[tex] \bf 2x^2-2x=1\implies 2x^2-2x-1=0\\\\\\~~~~~~~~~~~~\textit{quadratic formula}\\\\\stackrel{\stackrel{a}{\downarrow }}{2}x^2\stackrel{\stackrel{b}{\downarrow }}{-2}x\stackrel{\stackrel{c}{\downarrow }}{-1}=0\qquad \qquad x= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a}\\\\\\x=\cfrac{-(-2)\pm\sqrt{(-2)^2-4(2)(-1)}}{2(2)}\implies x=\cfrac{2\pm\sqrt{4+8}}{4} [/tex]
[tex] \bf x=\cfrac{2\pm\sqrt{12}}{4}~~\begin{cases}12=2\cdot 2\cdot 3\\\qquad 2^2\cdot 3\end{cases}\implies x=\cfrac{2\pm\sqrt{2^2\cdot 3}}{4}\\\\\\x=\cfrac{2\pm 2\sqrt{3}}{4}\implies x=\cfrac{2(1\pm\sqrt{3})}{4}\\\\\\x=\cfrac{1\pm\sqrt{3}}{2}\implies x=\begin{cases}\cfrac{1+\sqrt{3}}{2}\\\\\cfrac{1-\sqrt{3}}{2}\end{cases} [/tex]
