1. **State the problem:** Solve the quadratic equation $x^2 - 5x + 1 = 0$. 2. **Formula used:** The quadratic formula is used to solve equations of the form $ax^2 + bx + c = 0$: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=-5$, and $c=1$. 3. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 1 \times 1 = 25 - 4 = 21$$ 4. **Apply the quadratic formula:** $$x = \frac{-(-5) \pm \sqrt{21}}{2 \times 1} = \frac{5 \pm \sqrt{21}}{2}$$ 5. **Final answer:** $$x = \frac{5 + \sqrt{21}}{2} \quad \text{or} \quad x = \frac{5 - \sqrt{21}}{2}$$ These are the two solutions to the quadratic equation.