A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/hr more than the first speed. If it takes 3 hours to complete the total journey, what is its first speed?
Let the first speed of the train be x km/h
Time taken to cover 54 km = 
New speed of train = x + 6 km/h
Time taken to cover 63 km = 
According to given question
taking LCM
117x + 324 = 3(x2 + 6x)
117x + 324 = 3x2 + 18x
3x2 – 99x – 324 = 0
x2 – 33x – 108 = 0
Using the splitting middle term – the middle term of the general equation 
is divided in two such values that:
Product = a.c
For the given equation a = 1 b = – 33 c = – 108
= 1. – 108 = – 108
And either of their sum or difference = b
= – 33
Thus the two terms are – 36 and 3
Difference = – 36 + 3 = – 33
Product = – 36.3 = – 108
x2 – 36x + 3x – 108 = 0
x (x – 36) + 3(x – 36) = 0
(x – 36) (x + 3) = 0
x = 36 or x = – 3
x = 36 (speed cannot be negative)
Hence the first speed of the train is 36 km/hr