Question 986739
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Since  {{{x-2}}},  {{{2sqrt(3)}}} and  {{{x-3)}}}  are the first  3  terms of a geometric sequence,  you have a proportion


{{{2sqrt(3)/(x-2)}}} = {{{(x-3)/(2sqrt(3))}}}


(saying that the ratio of the second term to the first one is equal to the ratio of the third term to the second one). 


From the proportion,


(x-2)*(x-3) = {{{(2sqrt(3))^2}}} = 4*3 = 12.


Hence,


{{{x^2 - 2x - 3x + 6}}} = {{{12}}},


{{{x^2 - 5x - 6}}} = {{{0}}},


(x+1)*(x-6) = 0.


The roots are  -1  and  6. 


<B>Answer</B>. &nbsp;x = -1 &nbsp;or &nbsp;x = 6.