To solve the quadratic equation $x^2 - 3x + 1 = 0$ using the quadratic formula, we use:

$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}$

For this equation:

• $a = 1$ (coefficient of $x^2$)
• $b = -3$ (coefficient of $x$)
• $c = 1$ (constant term)

Substituting these values into the formula:

$x = \frac{{-(-3) \pm \sqrt{{(-3)^2 - 4 \cdot 1 \cdot 1}}}}{2 \cdot 1}$

Simplifying inside the square root:

$x = \frac{{3 \pm \sqrt{{9 - 4}}}}{2}$

$x = \frac{{3 \pm \sqrt{5}}}{2}$

This gives us two solutions:

1. $x = \frac{{3 + \sqrt{5}}}{2}$
2. $x = \frac{{3 - \sqrt{5}}}{2}$

So, the solutions are:

$x = \frac{{3 + \sqrt{5}}}{2} \quad \text{and} \quad x = \frac{{3 - \sqrt{5}}}{2}$

Would you like further details on these steps? Here are some related questions:

1. How does the discriminant affect the nature of the solutions?
2. Can the solutions be expressed as decimals?
3. How would you verify these solutions by substituting back into the equation?
4. What is the vertex form of a quadratic equation, and how does it relate to this problem?
5. How do you graph the quadratic function $x^2 - 3x + 1$?

Tip: Simplifying the square root when using the quadratic formula helps identify if the solutions are rational or irrational.