# A contractor is installing carpet in a new house. The rectangular living room has an area of 96 square feet, and the length is 12 feet less than three times the width of the room. What are the dimensions of the room?

Nov 20, 2016

The room is 8' x 12'

#### Explanation:

We need to set this up based on fact we learned earlier. Area equals length multiplied by width. This is where the algebra kicks in.

$A = l \cdot w$

We know that the length is 12' less than three times the width

$A = 96$
$l = 3 w - 12$
$w = w$

Plug these values into the original formula.

$A = l \cdot w$
$96 = \left(3 w - 12\right) \cdot w$
$96 = 3 {w}^{2} - 12 w$
$0 = 3 {w}^{2} - 12 w - 96$

Now we factor

$0 = 3 \left({w}^{2} - 4 w - 32\right)$
$0 = 3 \left(w + 4\right) \left(w - 8\right)$

One of the $w$'s has to make the equation equal to zero for it to be true. Ergo,

$w = - 4 , 8$

Since the width has to be a positive number, that leaves 8 as your width. Sub that into the length equation to solve for $l$.

$l = 3 w - 12$
$l = 3 \left(8\right) - 12$
$l = 24 - 12$
$l = 12$

$A = l \cdot w$
$96 = 8 \cdot 12$

:)