Question:The equation sqrt(9 - x^2) = 3 - x has two solutions. What is the value of the smaller solution?

1. APPLY CONSTRAINTS to establish the domain

• Given equation: \(\sqrt{9 - \mathrm{x}^2} = 3 - \mathrm{x}\)
• Domain restrictions needed:
  • For \(\sqrt{9 - \mathrm{x}^2}\) to be real: \(9 - \mathrm{x}^2 \geq 0\), so \(-3 \leq \mathrm{x} \leq 3\)
  • For \(3 - \mathrm{x} \geq 0\): \(\mathrm{x} \leq 3\)
  • Combined domain: \(-3 \leq \mathrm{x} \leq 3\)

2. INFER the solution strategy

• Since we have a square root equal to an expression, we can eliminate the radical by squaring both sides
• This will create a quadratic equation we can solve

3. SIMPLIFY by squaring both sides and solving

• Square both sides: \((\sqrt{9 - \mathrm{x}^2})^2 = (3 - \mathrm{x})^2\)
• This gives us: \(9 - \mathrm{x}^2 = 9 - 6\mathrm{x} + \mathrm{x}^2\)
• Rearranging: \(9 - \mathrm{x}^2 = 9 - 6\mathrm{x} + \mathrm{x}^2\)
• Combining like terms: \(-\mathrm{x}^2 = -6\mathrm{x} + \mathrm{x}^2\)
• Moving all terms to one side: \(0 = 2\mathrm{x}^2 - 6\mathrm{x}\)
• Factor: \(0 = 2\mathrm{x}(\mathrm{x} - 3)\)
• Solutions: \(\mathrm{x} = 0\) or \(\mathrm{x} = 3\)

4. APPLY CONSTRAINTS to verify solutions

• Check \(\mathrm{x} = 0\): \(\sqrt{9 - 0^2} = 3\) and \(3 - 0 = 3\) ✓
• Check \(\mathrm{x} = 3\): \(\sqrt{9 - 9} = 0\) and \(3 - 3 = 0\) ✓
• Both solutions satisfy the original equation and are within our domain

5. INFER the final answer

• We have two valid solutions: \(\mathrm{x} = 0\) and \(\mathrm{x} = 3\)
• The question asks for the smaller solution

Answer: 0

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS skill: Students forget to establish domain restrictions before solving, or fail to verify their solutions in the original equation.

When squaring both sides, students might find \(\mathrm{x} = 0\) and \(\mathrm{x} = 3\), but not verify these in \(\sqrt{9 - \mathrm{x}^2} = 3 - \mathrm{x}\). Since both happen to be valid here, they might get lucky, but this poor habit will hurt them on problems with extraneous solutions.

This may lead them to select the wrong solution if they don't properly verify, or causes confusion about which answer to choose.

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when expanding \((3 - \mathrm{x})^2\) or when rearranging the resulting equation.

For example, they might incorrectly expand \((3 - \mathrm{x})^2\) as \(9 - \mathrm{x}^2\) instead of \(9 - 6\mathrm{x} + \mathrm{x}^2\), leading to an equation like \(9 - \mathrm{x}^2 = 9 - \mathrm{x}^2\), which would suggest all values in the domain are solutions.

This leads to confusion and guessing among the answer choices.

The Bottom Line:

Radical equations require careful attention to domain restrictions and solution verification. The key insight is that squaring both sides can introduce extraneous solutions, making verification essential rather than optional.