x^2 - x - 1 = 0 What values satisfy the equation above?...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{x^2 - x - 1 = 0}\)
• This is a quadratic equation in standard form \(\mathrm{ax^2 + bx + c = 0}\)
• Coefficients: \(\mathrm{a = 1, b = -1, c = -1}\)

2. INFER the solution approach

• This quadratic doesn't factor easily (try to find two numbers that multiply to -1 and add to -1)
• Since factoring is difficult, use the quadratic formula
• The quadratic formula will give us the exact solutions

3. SIMPLIFY using the quadratic formula

• Apply \(\mathrm{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}\)
• Substitute our values: \(\mathrm{x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}}\)
• SIMPLIFY the expression:
  • \(\mathrm{x = \frac{1 \pm \sqrt{1 + 4}}{2}}\)
  • \(\mathrm{x = \frac{1 \pm \sqrt{5}}{2}}\)

4. Write both solutions

• \(\mathrm{x = \frac{1 + \sqrt{5}}{2}}\) and \(\mathrm{x = \frac{1 - \sqrt{5}}{2}}\)

Answer: C. \(\mathrm{x = \frac{1+\sqrt{5}}{2}}\) and \(\mathrm{x = \frac{1-\sqrt{5}}{2}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to factor the quadratic instead of recognizing they need the quadratic formula. They might try to find factors of -1 that add to -1, get frustrated when this doesn't work easily, and either guess or make factoring errors like \(\mathrm{(x-1)(x+1) = x^2 - 1}\), not \(\mathrm{x^2 - x - 1}\).

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need for the quadratic formula but make sign errors. The most common mistake is writing \(\mathrm{x = \frac{-1 \pm \sqrt{5}}{2}}\) instead of \(\mathrm{x = \frac{1 \pm \sqrt{5}}{2}}\), forgetting that \(\mathrm{-b = -(-1) = +1}\).

This may lead them to select Choice D (\(\mathrm{\frac{-1+\sqrt{5}}{2}}\) and \(\mathrm{\frac{-1-\sqrt{5}}{2}}\)).

The Bottom Line:

This problem tests whether students can recognize when factoring won't work efficiently and can correctly apply the quadratic formula without sign errors. The key insight is that not all quadratics factor nicely, making the quadratic formula essential.