For x neq 0, the equation x - 1/x = 1 holds. What is the larger solution of the equation?

1. TRANSLATE the problem information

• Given: \(\mathrm{x - \frac{1}{x} = 1}\), where \(\mathrm{x ≠ 0}\)
• Find: The larger solution

2. INFER the solution strategy

• The fraction \(\frac{1}{\mathrm{x}}\) makes this tricky to solve directly
• Since \(\mathrm{x ≠ 0}\), we can safely multiply both sides by x to eliminate the fraction
• This should convert it to a more manageable form

3. SIMPLIFY by clearing the fraction

• Multiply both sides by x: \(\mathrm{x(x - \frac{1}{x}) = x(1)}\)
• Left side: \(\mathrm{x^2 - 1}\)
• Right side: \(\mathrm{x}\)
• Result: \(\mathrm{x^2 - 1 = x}\)

4. SIMPLIFY to standard quadratic form

• Move all terms to one side: \(\mathrm{x^2 - x - 1 = 0}\)
• This is now in the form \(\mathrm{ax^2 + bx + c = 0}\) where \(\mathrm{a = 1, b = -1, c = -1}\)

5. INFER that we need the quadratic formula

• This quadratic doesn't factor easily, so use the quadratic formula
• \(\mathrm{x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}}\)

6. SIMPLIFY using the quadratic formula

• Substitute: \(\mathrm{x = \frac{1 ± \sqrt{1 + 4}}{2} = \frac{1 ± \sqrt{5}}{2}}\)
• This gives us two solutions:
  • \(\mathrm{x_1 = \frac{1 + \sqrt{5}}{2}}\)
  • \(\mathrm{x_2 = \frac{1 - \sqrt{5}}{2}}\)

7. CONSIDER ALL CASES to find the larger solution

• Since \(\mathrm{\sqrt{5} \gt 0}\), adding \(\mathrm{\sqrt{5}}\) gives a larger result than subtracting it
• Therefore \(\mathrm{\frac{1 + \sqrt{5}}{2} \gt \frac{1 - \sqrt{5}}{2}}\)
• We can verify: \(\mathrm{\frac{1 + \sqrt{5}}{2} ≈ 1.618}\) and \(\mathrm{\frac{1 - \sqrt{5}}{2} ≈ -0.618}\) (use calculator)

Answer: (D) \(\mathrm{\frac{1 + \sqrt{5}}{2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make algebraic errors when multiplying by x or rearranging terms. They might get \(\mathrm{x^2 + 1 = x}\) instead of \(\mathrm{x^2 - 1 = x}\), or incorrectly rearrange to \(\mathrm{x^2 + x - 1 = 0}\). These sign errors lead to wrong coefficients in the quadratic formula, producing incorrect solutions that don't match any answer choice. This leads to confusion and guessing.

Second Most Common Error:

Poor CONSIDER ALL CASES reasoning: Students correctly find both solutions but incorrectly identify which is larger. They might think \(\mathrm{\frac{1 - \sqrt{5}}{2}}\) is larger because they focus on the "1" part and ignore that \(\mathrm{\sqrt{5} \gt 2}\), making \(\mathrm{(1 - \sqrt{5})}\) negative. This may lead them to select Choice (C) \(\mathrm{\frac{1 - \sqrt{5}}{2}}\) instead of the correct larger solution.

The Bottom Line:

This problem tests whether students can systematically convert a rational equation to quadratic form without making sign errors, then correctly compare radical expressions to identify the maximum value.