1. **State the problem:** Solve the inequality $q^2 - q > 1$. 2. **Rewrite the inequality:** Move all terms to one side to set the inequality to zero: $$q^2 - q - 1 > 0$$ 3. **Find the roots of the quadratic equation:** Solve $q^2 - q - 1 = 0$ using the quadratic formula: $$q = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-1$, and $c=-1$. 4. **Calculate the discriminant:** $$\Delta = (-1)^2 - 4 \times 1 \times (-1) = 1 + 4 = 5$$ 5. **Find the roots:** $$q = \frac{1 \pm \sqrt{5}}{2}$$ 6. **Analyze the inequality:** Since the parabola opens upwards ($a=1>0$), the quadratic expression is greater than zero outside the roots. 7. **Write the solution:** $$q < \frac{1 - \sqrt{5}}{2} \quad \text{or} \quad q > \frac{1 + \sqrt{5}}{2}$$ **Final answer:** $$\boxed{q < \frac{1 - \sqrt{5}}{2} \text{ or } q > \frac{1 + \sqrt{5}}{2}}$$