Answer
$x=\dfrac{3\pm\sqrt{5}}{4}$
Work Step by Step
The expression $
\left( x-\dfrac{1}{2} \right)^2=\dfrac{x}{2}
$ simplifies to
\begin{array}{l}
x^2-x+\dfrac{1}{4}=\dfrac{x}{2}
\\\\
4x^2-4x+1=2x
\\\\
4x^2-6x+1=0
.\end{array}
Using the Quadratic Formula $\left(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\right)$, then,
\begin{array}{l}
x=\dfrac{-(-6)\pm\sqrt{(-6)^2-4(4)(1)}}{2(4)}\\\\
x=\dfrac{6\pm\sqrt{20}}{8}\\\\
x=\dfrac{6\pm\sqrt{4\cdot5}}{8}\\\\
x=\dfrac{6\pm2\sqrt{5}}{8}\\\\
x=\dfrac{3\pm\sqrt{5}}{4}
.\end{array}
