The length of a rectangular floor is 2 feet more than its width. The area of the floor is 168 square feet. Kim wants to use a rug in the middle of the room and leave a 2 foot border of the floor visible on all sides. What should the length (the longer side) of the rug be?

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To find the length of the rug, we first need to determine the dimensions of the rectangular floor.

Let's assume the width of the floor is x feet. Since the length is 2 feet more than the width, the length would be x + 2 feet.

The area of the floor is given as 168 square feet, so we can set up an equation to represent this:

Length * Width = Area

(x + 2) * x = 168

Expanding the equation, we get:

x^2 + 2x = 168

Rearranging the equation:

x^2 + 2x - 168 = 0

Now we can solve this quadratic equation to find the value of x using factoring, completing the square, or the quadratic formula. Factoring doesn't work in this case, so let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation, a = 1, b = 2, and c = -168. Plugging these values into the formula:

x = (-2 ± √(2^2 - 4 * 1 * -168)) / (2 * 1)

Simplifying:

x = (-2 ± √(4 + 672)) / 2

x = (-2 ± √676) / 2

x = (-2 ± 26) / 2

Now we have two possible solutions for x:

Case 1: x = (-2 + 26) / 2 = 24 / 2 = 12
Case 2: x = (-2 - 26) / 2 = -28 / 2 = -14

Since the width of a floor cannot be negative, we discard the negative solution. Therefore, the width of the floor is 12 feet.

Now, to find the length of the rug, we need to add 2 feet to the width:

Length = Width + 2 = 12 + 2 = 14 feet

Therefore, the length of the rug should be 14 feet.