To factor \((a - \frac{1}{2})^2\), it's helpful to recognize that this expression is already in the form of a perfect square binomial. However, I'll guide you through the process to understand why this is the case.

### Step-by-Step Guide to Factoring \((a - \frac{1}{2})^2\)

1. **Identify the Expression:**
   \[
   (a - \frac{1}{2})^2
   \]
   This is a binomial raised to the power of 2.

2. **Recall the Formula for a Perfect Square Binomial:**
   The formula for squaring a binomial \( (x - y)^2 \) is:
   \[
   (x - y)^2 = x^2 - 2xy + y^2
   \]

3. **Apply the Formula:**
   For the given expression \((a - \frac{1}{2})^2\), identify \(x = a\) and \(y = \frac{1}{2}\).

   Now apply the formula:
   \[
   (a - \frac{1}{2})^2 = a^2 - 2 \cdot a \cdot \frac{1}{2} + \left(\frac{1}{2}\right)^2
   \]

4. **Simplify Each Term:**
   - The first term \(a^2\) remains as \(a^2\).
   - The second term simplifies as follows:
     \[
     -2 \cdot a \cdot \frac{1}{2} = -a
     \]
   - The third term is:
     \[
     \left(\frac{1}{2}\right)^2 = \frac{1}{4}
     \]

5. **Combine the Simplified Terms:**
   Putting it all together, you get:
   \[
   a^2 - a + \frac{1}{4}
   \]

### Conclusion:

The factored form of \(a^2 - a + \frac{1}{4}\) is \((a - \frac{1}{2})^2\). Since the given expression was already in a factored form, the process shows why it represents a perfect square trinomial.

### Questions:
1. What is the factored form of \((b + \frac{1}{4})^2\)?
2. How would you expand \((a + 2)^2\) to its trinomial form?
3. Can you explain why \((a - \frac{1}{2})^2\) is a perfect square trinomial?
4. What steps would you take to factor \(x^2 + 4x + 4\)?
5. How do you factor a difference of squares like \(a^2 - \frac{1}{4}\)?

**Tip:** Always check if the middle term in a trinomial is twice the product of the square roots of the first and last terms to confirm it's a perfect square.

Learn how to factor the expression (a - 1/2)^2 step-by-step. Understand why (a - 1/2)^2 is a perfect square binomial using the formula (x - y)^2 = x^2 - 2xy + y^2. Suitable for high school students studying algebra and factoring.

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