SOLUTION: 7e^x=12-e^-x Solve algebraically. The first (-) is subtraction and the last (-) is a negative x. Thanks in advance.

Algebra ->  Equations -> SOLUTION: 7e^x=12-e^-x Solve algebraically. The first (-) is subtraction and the last (-) is a negative x. Thanks in advance.       Log On


   

Question 1096243: 7e^x=12-e^-x
Solve algebraically. The first (-) is subtraction and the last (-) is a negative x. Thanks in advance.
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
7e^x=12-e^-x
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multiply by e^x
7e^(2x) = 12e^x - 1
7e^(2x) - 12e^x + 1 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 7x%5E2%2B-12x%2B1+=+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=%28-12%29%5E2-4%2A7%2A1=116.

Discriminant d=116 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--12%2B-sqrt%28+116+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-12%29%2Bsqrt%28+116+%29%29%2F2%5C7+=+1.62645211530493
x%5B2%5D+=+%28-%28-12%29-sqrt%28+116+%29%29%2F2%5C7+=+0.0878335989807852

Quadratic expression 7x%5E2%2B-12x%2B1 can be factored:
7x%5E2%2B-12x%2B1+=+%28x-1.62645211530493%29%2A%28x-0.0878335989807852%29
Again, the answer is: 1.62645211530493, 0.0878335989807852. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+7%2Ax%5E2%2B-12%2Ax%2B1+%29

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e^x =~ 1.6264
x = ln(1.6264)
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e^x =~ 0.0878
x = ln(0.0878)