SOLUTION: Solve the equation To clarify on the 1/3^x the denominator has the exponent not the whole fraction, I'm not sure if that matters. Thank you!

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Question 974196: Solve the equation
To clarify on the 1/3^x the denominator has the exponent not the whole fraction, I'm not sure if that matters. Thank you!
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
3%5Ex+%2B+1%2F3%5Ex+=+12
Multiply by 3^x
3%5E%282x%29+%2B+1+=+12%2A3%5Ex
3%5E%282x%29+-+12%2A3%5Ex+%2B+1+=+0
Sub u for 3^x
u%5E2+-+12u+%2B+1+=+0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%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%2A1%2A1=140.

Discriminant d=140 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--12%2B-sqrt%28+140+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-12%29%2Bsqrt%28+140+%29%29%2F2%5C1+=+11.9160797830996
x%5B2%5D+=+%28-%28-12%29-sqrt%28+140+%29%29%2F2%5C1+=+0.0839202169003839

Quadratic expression 1x%5E2%2B-12x%2B1 can be factored:
1x%5E2%2B-12x%2B1+=+%28x-11.9160797830996%29%2A%28x-0.0839202169003839%29
Again, the answer is: 11.9160797830996, 0.0839202169003839. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-12%2Ax%2B1+%29

=======================
3^x = 11.919
x*log(3) = log(11.919)
x = log(11.919)/log(3)
===========================
Do the other 3^x root the same way.