Question:x - 3 = sqrt(-2x + 21)What is the solution set for the equation above?

1. INFER the solution strategy

• Given: \(\mathrm{x - 3 = \sqrt{-2x + 21}}\)
• Key insight: To solve radical equations, we need to eliminate the square root by squaring both sides
• Important: Squaring can introduce extraneous solutions, so we must check our answers

2. SIMPLIFY by squaring both sides

• Square the left side: \(\mathrm{(x - 3)^2 = x^2 - 6x + 9}\)
• Square the right side: \(\mathrm{(\sqrt{-2x + 21})^2 = -2x + 21}\)
• Result: \(\mathrm{x^2 - 6x + 9 = -2x + 21}\)

3. SIMPLIFY into standard quadratic form

• Move all terms to left side: \(\mathrm{x^2 - 6x + 9 + 2x - 21 = 0}\)
• Combine like terms: \(\mathrm{x^2 - 4x - 12 = 0}\)
• Factor: \(\mathrm{(x - 6)(x + 2) = 0}\)
• Solutions: \(\mathrm{x = 6}\) or \(\mathrm{x = -2}\)

4. APPLY CONSTRAINTS by checking solutions

• Check \(\mathrm{x = 6}\): \(\mathrm{6 - 3 = \sqrt{-2(6) + 21}}\) → \(\mathrm{3 = \sqrt{9}}\) → \(\mathrm{3 = 3}\) ✓
• Check \(\mathrm{x = -2}\): \(\mathrm{-2 - 3 = \sqrt{-2(-2) + 21}}\) → \(\mathrm{-5 = \sqrt{25}}\) → \(\mathrm{-5 = 5}\) ✗
• Since -5 ≠ 5, the solution \(\mathrm{x = -2}\) is extraneous

Answer: \(\mathrm{\{6\}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS reasoning: Students solve the quadratic correctly to get \(\mathrm{x = 6}\) and \(\mathrm{x = -2}\), but fail to check these solutions in the original equation. They don't realize that squaring both sides can introduce extraneous solutions.

Without verification, they assume both solutions are valid and select Choice C (\(\mathrm{\{-2, 6\}}\)) instead of recognizing that \(\mathrm{x = -2}\) makes the original equation impossible (since the left side is negative while the square root on the right side must be non-negative).

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make algebraic errors when expanding \(\mathrm{(x - 3)^2}\) or when rearranging terms to standard form. Common mistakes include sign errors or incorrect coefficient combinations.

This leads to an incorrect quadratic equation, producing wrong solutions entirely. This causes confusion and often leads to guessing among the answer choices.

The Bottom Line:

The key challenge is remembering that when you square both sides of an equation, you might create solutions that don't work in the original equation. Always check your answers by substituting back into the original equation - this verification step is not optional, it's essential!