dameon
50

Solve the equation for x and express the roots in simplest a+bi form: 3x^2=2x-1

$3x^2=2x-1\\\\3x^2-2x+1=0\\\\a=3;\ b=-2;\ c=1\\\\\Delta=b^2-4ac\\\\\Delta=(-2)^2-4\cdot3\cdot1=4-12=-6;\ \sqrt\Delta=\sqrt{-6}=\sqrt6\ i\\\\x_1=\frac{-b-\sqrt\Delta}{2a}\ and\ x_2=\frac{-+\sqrt\Delta}{2a}\\\\x_1=\frac{2-\sqrt6\ i}{2\cdot3}=\frac{2}{6}-\frac{\sqrt6\ i}{6}=\frac{1}{3}-\frac{\sqrt6}{6}\ i\\\\x_2=\frac{2+\sqrt6\ i}{2\cdot3}=\frac{2}{6}+\frac{\sqrt6\ i}{6}=\frac{1}{3}+\frac{\sqrt6}{6}\ i$