Question 133322
how would you solve for x in this equation:
{{{3x + x*sqrt(3) = 3 + 3sqrt(3)}}} 
I'm studying for the GMAT and one practice book had this question but didn't give anything but the solution (x={{{sqrt(3)}}}) and it has been bugging me something terrible. 
<pre><font size = 4 color = "indigo"><b> 
Factor x out on the left side:
 
{{{x(3 + sqrt(3)) = 3 + 3*sqrt(3)}}}
 
Divide both sides by {{{(3 + sqrt(3))}}}
 
{{{
(x(3 + sqrt(3)))
/((3+sqrt(3)))
 = 
(3 + 3*sqrt(3))
/((3+sqrt(3)))
}}}
 
Cancel on the left:
 
{{{
(x(cross(3 + sqrt(3))))
/((cross(3+sqrt(3))))
 = 
(3 + 3*sqrt(3))
/((3+sqrt(3)))
}}}
 
{{{
x = 
(3 + 3*sqrt(3))
/((3+sqrt(3)))
}}}
 
Put parentheses around the top because we
are going to be multiplying:
 
{{{
x = 
((3 + 3*sqrt(3)))
/((3+sqrt(3)))
}}}
 
Rationalize the denominator by multiplying
by the conjugate of the denominator over
itself.  That is, multiply by {{{((3-sqrt(3)))/((3-sqrt(3))))}}}, which
can't affect the value since it equals 1.
 
{{{x = 
((3 + 3*sqrt(3)))
/((3+sqrt(3)))
}}}{{{((3-sqrt(3)))/((3-sqrt(3))))}}}
 
{{{
x = 
(  (3 + 3*sqrt(3))(3-sqrt(3))         )
/( (3+sqrt(3))(3-sqrt(3))  )
}}}
 
Now use the "FOIL" principle on the
top and bottom:
 
{{{x = (9-3*sqrt(3) +9*sqrt(3) - 3*(sqrt(3))^2)
 
/(9-3*sqrt(3)+3*sqrt(3) - (sqrt(3))^2)}}}
 
Combine the two middle terms in the top:
 
{{{x = (9+6*sqrt(3) - 3*(sqrt(3))^2)
 
/(9-3*sqrt(3)+3*sqrt(3) - (sqrt(3))^2)}}}
 
The middle two terms on the bottom cancel:
 
{{{x = (9+6*sqrt(3) - 3*(sqrt(3))^2)
 
/(9-cross(3*sqrt(3)+3*sqrt(3)) - (sqrt(3))^2)}}}
 
{{{x = (9+6*sqrt(3) - 3*(sqrt(3))^2)
 
/(9 - (sqrt(3))^2)}}}
 
Replace {{{(sqrt(3))^2}}} by {{{3}}}
 
{{{x = (9+6*sqrt(3) - 3*3)
 
/(9 - 3)}}}
 
{{{x = (9+6*sqrt(3) - 9)
 
/6}}}
 
{{{x = (6*sqrt(3))
 
/6}}}
 
Cancel the {{{6}}}'s
 
{{{x = (cross(6)*sqrt(3))
 
/cross(6)}}}

{{{x = sqrt(3)}}}

Edwin</pre>