SOLUTION: please help me solve this using substitution y=3^X, solve the equation 3^x=2+3^-x.

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Question 1075860: please help me solve this
using substitution y=3^X, solve the equation 3^x=2+3^-x.
Found 2 solutions by ikleyn, Alan3354:
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
O, great !!! This is entirely different and fully correct request. 

After substituting  y = 3%5Ex, your equation becomes

y = 2+%2B+1%2Fy

Multiply by "y" both sides. You will get

y%5E2 = 2y+%2B+1,  or

y%5E2+-+2y+-1 = 0.


Apply the quadratic formula to solve the quadratic equation. You will get

y%5B1%2C2%5D = %282+%2B-+sqrt%284+%2B4%29%29%2F2 = %282%2B-+sqrt%288%29%29%2F2 = 1+%2B-+sqrt%282%29.


The negative root does not make sense, since y = 3%5Ex must be positive. 
It is EXTRANEOUS root for our problem.

The only positive root y = 1+%2B+sqrt%282%29 survives.


So, we have 3%5Ex = 1%2Bsqrt%282%29, which implies

x = log%283%2C+%281%2Bsqrt%282%29%29%29.

Solved.



Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
solve the equation 3^x=2+3^-x.
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3^x = 2 + 3^-x
3^x - 2 - 3^-x = 0
Mutliply thru by 3^x
3^(2x) - 2*3^x - 1 = 0
Sub y for 3^x
y^2 - 2y - 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-2x%2B-1+=+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-1=8.

Discriminant d=8 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--2%2B-sqrt%28+8+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-2%29%2Bsqrt%28+8+%29%29%2F2%5C1+=+2.41421356237309
x%5B2%5D+=+%28-%28-2%29-sqrt%28+8+%29%29%2F2%5C1+=+-0.414213562373095

Quadratic expression 1x%5E2%2B-2x%2B-1 can be factored:
1x%5E2%2B-2x%2B-1+=+%28x-2.41421356237309%29%2A%28x--0.414213562373095%29
Again, the answer is: 2.41421356237309, -0.414213562373095. 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-1+%29

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Ignore the negative solution
3^x = 1 + sqrt(2)
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x*log(3) = log(1 + sqrt(2))
x = log(1 + sqrt(2))/log(3)
x =~ 0.80226