# The width and the length of a rectangle are consecutive even integers. If the width is decreased by 3 inches. then the area of the resulting rectangle is 24 square inches What is the area of the original rectangle?

Nov 20, 2017

$48 \text{ square inches}$

#### Explanation:

$\text{let the width } = n$

$\text{then length } = n + 2$

$n \text{ and "n+2color(blue)" are consecutive even integers}$

$\text{the width is decreased by " 3" inches}$

$\Rightarrow \text{width } = n - 3$

$\text{area "="length "xx" width}$

$\Rightarrow \left(n + 2\right) \left(n - 3\right) = 24$

$\Rightarrow {n}^{2} - n - 6 = 24$

$\Rightarrow {n}^{2} - n - 30 = 0 \leftarrow \textcolor{b l u e}{\text{in standard form}}$

$\text{the factors of - 30 which sum to - 1 are + 5 and - 6}$

$\Rightarrow \left(n - 6\right) \left(n + 5\right) = 0$

$\text{equate each factor to zero and solve for n}$

$n - 6 = 0 \Rightarrow n = 6$

$n + 5 = 0 \Rightarrow n = - 5$

$n > 0 \Rightarrow n = 6$

$\text{the original dimensions of the rectangle are}$

$\text{width } = n = 6$

$\text{length } = n + 2 = 6 + 2 = 8$

$6 \text{ and "8" are consecutive even integers}$

$\Rightarrow \text{original area "=8xx6=48" square inches}$