Question

Solve this quadratic equation using the quadratic formula. 2x2 - 2x = 1

Answer 1

`\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√((-2)^2-4(2)(-1))}{2(2)}\implies x=\cfrac{2\pm√(4+8)}{4}`

`\bf x=\cfrac{2\pm√(12)}{4}~~\begin{cases}12=2\cdot 2\cdot 3\\\qquad 2^2\cdot 3\end{cases}\implies x=\cfrac{2\pm√(2^2\cdot 3)}{4}\\\\\\x=\cfrac{2\pm 2√(3)}{4}\implies x=\cfrac{2(1\pm√(3))}{4}\\\\\\x=\cfrac{1\pm√(3)}{2}\implies x=\begin{cases}\cfrac{1+√(3)}{2}\\\\\cfrac{1-√(3)}{2}\end{cases}`