Answer
The solution is $x=\frac{1}{2}$.
Work Step by Step
$2x=\sqrt 2-2x$
Square both sides:
$4x^2=2-2x$
$4x^2+2x-2=0$
$(2x-1)(x+1)=0$
$2x-1=0$ or $x+1=0$
$x=\frac{1}{2}$ or $x=-1$
Check:
$2(\frac{1}{2})=\sqrt 2-2(\frac{1}{2})$
$1=1$
$2(-1)=\sqrt 2-2(-1)$
$-2=2$
The solution is $x=\frac{1}{2}$.
