Represent the following situations in the form of quadratic equations:
(i) The area of a rectangular plot is 528m². The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan's present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

i)Let the breadth be x m and the length will be 2x+1 m.
Area = l × b
Area = x(2x+1) = 528
2x² + x - 528 = 0
2x² + 33x - 32x - 528 = 0
⇒2x(x - 16) + 33(x - 16) = 0
⇒(2x + 33)(x - 16) = 0
⇒x = 16, -33/2
Breadth = 16m and length = 33m
ii)Let one number be x then the next number will be x+1
x(x+1) = 306
⇒x² + x - 306 = 0
⇒x² + 18x - 17x - 306 = 0
⇒x(x + 18) - 17(x + 18) = 0
⇒(x + 18)(x - 17) = 0
⇒x = 17, -18
The numbers are 17 and 18.
iii)Let Rohan's present age = x yrs.
Then his mother's present age = x + 26 yrs
After 3 yrs
Rohan's age = x + 3 yrs
His mother's age = x + 29 yrs
(x + 3)(x + 29) = 360
⇒x² + 32x + 87 - 360 = 0
⇒x² + 32x - 273 = 0
⇒x² + 39x - 7x - 273 = 0
⇒x(x + 39) - 7(x + 39) = 0
⇒(x + 39)(x - 7) = 0
⇒x = 7, -39
So, Rohan's present age = 7 yrs.
iv)Let the speed of the train = x km/hr
480/(x - 8) - 480/x = 3
⇒480x - 480x + 3840 = 3x(x - 8)
⇒3x² - 24x - 3840 = 0
⇒x² - 8x - 1280 = 0
⇒x² - 40x + 32x - 1280 = 0
⇒x(x - 40) + 32(x - 40) = 0
⇒(x + 32)(x - 40) = 0
⇒x = 40, -32
The speed of the train is 40 km/hr.