Solve for Square Side Length: When the Extended Rectangle's Area Matches the Square

To solve this problem, we'll follow these steps:

• Step 1: Identify the areas for both square and rectangle using given dimensions.
• Step 2: Equate the two areas as the problem states they are equal.
• Step 3: Solve the resulting equation for $x$.
• Step 4: Check that the condition $x > 1$ is satisfied and select the correct answer choice.

Now, let's work through the solution:

Step 1: The area of the square with side length $x$ is given by:

$x^2$.

For the rectangle, where one side is extended by 3 cm and an adjacent side is shortened by 1 cm, we have:

Original length and width of the rectangle are $x + 3$ and $x - 1$, respectively.

The area of the rectangle becomes:

$(x + 3)(x - 1)$.

Step 2: As per the problem, these two areas are equal:

$x^2 = (x + 3)(x - 1)$.

Step 3: Expanding the right-hand side of the equation:

$(x + 3)(x - 1) = x^2 + 3x - x - 3 = x^2 + 2x - 3$.

Now, equate and simplify:

$x^2 = x^2 + 2x - 3$.

Subtract $x^2$ from both sides:

$0 = 2x - 3$.

Adding 3 to both sides gives:

$3 = 2x$.

Divide both sides by 2 to solve for $x$:

$x = \frac{3}{2}$.

Step 4: We check the condition $x > 1$ and find $x = \frac{3}{2} \approx 1.5$ which satisfies it.

Therefore, the side length of the square is $x = \frac{3}{2}$ cm.