To factor the quadratic expression \( x^2 - 4x + 1 \), we will follow these steps:

1. **Identify coefficients:**  
   The given quadratic expression is \( x^2 - 4x + 1 \), where:
   - The coefficient of \( x^2 \) (the leading coefficient) is \( a = 1 \).
   - The coefficient of \( x \) is \( b = -4 \).
   - The constant term is \( c = 1 \).

2. **Use the quadratic formula to find the roots:**  
   The roots of the quadratic expression can be found using the quadratic formula:
   \[
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Substituting \( a = 1 \), \( b = -4 \), and \( c = 1 \):
   \[
   x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(1)}}{2(1)}
   \]
   \[
   x = \frac{4 \pm \sqrt{16 - 4}}{2}
   \]
   \[
   x = \frac{4 \pm \sqrt{12}}{2}
   \]
   \[
   x = \frac{4 \pm 2\sqrt{3}}{2}
   \]
   \[
   x = 2 \pm \sqrt{3}
   \]

3. **Express the quadratic in factored form:**
   The quadratic can be factored as:
   \[
   x^2 - 4x + 1 = (x - (2 + \sqrt{3}))(x - (2 - \sqrt{3}))
   \]

So, the factored form of \( x^2 - 4x + 1 \) is:
\[
(x - (2 + \sqrt{3}))(x - (2 - \sqrt{3}))
\]

Would you like further details or have any questions?

Here are 5 related questions you might consider:
1. How do you factor a quadratic expression when the discriminant is a perfect square?
2. What is the importance of the discriminant in the quadratic formula?
3. How can you complete the square for the expression \( x^2 - 4x + 1 \)?
4. How do you determine the vertex of a parabola given a quadratic expression?
5. How does the factorization of a quadratic relate to its graph?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before attempting to factor a quadratic. If it's a perfect square, the quadratic can be factored using real numbers.

Learn how to factor the quadratic expression x^2 - 4x + 1 using the quadratic formula. This detailed solution explains each step and demonstrates how to find the roots and express the equation in its factored form. Perfect for students in Grades 9-12 exploring key algebraic concepts.

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