# The length of a rectangle is 2 centimeters less than twice the width. If the area is 84 square centimeters how do you find the dimensions of the rectangle?

Nov 14, 2016

width = 7 cm
length = 12 cm

#### Explanation:

It is often helpful to draw a quick sketch.

Let length be $L$
Let width be $w$

Area $= w L$

$= w \left(2 w - 2\right)$

$= 2 {w}^{2} - 2 w \text{ "=" } 84 c {m}^{2}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine } w}$

Subtract 84 from both sides

$0 = 2 {w}^{2} - 2 w - 84 \text{ " larr" this is a quadratic}$

I take one look at this and think: 'can not spot how to factorise so use the formula.'

Compare to $y = a {x}^{2} + b x + c \text{ }$ where $\text{ } x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

So for our equation we have:

$a = 2 \text{; "b=-2"; } c = - 84$

$\implies w = \frac{2 \pm \sqrt{{2}^{2} - 4 \left(2\right) \left(- 84\right)}}{2 \left(2\right)}$

$w = \frac{2 \pm \sqrt{676}}{4}$

$w = \frac{2}{4} \pm \frac{26}{4}$

To have $w$ as a negative value is not logical so go for:

" "color(green)(ul(bar(|color(white)(.)w=1/2+6 1/2=7 cmcolor(white)(.)|))

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Determine } L}$

$L = 2 w - 2$ so substitute for $w$ giving:

$L = 2 \left(7\right) - 2 = 12 c m$

" "color(green)(ul(bar(|color(white)(./.)L=2(7)-2 = 12 cmcolor(white)(.)|))