1. **State the problem:** Solve the quadratic equation $x^2 - 4x + 1 = 0$. 2. **Formula used:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 1$, $b = -4$, and $c = 1$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-4)^2 - 4 \times 1 \times 1 = 16 - 4 = 12$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-4) \pm \sqrt{12}}{2 \times 1} = \frac{4 \pm \sqrt{12}}{2}$$ 6. **Simplify the square root:** $$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$$ 7. **Substitute back:** $$x = \frac{4 \pm 2\sqrt{3}}{2}$$ 8. **Simplify the fraction:** $$x = \frac{\cancel{2} \times 2 \pm \cancel{2} \times \sqrt{3}}{\cancel{2}} = 2 \pm \sqrt{3}$$ **Final answer:** $$x = 2 + \sqrt{3} \quad \text{or} \quad x = 2 - \sqrt{3}$$