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Algebra · Quadratic-Equations

Solve r^2 - r - 1 = 0

r^{2} - r - 1 = 0

Step by step

  1. r^{2} - r - 1 = 0

    Start from the equation as given.

  2. a = 1,\quad b = -1,\quad c = -1

    Read off the coefficients of the standard form ax² + bx + c = 0.

  3. \Delta = b^2 - 4ac = (-1)^2 - 4(1)(-1) = 5

    It does not factor nicely, so use the discriminant Δ = b² − 4ac.

  4. r = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{1 \pm \sqrt{5}}{2}

    Δ > 0, so there are two distinct real roots. Substitute into the quadratic formula.

  5. r = \frac{1}{2} + \frac{\sqrt{5}}{2} \approx 1.6180 \quad\text{or}\quad r = \frac{1}{2} - \frac{\sqrt{5}}{2} \approx -0.61803

    Simplify each root.

Reveal the answer
r = \frac{1}{2} + \frac{\sqrt{5}}{2} \quad\text{or}\quad r = \frac{1}{2} - \frac{\sqrt{5}}{2}