSIDER ALL CASES reasoning: Finding only one solution to the quadratic equation

Some students solve \((\mathrm{x} - 10)(\mathrm{x} + 3) = 0\) but only find \(\mathrm{x} = 10\), missing the \(\mathrm{x} = -3\) solution. Since the problem asks for the smallest solution, they would incorrectly conclude the answer is \(10\). This may lead them to select an incorrect answer choice if \(10\) were among the options.

The Bottom Line:

This problem tests whether students can systematically work through a multi-step algebraic process while maintaining accuracy at each step. The key challenge is executing several algebraic manipulations correctly in sequence, then ensuring all solutions are found and properly compared.