Answer
The rug is $9$ft by $12$ft
Work Step by Step
The room is $12$ft by $15$ft, and the rug has an even gap $x$, all around the walls resulting in a rug with and area of $108$ft$^2$
Since there is a gap on each side of the rug,
Rug width = width of the room minus 2x = $12-2x$
Rug length = length of the room minus 2x = $15-2x$
We know that the formula for area of a rectangle is $(width)(length)=area$ so:
$(12-2x)(15-2x)=108$
$180-24x-30x+4x^2=108$
$4x^2 -54x+180=108$
If we rewrite in standard quadratic equation form $ax^2 + bx +c=0$,
$4x^2-54x+72=0$
where $a=4$, $b=-54$, and $c=72$,
we can solve by applying the quadratic formula: $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{(-)(-54)\pm\sqrt{(-54)^2-4(4)(72)}}{2(4)}$
$x=\dfrac{54\pm\sqrt{2916-1152}}{8}$
$x=\dfrac{54\pm\sqrt{1764}}{8}$
$x=\dfrac{54\pm42}{8}$
$x=\dfrac{54+42}{8}$ or $x=\dfrac{54-42}{8}$
$x=\dfrac{96}{8}$ or $x=\dfrac{12}{8}$
$x=12$ or $x=1.5$
Now we test our $x$ values in our formula, $(12-2x)(15-2x)=108$
If $x=12$:
$[12-2(12)]\times[15-2(12)]=108$
$(12-24)(15-24)=108$
$(-12)(-9)=108$
which is mathematically correct, but the rug CANNOT have a negative length or width
If $x=1.5$:
$[12-2(1.5)]\times[15-2(1.5)]=108$
$(12-3)(15-3)=108$
$(9)(12)=108$
So the dimensions of the rug are $9$ft by $12$ft
