The area of a square is given as A = 9x^2 − 6x + 1.  Find the perimeter of the square.

To find the perimeter of the square, we first need to determine the side length of the square. The area of a square is given by \( A = s^2 \), where \( s \) is the side length.

Here, the area \( A \) is given by the quadratic expression \( 9x^2 - 6x + 1 \). Therefore, we have:

\[
s^2 = 9x^2 - 6x + 1
\]

To find \( s \), we take the square root of both sides:

\[
s = \sqrt{9x^2 - 6x + 1}
\]

Next, we need to express the perimeter of the square. The perimeter \( P \) of a square is given by:

\[
P = 4s
\]

So, substituting the expression for \( s \):

\[
P = 4 \times \sqrt{9x^2 - 6x + 1}
\]

Thus, the perimeter of the square is \( P = 4 \sqrt{9x^2 - 6x + 1} \).

Would you like further details on how to simplify or evaluate this expression, or any other questions?

Here are 5 related questions:
1. How can we simplify \( \sqrt{9x^2 - 6x + 1} \)?
2. What is the value of \( s \) when \( x = 1 \)?
3. Can the quadratic expression \( 9x^2 - 6x + 1 \) be factored further?
4. What happens to the perimeter if \( x \) is doubled?
5. How does the area change as \( x \) increases?

**Tip:** When dealing with square roots of quadratic expressions, it's helpful to check if the quadratic can be factored into a perfect square, which simplifies calculations.

Learn how to find the perimeter of a square when the area is given by the quadratic expression 9x^2 − 6x + 1. This problem involves calculating the side length of the square using algebraic manipulation and then applying the formula for perimeter. Suitable for students in Grades 10-12.

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