Two people can paint a house in 21 hours. Working individually one of the two people can paint the house in 6 hours more than it takes the other person to paint the house. How long would it take each person to paint the house. How long would it take each person working individually to paint the house.

Problem : Two people can paint a house in 21 hours. Working individually one of the two people can paint the house in 6 hours more than it takes the other person to paint the house. How long would it take each person to paint the house. How long would it take each person working individually to paint the house. 

Doubt by Muskan

Solutions : 

Let the time taken by other person to paint the house individually = x hours
and the time taken by the first person = (x + 6) hours

Amount of work done by other person in 1 hour = 1/x
Amount of work done by other person in 21 hours = 21/x 

Amount of work done by first person in 1 hour = 1/(x+6)
Amount of work done by first person in 21 hours = 21/(x+6)

ATQ

21/x + 21/(x+6) = 1
21 [ 1/x + 1/(x+6)] = 1
[1/x + 1/(x+6)] = 1/21
[(x+6) + x] / x(x+6) = 1/21
21 (2x + 6) = x (x+6)
42x + 126 = x² + 6x
x² + 6x - 42x - 126 = 0
x² - 36x - 126 = 0

a = 1
b = -36
c = -126
D = b²-4ac 
D = (-36)² - 4(1)(-126)
D = 1296 +  504
D = 1800

Using Quadratic Formula 

x = (-b±√D)/2a
x = [-(-36) ± √1800]/2(1)
x = (36 ± 10√18)/2
x = 2(18±5√18)/2
x = 18 ± 5√18
x = 18 ± 15√2

When x = 18 - 15√2
x = 18 - 15 (1.414)
x = 18 - 21.21
x = -3.21
which is not possible. 

When x = 18 + 15√2
x = 18 + 15 (1.414)
x = 18 + 21.21
x = 39.21

Time taken by two painters would be 39.21 hours and 45.21 hours (39.21 + 6)