SOLUTION: ln x + ln (x-2) = 1

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Question 427145: ln x + ln (x-2) = 1
Answer by nerdybill(7384) About Me  (Show Source):
You can put this solution on YOUR website!
Applying log rules:
ln x + ln (x-2) = 1
ln x(x-2) = 1
x(x-2) = e^1
x^2-2x = e^1
x^2-2x - e^1 = 0
x^2-2x - 2.718 = 0
applying the quadratic formula we get:
x = {2.928, -0.928}
you can throw out the negative solution leaving:
x = 2.928
.
details of quadratic follows:
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-2x%2B-2.718+=+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=%28-2%29%5E2-4%2A1%2A-2.718=14.872.

Discriminant d=14.872 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+14.872+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+14.872+%29%29%2F2%5C1+=+2.92821160664487
x%5B2%5D+=+%28-%28-2%29-sqrt%28+14.872+%29%29%2F2%5C1+=+-0.928211606644872

Quadratic expression 1x%5E2%2B-2x%2B-2.718 can be factored:
1x%5E2%2B-2x%2B-2.718+=+1%28x-2.92821160664487%29%2A%28x--0.928211606644872%29
Again, the answer is: 2.92821160664487, -0.928211606644872. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-2%2Ax%2B-2.718+%29