To solve this problem, we need to find the value of $x$ for which the area of the walkway equals the area of the pool.

Step 1: Find the Area of the Pool

The pool is rectangular with dimensions $12 \, \text{meters} \times 8 \, \text{meters}$. $\text{Area of the pool} = 12 \times 8 = 96 \, \text{square meters}$

Step 2: Determine the Dimensions of the Outer Rectangle (Pool + Walkway)

The walkway surrounds the pool on all sides with a width of $x$ meters. Therefore:

• The length of the outer rectangle (including the walkway) is $12 + 2x$.
• The width of the outer rectangle (including the walkway) is $8 + 2x$.

Step 3: Calculate the Area of the Outer Rectangle

The area of the outer rectangle is: $\text{Area of the outer rectangle} = (12 + 2x)(8 + 2x)$

Expanding this: $(12 + 2x)(8 + 2x) = 96 + 24x + 16x + 4x^2 = 96 + 40x + 4x^2$

So, the area of the outer rectangle is: $4x^2 + 40x + 96$

Step 4: Calculate the Area of the Walkway

The area of the walkway is the area of the outer rectangle minus the area of the pool: $\text{Area of the walkway} = (4x^2 + 40x + 96) - 96 = 4x^2 + 40x$

Step 5: Set Up the Equation

We want the area of the walkway to equal the area of the pool, so: $4x^2 + 40x = 96$

Step 6: Solve for $x$

Divide each term by 4 to simplify: $x^2 + 10x = 24$

Rewrite as a standard quadratic equation: $x^2 + 10x - 24 = 0$

Solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ where $a = 1$, $b = 10$, and $c = -24$.

[ x = \frac{-10 \pm \sqrt{10^2 + 4 \cdot 24}}{2 \cdot