In order to find the value of x, we first need to calculate the area of the original plot.
Area = length x width
Area = 200 meters x 300 meters
Area = 60,000 square meters
Since the area of the new plot is double the original plot, the area is now 120,000 square meters.
Now, we can set up an equation to find the value of x:
(200 + x)(300 + x) = 120,000
Expanding the equation:
60,000 + 200x + 300x + x^2 = 120,000
x^2 + 500x - 60,000 = 0
Now we can use the quadratic formula to solve for x:
x = (-500 ± √(500^2 - 4(-60,000)) / 2
x = (-500 ± √(250,000 + 240,000)) / 2
x = (-500 ± √490,000) / 2
x = (-500 ± 700) / 2
x = (200 or -300)
Since the length and width can't be negative, the value of x is 200.
Thus, x = 200 meters.