Question 1207342
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Find x, 3333²ˣ + 3333⁴ˣ = 11112222.
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<pre>
Let y = {{{3333^(2x)}}} 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 = {{{(-1 +- sqrt(1^2-4*1*(-11112222)))/2}}} = {{{(-1 +- 6667)/2}}}.


Since y = {{{3333^(2x)}}}  is positive, we accept the positive root 

    x = {{{(-1+6667)/2}}} = 3333

and discard the negative root.


So,  {{{3333^(2x)}}} = 3333;  hence,  2x = 1  and  x= 1/2 = 0.5.


<U>ANSWER</U>.  x = 1/2 = 0.5.
</pre>

Solved.