To get to work sam jogs 3 kilometers to the train, then rides the remaining 5 kilometers. if the train goes 40 km per hour faster than sam’s constant rate of jogging and the entire trip takes 30 minutes, how fast does sam jog?

Let
r = Sam's jogging rate
r+40 = train's rate

The rate refers to a speed of any moving body.
Know first the relationship of distance, time, and rate. The time is the distance divided by the rate.
By analyzing the problem, the time for Sam and the time for train have the total time of 30 minutes or 1/2 hours. Expressing it into an equation yields
t(Sam) + t(train) = 1/2 hours
The distance covered by Sam is 3 km, and the distance covered by train is 5 km. So
3/s + 5/(s+40) = 1/2
Solving for s,
3(s+40) + 5s = (1/2)*(s(s+40))
3s + 120 + 5s = (1/2)s² + 20s
8s + 120 = (1/2)s² + 20s
(1/2)s² + 12s - 120 = 0

The solution is turned out into a quadratic equation. If you know the quadratic formula, you get for x, for instance the two values of x are
s = 7.59 and s = -31.59

The value s = -31.59 is meaningless. Therefore Sam's jogging rate is 7.59 km/hr.