SOLUTION: Find x, 3333²ˣ + 3333⁴ˣ = 11112222

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Question 1207342: Find x,
3333²ˣ + 3333⁴ˣ = 11112222
Answer by ikleyn(52795) About Me  (Show Source):
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Find x, 3333²ˣ + 3333⁴ˣ = 11112222.
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Let y = 3333%5E%282x%29 be new variable. Then the given equation takes the form

    y^2 + y = 11112222.


Simplify and find y

    y^2 + y - 11112222 = 0.


Use the quadratic formula

    y = %28-1+%2B-+sqrt%281%5E2-4%2A1%2A%28-11112222%29%29%29%2F2 = %28-1+%2B-+6667%29%2F2.


Since y = 3333%5E%282x%29  is positive, we accept the positive root 

    x = %28-1%2B6667%29%2F2 = 3333

and discard the negative root.


So,  3333%5E%282x%29 = 3333;  hence,  2x = 1  and  x= 1/2 = 0.5.


ANSWER.  x = 1/2 = 0.5.

Solved.