Calculate Rectangle Area: Solve for X Given X-6 Dimensions

Given a rectangle whose side is smaller by 6 than the other side. We mark the area of the rectangle with S

and the large side with X

Check the correct argument:

To solve this problem, we need to express the area of the rectangle in terms of $X$, accounting for the relationship between its sides.

Step 1: Calculate the area of the rectangle.
The area $s$ of a rectangle is obtained by multiplying the length by the width:

$s = X \times (X - 6)$

Expanding the expression gives us:

$s = X^2 - 6X$

We want to express this in a form that matches the given answer choices. Recognizing the square of a binomial will help us reformulate:

$s = X^2 - 6X$ can be related to a square of a binomial by adjusting it:

Step 2: Recast into a recognizable square:
We want to find a relation to $(x-3)^2$, so bear in mind:

$(X - 3)^2 = X^2 - 6X + 9$

Therefore, rearranging in the form of $(X - 3)^2$, we derive:

$s = (X - 3)^2 - 9$

Therefore, the area of the rectangle, expressed in a way to match the correct choices, is $s = (x - 3)^2 - 9$, which corresponds to choice 3 from the given options.

The correct choice is: Choice 3: $s=(x-3)^2-9$.