Let speed of train be x km/h

Time taken by train to cover 480 km = 480x hours

If, speed had been 8km/h less then time taken would be (480x−8) hours

According to given condition, if speed had been 8km/h less then time taken is 3 hours less.

Therefore, 480x – 8 = 480x + 3

⇒ 480 (1x – 8 − 1x) = 3

⇒ 480 (x – x + 8) (x) (x − 8) = 3

⇒ 480 × 8 = 3 (x) (x − 8)

⇒ 3840 = 3x²-24x

⇒ 3x²-24x-3840= 0

Dividing equation by 3, we get

⇒x²-8x-1280= 0

This is a required Quadratic Equation.

Problem: Find the speed of a train that travels a distance of 480 km at a uniform speed. If the speed had been 8 km/hr less, then it would have taken 3 hours more to cover the same distance.

Solution:
Let the original speed of the train be x km/hr.

Step 1: Formulate the equation using the given information.

According to the problem, we have the following information:

Distance travelled by the train = 480 km
Original speed of the train = x km/hr
New speed of the train (8 km/hr less than the original speed) = (x-8) km/hr
Additional time taken to cover the distance at the new speed = 3 hours

Using the formula,

Time = Distance / Speed

we can write two equations as follows:

Time taken to cover the distance at the original speed = 480/x hours
Time taken to cover the same distance at the new speed = 480/(x-8) hours

We know that the additional time taken at the new speed is 3 hours. Therefore, we can write the following equation:

480/(x-8) - 480/x = 3

Simplifying the above equation, we get:

480x - 3840 - 480x + 8x² = 3x(x-8)
8x² - 24x - 3840 = 0
x² - 3x - 480 = 0

Step 2: Solve the quadratic equation to find the value of x.

Using the quadratic formula, we get:

x = (3 ± √(3² + 4*480))/2
x = (3 ± 33)/2

We can ignore the negative value of x and take the positive value, which is:

x = (3 + 33)/2
x = 18

Therefore, the speed of the train is 18 km/hr.

Step 3: Verify the solution.

We can verify the solution by checking if the time taken to cover the distance at the original speed and the new speed is correct.

Time taken to cover the distance at the original speed = 480/18 = 26.67 hours
Time taken to cover the same distance at the new speed = 480/10 = 48 hours

We can see that the time taken at the new speed is 3 hours more than the time taken at the original speed, which confirms that our solution is correct.

Conclusion: The speed of the train is 18 km/hr.