Question 1127154
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<pre>
The equation is

{{{3/z}}} - {{{3/sqrt(z)}}} + 1 = 0.


Multiply by z both sides. You will get


{{{3 - 3*sqrt(z) + z}}} = 0,   or, equivalently,

{{{z - 3*sqrt(z) + 3}}} = 0.


Now introduce new variable  u = {{{sqrt(z)}}}.  The last equation will take the form

{{{u^2 - 3u + 3}}} = 0.


Apply the quadratic formula


{{{u[1,2]}}} = {{{(3 +- sqrt(3^2 - 4*3*1))/2}}} = {{{(3 +- sqrt(-3))/2}}} = {{{(3 +- i*sqrt(3))/2}}}.


Now consider separately both cases


a)  u = {{{sqrt(z)}}} = {{{(3 + i*sqrt(3))/2}}}.  Hence, squaring, you get
 
    z = {{{((3+i*sqrt(3))/2)^2}}} = {{{(9 + 6i*sqrt(3) -3)/4}}} = {{{(6+6i*sqrt(3))/4}}} = {{{(3 + 3i*sqrt(3))/2}}},    


and


b)  u = {{{sqrt(z)}}} = {{{(3 - i*sqrt(3))/2}}}.  Hence, squaring, you get
 
    z = {{{((3-i*sqrt(3))/2)^2}}} = {{{(9 - 6i*sqrt(3) -3)/4}}} = {{{(6-6i*sqrt(3))/4}}} = {{{(3 - 3i*sqrt(3))/2}}}.


<U>Answer</U>.  There are two solutions.  They are complex numbers

         {{{(3 + 3i*sqrt(3))/2}}}  and  {{{(3 - 3i*sqrt(3))/2}}}.
</pre>

Solved.


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Surely, it can be done in many ways and on different paths.


I showed here one of the simplest paths/ways.


Other people may prefer other ways.