1. Stating the problem: Solve the quadratic equation $$2x^{2}-7x+1=0$$. 2. Identify coefficients: Here, $a=2$, $b=-7$, and $c=1$. 3. Use the quadratic formula: $$x=\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$$. 4. Calculate the discriminant: $$\Delta = b^{2} - 4ac = (-7)^{2} - 4 \cdot 2 \cdot 1 = 49 - 8 = 41$$. 5. Substitute values into the formula: $$x= \frac{-(-7) \pm \sqrt{41}}{2 \cdot 2} = \frac{7 \pm \sqrt{41}}{4}$$. 6. Present the two solutions: $$x= \frac{7 + \sqrt{41}}{4} \quad \text{or} \quad x= \frac{7 - \sqrt{41}}{4}$$. These are the exact roots of the quadratic equation.