To factor the quadratic expression \( 4y^2 - 4y + 1 \), let's follow these steps:

1. **Recognize the quadratic structure**: The given expression is in the form of \( ay^2 + by + c \) where \( a = 4 \), \( b = -4 \), and \( c = 1 \).

2. **Check if it's a perfect square trinomial**: A perfect square trinomial has the form \( (py + q)^2 \), which expands to \( p^2y^2 + 2pqy + q^2 \).

3. **Compare the given expression with the perfect square form**:
   - \( p^2 = 4 \), so \( p = 2 \).
   - \( q^2 = 1 \), so \( q = 1 \).
   - Check if \( 2pq = -4 \). Here, \( 2(2)(1) = 4 \), and since we have a negative sign in the original equation, we need \( q = -1 \).

4. **Write the factored form**:
   \[
   4y^2 - 4y + 1 = (2y - 1)^2
   \]

So, the factored form of the expression is \( \boxed{(2y - 1)^2} \).

Would you like more details or have any questions?

Here are some related questions to explore further:

1. How do you factor a quadratic expression in general?
2. What are the steps to recognize a perfect square trinomial?
3. How can you use the quadratic formula to factor expressions?
4. What is the difference between factoring by grouping and factoring by completing the square?
5. How do you expand and simplify \( (2y - 1)^2 \) to verify the result?

**Tip**: Always check your factored form by expanding it to ensure it matches the original expression.

Learn how to factor the quadratic expression 4y^2 - 4y + 1 with a detailed step-by-step solution. Understand the concept of perfect square trinomials and their application in factoring. Suitable for students in Grades 9-12 studying algebra and quadratic equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation