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Let's factor the quadratic expression \( 4x^2 - 4x + 1 \).

We are dealing with a trinomial of the form \( ax^2 + bx + c \), where:
- \( a = 4 \),
- \( b = -4 \),
- \( c = 1 \).

### Step 1: Use the quadratic formula to find the roots
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Substitute the values \( a = 4 \), \( b = -4 \), and \( c = 1 \) into the formula:

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(1)}}{2(4)}
\]
\[
x = \frac{4 \pm \sqrt{16 - 16}}{8}
\]
\[
x = \frac{4 \pm \sqrt{0}}{8}
\]
\[
x = \frac{4}{8} = \frac{1}{2}
\]

Since there is only one root (a repeated root), the factorization of the quadratic is:
\[
4x^2 - 4x + 1 = \left( 2x - 1 \right)^2
\]

### Final Answer:
\[
4x^2 - 4x + 1 = (2x - 1)^2
\]

Would you like further details or have any questions about this? Here are some related questions for further exploration:

1. How can the quadratic formula be applied to other quadratic expressions?
2. What does it mean when a quadratic equation has only one repeated root?
3. Can all quadratic expressions be factored in the same way?
4. How can the discriminant help in determining the nature of the roots?
5. What is the geometric interpretation of a quadratic expression with a perfect square factorization?

**Tip:** When factoring quadratics, always check for the discriminant (\(b^2 - 4ac\)) to quickly determine the nature of the roots (real, repeated, or complex).

Learn how to factor the quadratic equation 4x^2 - 4x + 1 using the quadratic formula. This example demonstrates step-by-step algebraic methods and is suitable for students in Grades 9-11.

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