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!