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.