SOLUTION: Working alone, Steve can paint a house in 2 hours less than Gary. Working together, they can paint a house in 10 hours. How long would it take Gary to paint the house by himself?

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Question 33180: Working alone, Steve can paint a house in 2 hours less than Gary. Working together, they can paint a house in 10 hours. How long would it take Gary to paint the house by himself?
Answer by stanbon(48552) About Me  (Show Source):
You can put this solution on YOUR website!
Steve Data:
time=x-2 hrs/job rate = 1/(x-2) job/hr
Gary Data:
time= x hrs/job rate= 1/x job/hr
Working together Data:
time= 10 hrs/job rate=1/10 job/hr.
EQUATION:
rate + rate = 1/10
1/(x-2) + 1/x = 1/10
10x +10(x-2)= x(x-2)
20x-20 = x^2-2x
x^2-22x+20=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-22x%2B20+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-22%29%5E2-4%2A1%2A20=404.

Discriminant d=404 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--22%2B-sqrt%28+404+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-22%29%2Bsqrt%28+404+%29%29%2F2%5C1+=+21.0498756211209
x%5B2%5D+=+%28-%28-22%29-sqrt%28+404+%29%29%2F2%5C1+=+0.95012437887911

Quadratic expression 1x%5E2%2B-22x%2B20 can be factored:
1x%5E2%2B-22x%2B20+=+1%28x-21.0498756211209%29%2A%28x-0.95012437887911%29
Again, the answer is: 21.0498756211209, 0.95012437887911. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-22%2Ax%2B20+%29

From this you see x=21.05 hrs (amount of time Gary requires to do the job alone. It can't be x=0.95 for then Steve's time(x-2) would be negative.
Cheers,
Stan H.