Question

Not drawn
to scale
(x - 2)cm
D
The diagram shows two joined rectangles.
The total area of the compound shape
ABCDEF is 36 cm
2x cm
c
By considering the areas of the two rectangles,
show that 2x2 - 5x - 18 = 0 and hence find B
the value of length AB.
x cm
A
+
(x - 4)cm
Note: Please make sure your final line only shows your final answer written as AB = ...

Answer 1

First find the areas of the two separate rectangles where:

Area of rectangle = length x width

Area of the small rectangle = x(x–4)

Area of small rectangle = $x^{2} -4x$

Area of big rectangle = (x–2)(2x+x)

Area of big rectangle = (x–2)(3x)

Area of big rectangle = $3x^{2} -6x$

The total area of the compound shape is 36, therefore:

$3x^{2} -6x +x^{2} -4x=36$

$4x^{2} -10x-36=0$ (collecting like-terms and by bringing the +36 to the other side)

By dividing this by 2, you get:

$2(2x^{2} -5x-18)=0$

$2x^{2} -5x-18=0$

To find the length of AB (x), solve the quadratic by either factorising, completing the square or by using the quadratic formula, etc to solve for x.

$2x^{2} -5x-18=0$

$(2x-9)(x+2)=0$

$x=\frac{9}{2}$

$x=-2$

However, a length cannot be negative, so x must be $\frac{9}{2}$ or 4.5cm, so length AB = 4.5cm.

Hope this helps :)