SOLUTION: Solve for x 3^x+1 - 4 + 1/3^x =0

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Question 1036472: Solve for x 3^x+1 - 4 + 1/3^x =0
Found 2 solutions by Alan3354, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
Solve for x 3^x+1 - 4 + 1/3^x =0
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I'll assume it's 3^(x+1)
---
Is it (1/3)^x ?
or 1/(3^x) ?
==============
Parentheses are free.

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
Solve for x 3^x+1 - 4 + 1/3^x =0
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I will read it as

 +  = .

Rewrite it in this way

 +  = .    (1)

Introduce new variable u = . Then the equation (1) takes the form

 = .

Multiply both sides by "u" to rid of denominator. You will get

 = .

Factor the left side:

(3u-1)*(u-1) = 0.    (2)

Then the equation (2) splits in two independent equations:


1)  3u - 1 = 0  --->  u = .  Then   = .  Hence, x = -1 is the solution of the original equation (1).

2)  u - 1 = 0  --->  u = 1.  Then  = 1.  Hence, x = 0 is the solution of the original equation (1).

Answer.  The solutions of the original equation are x = 0  and  x = -1.

The lesson to learn from this solution is the method of introducing a new variable.


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