Question 1150082
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Problem
<img src = "https://i.imgur.com/4nXUmA9.jpg" width = "200">


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The given equation
{{{(sqrt(3x) + sqrt(3/x) - 2sqrt(3))/(sqrt(3x) + sqrt(3/x)) = (sqrt(3x) + sqrt(3/x))/(sqrt(3x) + sqrt(3/x)+sqrt(3))}}}
has the term {{{sqrt(3x) + sqrt(3/x)}}} show up a bunch of times as shown by the red marker


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


So we can replace every copy of {{{sqrt(3x) + sqrt(3/x)}}} with some other variable temporarily. Lets call that y.
Let {{{y = sqrt(3x) + sqrt(3/x)}}}


The given equation
{{{(sqrt(3x) + sqrt(3/x) - 2sqrt(3))/(sqrt(3x) + sqrt(3/x)) = (sqrt(3x) + sqrt(3/x))/(sqrt(3x) + sqrt(3/x)+sqrt(3))}}}
simplifies greatly to 
{{{(y - 2sqrt(3))/(y) = (y)/(y+sqrt(3))}}}


Recall that the range of the square root function is nonnegative. In other words, it is impossible to get a negative value output from {{{sqrt(x)}}}. Adding two nonnegative square roots in {{{y = sqrt(3x) + sqrt(3/x)}}} like this means that {{{y > 0}}}. We'll use this fact later.


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Let's solve for y


{{{(y - 2sqrt(3))/(y) = (y)/(y+sqrt(3))}}}


{{{(y - 2sqrt(3))(y+sqrt(3)) = y*y}}} Cross multiply


{{{(y - 2sqrt(3))(y+sqrt(3)) = y^2}}}


{{{A(y+sqrt(3)) = y^2}}} Temporarily, let {{{A = (y - 2sqrt(3))}}}. This helps us distribute in the next step


{{{A*y + A*sqrt(3) = y^2}}}


{{{y*A + sqrt(3)*A = y^2}}}


{{{y(y - 2sqrt(3))+sqrt(3)(y - 2sqrt(3)) = y^2}}} Replace A with {{{(y - 2sqrt(3))}}} again.


{{{y(y) + y(-2sqrt(3))+sqrt(3)(y) + sqrt(3)*(-2sqrt(3)) = y^2}}} Distribute again


{{{y^2 - 2y*sqrt(3) + y*sqrt(3)-6 = y^2}}}


{{{y^2 - y*sqrt(3)-6 = y^2}}}


{{{y^2 - y*sqrt(3)-6-y^2 = y^2-y^2}}} Subtract y^2 from both sides


{{{-y*sqrt(3)-6 = 0}}}


{{{-y*sqrt(3)-6+y*sqrt(3) = 0+y*sqrt(3)}}} Add y*sqrt(3) to both sides


{{{-6 = y*sqrt(3)}}}


{{{y*sqrt(3) = -6}}}


{{{(y*sqrt(3))/(sqrt(3)) = -6/(sqrt(3))}}} Divide both sides by sqrt(3)


{{{y = -6/(sqrt(3))}}} Divide both sides by sqrt(3)


{{{y = -3.46410161513776}}} we've run into a problem


The result we got {{{y = -3.46410161513776}}} contradicts {{{y > 0}}}


There are no solutions for y, which means that <font color=red size=4>there are no solutions for x</font>.
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