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The problem provides the area of a square garden as \( A = 36x^2 - 60x + 25 \), and asks for the perimeter when \( x = 3 \) feet.

### Step 1: Recognize the formula for the area of a square
The area \( A \) of a square is given by the formula:
\[
A = \text{side}^2
\]
Thus, the expression \( 36x^2 - 60x + 25 \) represents the square of the side length of the garden.

### Step 2: Solve for the side length
To find the side length, take the square root of the quadratic expression for the area:
\[
\text{side} = \sqrt{36x^2 - 60x + 25}
\]
We can recognize that the quadratic expression is a perfect square trinomial:
\[
36x^2 - 60x + 25 = (6x - 5)^2
\]
So, the side length is:
\[
\text{side} = |6x - 5|
\]
Since the side length must be positive, we drop the absolute value for simplicity:
\[
\text{side} = 6x - 5
\]

### Step 3: Substitute \( x = 3 \)
Now, substitute \( x = 3 \) into the expression for the side length:
\[
\text{side} = 6(3) - 5 = 18 - 5 = 13 \text{ feet}
\]

### Step 4: Calculate the perimeter
The perimeter \( P \) of a square is four times the side length:
\[
P = 4 \times \text{side} = 4 \times 13 = 52 \text{ feet}
\]

Thus, the perimeter of the garden is **52 feet**.

Would you like further details on any step, or have any other questions?

Here are 5 related questions:
1. How do you factor a perfect square trinomial?
2. What is the formula for the perimeter of other polygons like rectangles or triangles?
3. Can we solve the same problem using a different approach, like completing the square?
4. How does the concept of absolute value apply to side length?
5. What would the area of the garden be if \( x = 4 \)?

**Tip**: When dealing with perfect square trinomials, always check if the quadratic can be factored as a binomial square—it simplifies the process a lot!

The area of a square garden is 36x^2 - 60x + 25. What is the perimeter of the garden if x = 3 feet?

This math problem explores how to find the perimeter of a square garden when given the quadratic area expression A = 36x^2 - 60x + 25. Learn how to solve for the side length using algebraic methods and apply geometry formulas to find the perimeter. Suitable for students in Grades 8-10.

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