A real number x satisfies x + 1/x = 12/5, where x neq 0. Which of the following is a...

1. TRANSLATE the rational equation to eliminate fractions

• Given: \(\mathrm{x + \frac{1}{x} = \frac{12}{5}}\), where \(\mathrm{x ≠ 0}\)

• Strategy: Multiply both sides by x to clear the fraction:
  \(\mathrm{x^2 + 1 = \frac{12}{5}x}\)

• Then multiply both sides by 5 to eliminate the remaining fraction:
  \(\mathrm{5x^2 + 5 = 12x}\)

2. SIMPLIFY to standard quadratic form

• Rearrange all terms to one side:
  \(\mathrm{5x^2 - 12x + 5 = 0}\)

• Now we have a quadratic equation in standard form \(\mathrm{ax^2 + bx + c = 0}\)
  where \(\mathrm{a = 5, b = -12, c = 5}\)

3. APPLY the quadratic formula

• Use \(\mathrm{x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}}\)

• Substitute:
  \(\mathrm{x = \frac{12 ± \sqrt{144 - 100}}{10}}\)
  \(\mathrm{x = \frac{12 ± \sqrt{44}}{10}}\)

4. SIMPLIFY the radical expression

• Factor the radicand:
  \(\mathrm{\sqrt{44} = \sqrt{4 · 11} = 2\sqrt{11}}\)

• Therefore:
  \(\mathrm{x = \frac{12 ± 2\sqrt{11}}{10}}\)
  \(\mathrm{x = \frac{6 ± \sqrt{11}}{5}}\)

• The two solutions are \(\mathrm{\frac{6 + \sqrt{11}}{5}}\) and \(\mathrm{\frac{6 - \sqrt{11}}{5}}\)

5. APPLY CONSTRAINTS to match answer choices

• Check which solution appears in the answer choices
• Only \(\mathrm{\frac{6 + \sqrt{11}}{5}}\) is listed as option (D)

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students attempt to substitute each answer choice back into the original equation rather than solving algebraically.

While this can work, it's time-consuming and error-prone, especially with expressions involving radicals like \(\mathrm{\frac{6 + \sqrt{11}}{5}}\). Students may make arithmetic errors when computing \(\mathrm{x + \frac{1}{x}}\) for complex expressions, or give up partway through the checking process.

This leads to confusion and guessing among the remaining choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the quadratic \(\mathrm{5x^2 - 12x + 5 = 0}\) but make calculation errors in the quadratic formula.

Common mistakes include: computing the discriminant as \(\mathrm{\sqrt{144 - 20} = \sqrt{124}}\) instead of \(\mathrm{\sqrt{144 - 100} = \sqrt{44}}\), or failing to simplify \(\mathrm{\sqrt{44}}\) to \(\mathrm{2\sqrt{11}}\). These errors lead to incorrect final expressions that don't match any answer choice.

This causes them to get stuck and guess.

The Bottom Line:

The key challenge is recognizing that clearing denominators transforms a rational equation into a more manageable quadratic equation. Students who try to work directly with the fractional form or who substitute answer choices often struggle with the complex arithmetic involved.