The factorized form of `x^2 - x + (1)/(4)`is

Options

A. `(x - (1)/(2) ) (x - (1)/(2) )`
B. `(2 x -2) (x - (1)/(2))`
C. `(2 x-2 ) (2x-2)`
D. `(x-2) (x-2)`

Answer

To factorize the expression \( x^2 - x + \frac{1}{4} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the expression**: We start with the expression \( x^2 - x + \frac{1}{4} \).

2. **Rewrite the expression**: Notice that \( \frac{1}{4} \) can be expressed as \( \left(\frac{1}{2}\right)^2 \). Thus, we can rewrite the expression as:
\[ x^2 - x + \left(\frac{1}{2}\right)^2 \]

3. **Recognize the form**: The expression now resembles the perfect square trinomial form \( a^2 - 2ab + b^2 \), where \( a = x \) and \( b = \frac{1}{2} \).

4. **Apply the formula**: Recall that the perfect square trinomial can be factored as:
\[ (a - b)^2 = a^2 - 2ab + b^2 \]
In our case, we have:
\[ x^2 - 2 \cdot x \cdot \frac{1}{2} + \left(\frac{1}{2}\right)^2 \]
This can be factored into:
\[ \left(x - \frac{1}{2}\right)^2 \]

5. **Final factorized form**: Therefore, the factorized form of the expression \( x^2 - x + \frac{1}{4} \) is:
\[ \left(x - \frac{1}{2}\right)^2 \]