Question

The area of a rectangle is represented by (x^2-14x-32) square units . If the width of the rectangle is represented by (x+2) units , which expression represents the length of the rectangle

Answer 1

The expression for length of rectangle is (x - 16) units

Solution:

Given that area of a rectangle is represented by $x^2-14x-32$ square units

width of the rectangle is represented by (x+2) units

To find: length of the rectangle

The area of rectangle is given as:

$\text { area of rectangle }=\text { length } \times \text { width }$

Substituting the values in formula, we get

$x^{2}-14x-32=\text{length} \times(x+2)$  ---- eqn 1

Let us first factorise the L.H.S of eqn 1

$x^{2}-14x-32$

Break the expression into groups

$=\left(x^2+2x\right)+\left(-16x-32\right)$

$\mathrm{Factor\:out\:}x\mathrm{\:from\:}x^2+2x\mathrm{:\quad }x\left(x+2\right)$

$\mathrm{Factor\:out\:}-16\mathrm{\:from\:}-16x-32\mathrm{:\quad }-16\left(x+2\right)$

$=x\left(x+2\right)-16\left(x+2\right)$

$\mathrm{Factor\:out\:common\:term\:}x+2$

$=\left(x+2\right)\left(x-16\right)$

Now substitute the above value in eqn 1

$(x + 2)(x - 16) = length \times (x + 2)$

$length = \frac{(x + 2)(x - 16)}{x + 2}$

Length = x - 16

Thus the expression for length of rectangle is (x - 16) units