Let the speed of train = x Km/h
Total distance = 480 Km
Therefore, time taken = 480/x hours
If the speed had been 8 Km/h less, than time taken = 480/x-8 hours
According to the questions,
480/x-8 – 480/x = 3
⇒ 480x – 480(x – 8)/(x – 8)x = 3
⇒ 480x – 480x + 3640 = 3(x – 8)x
⇒ 3640 = 3x² – 24x
⇒ 3x² – 24x – 3640 = 0
Hence, the speed of train satisfies the quadratic equation 3x² – 24x – 3640 = 0.

Lets, the number of article = x
Therefore, the cost of one article = 2x + 3
According to the question, the total cost x(2x + 3) = 90
⇒ 2x² + 3x = 90
⇒ 2x² + 3x – 90 = 0
⇒ 2x² + 15x – 12x – 90 = 0
⇒ x(2x + 15) – 6 (2x + 15) = 0
⇒ (2x + 15) (x – 6) = 0
⇒ (2x + 15) = 0 or (x – 6) = 0
Either x = – 15/2 or x = 6
But, x ≠ = – 15/2, x as is number of articles.
Therefore, x = 6 and the cost of each article 2x + 3 = 2 × 6 + 3 = 15
Hence, the number of articles = 6 and the cost of each article is Rs 15.

100 x² – 20x + 1 = 0
Solving the quadratic equation, we get,
100 x² – 20x + 1 = 0
⇒ 100x² – 10x – 10x + 1 = 0
⇒ 10x(10x – 1) – 10(10x – 1) = 0
⇒ (10x -1) (10x – 1) = 0
⇒ (10x – 1) = 0 or (10x – 1) = 0
Either x = 1/10 or x = 1/10
Hence, the roots of given quadratic equation are 1/10 and 1/10.