SOLUTION: Working together, a man and a woman can do the job for 8 hours. If either had to do it alone, it would take a woman 2 hours longer than a man. How long would it take either person

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Working together, a man and a woman can do the job for 8 hours. If either had to do it alone, it would take a woman 2 hours longer than a man. How long would it take either person       Log On


   

Question 903013: Working together, a man and a woman can do the job for 8 hours. If either had to do it alone, it would take a woman 2 hours longer than a man. How long would it take either person working alone?
Found 2 solutions by ewatrrr, richwmiller:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!
1%2Fx+%2B+1%2F%28x%2B2%29+=+1%2F8 |Multiplying thru by 8x(x+2) so as all denominators = 1
8(x+2) + 8x = x(x+2)
x^2 -14x - 14 = 0 (Toss out the negative solution for unit measure)
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-14x%2B-14+=+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=%28-14%29%5E2-4%2A1%2A-14=252.

Discriminant d=252 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--14%2B-sqrt%28+252+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-14%29%2Bsqrt%28+252+%29%29%2F2%5C1+=+14.9372539331938
x%5B2%5D+=+%28-%28-14%29-sqrt%28+252+%29%29%2F2%5C1+=+-0.937253933193772

Quadratic expression 1x%5E2%2B-14x%2B-14 can be factored:
1x%5E2%2B-14x%2B-14+=+1%28x-14.9372539331938%29%2A%28x--0.937253933193772%29
Again, the answer is: 14.9372539331938, -0.937253933193772. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-14%2Ax%2B-14+%29

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!

1%2Fx+%2B+1%2F%28x%2B2%29+=+1%2F8
Multiply by 8x(x+2) to remove denominators
8(x+2) + 8x = x(x+2)
x^2 -14x - 16 = 0
(Reject the negative solution for time)
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-14x%2B-16+=+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=%28-14%29%5E2-4%2A1%2A-16=260.

Discriminant d=260 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--14%2B-sqrt%28+260+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-14%29%2Bsqrt%28+260+%29%29%2F2%5C1+=+15.0622577482985
x%5B2%5D+=+%28-%28-14%29-sqrt%28+260+%29%29%2F2%5C1+=+-1.06225774829855

Quadratic expression 1x%5E2%2B-14x%2B-16 can be factored:
1x%5E2%2B-14x%2B-16+=+1%28x-15.0622577482985%29%2A%28x--1.06225774829855%29
Again, the answer is: 15.0622577482985, -1.06225774829855. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-14%2Ax%2B-16+%29

Solved by pluggable solver: COMPLETING THE SQUARE solver for quadratics
Read this lesson on completing the square by prince_abubu, if you do not know how to complete the square.
Let's convert 1x%5E2%2B-14x%2B-16=0 to standard form by dividing both sides by 1:
We have: 1x%5E2%2B-14x%2B-16=0. What we want to do now is to change this equation to a complete square %28x%2Bsomenumber%29%5E2+%2B+othernumber. How can we find out values of somenumber and othernumber that would make it work?
Look at %28x%2Bsomenumber%29%5E2: %28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2. Since the coefficient in our equation 1x%5E2%2Bhighlight_red%28+-14%29+%2A+x%2B-16=0 that goes in front of x is -14, we know that -14=2*somenumber, or somenumber+=+-14%2F2. So, we know that our equation can be rewritten as %28x%2B-14%2F2%29%5E2+%2B+othernumber, and we do not yet know the other number.
We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that %28x%2B-14%2F2%29%5E2+%2B+othernumber is equivalent to our original equation 1x%5E2%2B-14x%2Bhighlight_green%28+-16+%29=0.


The highlighted red part must be equal to -16 (highlighted green part).

-14%5E2%2F4+%2B+othernumber+=+-16, or othernumber+=+-16--14%5E2%2F4+=+-65.
So, the equation converts to %28x%2B-14%2F2%29%5E2+%2B+-65+=+0, or %28x%2B-14%2F2%29%5E2+=+65.

Our equation converted to a square %28x%2B-14%2F2%29%5E2, equated to a number (65).

Since the right part 65 is greater than zero, there are two solutions:

system%28+%28x%2B-14%2F2%29+=+%2Bsqrt%28+65+%29%2C+%28x%2B-14%2F2%29+=+-sqrt%28+65+%29+%29
, or

system%28+%28x%2B-14%2F2%29+=+8.06225774829855%2C+%28x%2B-14%2F2%29+=+-8.06225774829855+%29
system%28+x%2B-14%2F2+=+8.06225774829855%2C+x%2B-14%2F2+=+-8.06225774829855+%29
system%28+x+=+8.06225774829855--14%2F2%2C+x+=+-8.06225774829855--14%2F2+%29

system%28+x+=+15.0622577482985%2C+x+=+-1.06225774829855+%29
Answer: x=15.0622577482985, -1.06225774829855.