Question: x - 2 = sqrt(3x - 2) What value of x is a solution to the given equation?...

1. INFER the solving strategy

• Given: \(\mathrm{x - 2 = \sqrt{3x - 2}}\)
• Key insight: To solve equations with square roots, eliminate the radical by squaring both sides
• Important: Squaring can introduce extraneous solutions, so we must check all solutions

2. SIMPLIFY by squaring both sides

• Square the left side: \(\mathrm{(x - 2)^2 = x^2 - 4x + 4}\)
• Square the right side: \(\mathrm{(\sqrt{3x - 2})^2 = 3x - 2}\)
• Result: \(\mathrm{x^2 - 4x + 4 = 3x - 2}\)

3. SIMPLIFY to standard quadratic form

• Move all terms to one side: \(\mathrm{x^2 - 4x + 4 - 3x + 2 = 0}\)
• Combine like terms: \(\mathrm{x^2 - 7x + 6 = 0}\)

4. SIMPLIFY by factoring the quadratic

• Need two numbers that multiply to 6 and add to -7
• Those numbers are -6 and -1: \(\mathrm{(x - 6)(x - 1) = 0}\)
• Solutions: \(\mathrm{x = 6}\) or \(\mathrm{x = 1}\)

5. APPLY CONSTRAINTS by checking both solutions

Check \(\mathrm{x = 6}\):

• Left side: \(\mathrm{6 - 2 = 4}\)
• Right side: \(\mathrm{\sqrt{3(6) - 2} = \sqrt{16} = 4}\)
• \(\mathrm{4 = 4}\) ✓ Valid solution

Check \(\mathrm{x = 1}\):

• Left side: \(\mathrm{1 - 2 = -1}\)
• Right side: \(\mathrm{\sqrt{3(1) - 2} = \sqrt{1} = 1}\)
• \(\mathrm{-1 = 1}\) ✗ Extraneous solution

Answer: 6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak APPLY CONSTRAINTS: Students find both \(\mathrm{x = 6}\) and \(\mathrm{x = 1}\) from the quadratic but fail to check these solutions in the original equation.

Without checking, they might see that both 1 and 6 appear in the answer choices and select the smaller value thinking it's more likely correct, or they might get confused about which one to choose. This leads them to randomly select Choice A (1) instead of recognizing that \(\mathrm{x = 1}\) is extraneous.

Second Most Common Error:

Poor INFER reasoning: Students attempt to solve the equation without eliminating the square root first, trying to isolate x directly or making algebraic errors because they don't recognize the need to square both sides.

This approach leads to getting stuck early in the problem, causing confusion about how to proceed systematically. This leads to abandoning the systematic solution and guessing.

The Bottom Line:

This problem tests whether students understand that squaring both sides of an equation can introduce false solutions. The algebraic manipulation is straightforward, but the crucial step is checking solutions - many students skip this verification step and select incorrect answers.