It takes a painter and his friend 12 hours to paint a room. If the painter was working alone, it would take him 18 hours less than if his friend was working alone. How long does it take the painter to paint the room by himself?

18 hours

Step-by-step explanation:

The equation for time working is

1/a + 1/b = 1/c

where a and b are the times working alone and c is the time working together

1/b-18 + 1/b = 1/12

Multiply by 12b (b-18) to clear the fractions

12b (b-18) * (1/b-18 + 1/b) = 1/12 * 12b (b-18)

12b + 12 (b-18) = b (b-18)

Distribute

12b + 12b - 216 = b^2-18b

Combine like terms

24b-216=b^2-18b

Subtract 24b from each side

24b-24b - 216 = b^2 - 18b-24b

-216 = b^2 - 42b

Add 216 to each side

0 = b^2 - 42b + 216

Factor

0 = (b-36) (b-6)

b = 36

b=6 is not possible since this is less time then when they worked together

Using the zero product property

The painter takes b-18 = 36-18 = 18 hours

The friend takes 36 hours

b=36 b=6

18 hours

Step-by-step explanation:

Combined time = (T1 * T2 / (T1 + T2)

Let x be the time friend takes

12 = [x (x - 18) ]/[x + x - 18]

12 (2x - 18) = x² - 18x

24x - 216 = x² - 18x

x² - 42x + 216 = 0

x² - 6x - 36x + 216 = 0

x (x - 6) - 36 (x - 6) = 0

(x - : 6) (x - 36) = 0

x = 6, 36

His time:

x - 18

6 - 18 = - 12 (not possible)

36 - 18 = 18 hours