Step-by-step explanation:
To factorize the quadratic equation $2x^2 - 5x - 18 = 0$, we can use the area of the compound shape ABCDEF to find the factors.
The total area of the compound shape ABCDEF is given as $36cm^2$. We can express this area in terms of the areas of the two rectangles. Let's denote the length of rectangle AB as x and the width as (2x-9) and the length of rectangle EF as 3 and the width as (2x-9). Then, the total area is the sum of the areas of the two rectangles:
Area of ABCD = x * (2x-9)
Area of DEFC = 3 * (2x-9)
So, the total area is:
$36 = x(2x-9) + 3(2x-9)$
$36 = 2x^2 - 9x + 6x - 27$
$36 = 2x^2 - 3x - 27$
Now, we have the equation $2x^2 - 3x - 27 = 36$. Rearranging, we get:
$2x^2 - 3x - 63 = 0$
Comparing this with the given equation $2x^2 - 5x - 18 = 0$, we can see that they are equivalent. Therefore, the factors of $2x^2 - 5x - 18$ are (2x+9)(x-6).
To find the value of length AB, we can set x = 6 in the expression for the length of AB:
AB = x = 6 cm
So, the correct value of length AB is 6 cm.
