To find the measurement of the side of a square given its area, we need to take the square root of the area.

Given the area of the square is:

$\text{Area} = (4x^2 - 4x + 1) \text{ cm}^2$

The side length $s$ of the square is the square root of the area:

$s = \sqrt{4x^2 - 4x + 1}$

Now, let's simplify the expression inside the square root. The expression $4x^2 - 4x + 1$ is a perfect square trinomial. We can factor it as:

$4x^2 - 4x + 1 = (2x - 1)^2$

Therefore, the side length $s$ becomes:

$s = \sqrt{(2x - 1)^2}$

Since we are dealing with lengths, which are non-negative, we take the positive square root:

$s = |2x - 1|$

Thus, the measurement of the side of the square is:

$\boxed{|2x - 1| \text{ cm}}$

This means the side length is $2x - 1$ if $2x - 1 \geq 0$, or $1 - 2x$ if $2x - 1 < 0$.

Would you like more details or have any questions?

Related Questions:

1. What happens if $x = \frac{1}{2}$? What would be the side length?
2. How do you find the side length if the area is expressed differently?
3. How can you determine if the side length is a whole number for specific values of $x$?
4. What if the area is not a perfect square trinomial? How would you find the side length?
5. Can you derive the formula for the area of a square given any side length?

Tip: When solving for unknowns, always check if the expression can be simplified, such as factoring to identify perfect squares.