Hello, Jerry99!

Did you make a sketch?

$\displaystyle \text{A rug is fit into a room so that a border of even width is left on all sides.}$

$\displaystyle \text{If the room is 12 ft}\tomes\text{ 15 ft and the area of the rug is 108 ft}^2,$

. . $\displaystyle \text{how wide will the border be?}$

Code:

: - - - 15 - - - :
- *-----------------* -
: | : : | x
: | - *---------* - | -
: | | | | :
12| | | |12-2x
: | | | | :
: | - *---------* - | -
: | : : | x
. *-----------------* -
: x : 15-2x : x :

The room is 15 by 11 feet.

The border is $\displaystyle \,x$ feet wide.

The rug is $\displaystyle (15-2x)$ by $\displaystyle (12-2x)$ feet.

The area of the rug is 108 ft$\displaystyle ^2.$

There is our equation! . . . $\displaystyle (15-2x)(12-2x) \:=\:108$

This simplifies to: .$\displaystyle 2x^2 - 27x + 36 \:=\:0 \quad\Rightarrow\quad (2x-3)(x-12) \:=\:0$

Hence: .$\displaystyle x \:=\:\frac{3}{2},\;\rlap{///////}x \:=\:12$

The width of the border is $\displaystyle 1\frac{1}{2}$ feet.