Question 1207666
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Your starting equation is THIS

    {{{x^2}}} + {{{sqrt(3)*x^2}}} - 3 = 0.


It is the same as

    {{{x^2}}} + {{{sqrt(3)*x^2}}} = 3.


Factor left side

    {{{x^2*(1+sqrt(3))}}} = 3.


Divide both sides by  {{{(1+sqrt(3))}}}  and express x^2 explicitly

    x^2 = {{{3/(1+sqrt(3))}}}.


Get rid of irrationality in the denominator

    x^2 = {{{3/(sqrt(3)+1)}}} = {{{(3/(sqrt(3)+1))*((sqrt(3)-1)/(sqrt(3)-1))}}} = {{{(3*(sqrt(3)-1))/(3-1)}}} = {{{(3/2)*(sqrt(3)-1)}}}.


Now take square root of both sides and find x

    x = {{{sqrt((3/2)*(sqrt(3)-1))}}} = 1.047891...


<U>ANSWER</U>.  x = {{{sqrt((3/2)*(sqrt(3)-1))}}} = 1.047891  (rounded).
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Solved.