To solve the quadratic equation \( x^2 - 3x + 1 = 0 \) using the quadratic formula, we can use the formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = -3 \), and \( c = 1 \).
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 1 = 9 - 4 = 5 \]
Now, we can substitute the values of \( a \), \( b \), and the discriminant into the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{5}}{2 \cdot 1} \]
This simplifies to:
\[ x = \frac{3 \pm \sqrt{5}}{2} \]
Thus, the exact solutions are:
\[ x = \frac{3 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{3 - \sqrt{5}}{2} \]