1. We are asked to solve the quadratic equation $$x^2 - 3x + 1 = 0$$. 2. To solve this, we use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=1$$, $$b=-3$$, and $$c=1$$. 3. Substitute the values into the formula: $$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \times 1 \times 1}}{2 \times 1} = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2}$$. 4. Thus, the two solutions are: $$x = \frac{3 + \sqrt{5}}{2}$$ and $$x = \frac{3 - \sqrt{5}}{2}$$. 5. These are the roots of the quadratic equation.