To factor the quadratic expression \( x^2 - 3x + 1 \), we will use the quadratic formula, since this expression does not factor neatly with simple integers.

The quadratic formula is:

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

For the quadratic expression \( x^2 - 3x + 1 \), the coefficients are:
- \( a = 1 \)
- \( b = -3 \)
- \( c = 1 \)

Now, substitute these values into the quadratic formula:

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

Simplifying:

\[
x = \frac{3 \pm \sqrt{9 - 4}}{2} = \frac{3 \pm \sqrt{5}}{2}
\]

Thus, the two solutions are:

\[
x = \frac{3 + \sqrt{5}}{2} \quad \text{or} \quad x = \frac{3 - \sqrt{5}}{2}
\]

Therefore, the factorization of \( x^2 - 3x + 1 \) is:

\[
(x - \frac{3 + \sqrt{5}}{2})(x - \frac{3 - \sqrt{5}}{2})
\]

Would you like more details on the steps or the method?

Here are 5 related questions:
1. How can the quadratic formula be derived?
2. Can we factor other quadratics with irrational roots similarly?
3. What happens if the discriminant in the quadratic formula is negative?
4. How do we factor perfect square trinomials?
5. How can we complete the square for the expression \( x^2 - 3x + 1 \)?

**Tip:** When the discriminant (\( b^2 - 4ac \)) is not a perfect square, the quadratic won't factor nicely with rational numbers, so we use the quadratic formula to find the roots.

Explore the detailed solution for factorizing the quadratic expression x^2 - 3x + 1 using the quadratic formula. Learn how to apply algebraic techniques and the quadratic formula theorem to find the roots. Suitable for students in Grades 9-12.

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