Answer
$x=\left\{ \dfrac{1-i\sqrt{11}}{6},\dfrac{1+i\sqrt{11}}{6} \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
3x^2-x=-1
,$ convert it to the form $ax^2+bx+c=0$. Then use the Quadratic Formula.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the equation above is equivalent to
\begin{array}{l}\require{cancel}
3x^2-x+1=0
.\end{array}
In the equation above, $a=
3
,$ $b=
-1
,$ and $c=
1
.$ Using the Quadratic Formula which is given by $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a},$ then
\begin{array}{l}\require{cancel}
x=\dfrac{-(-1)\pm\sqrt{(-1)^2-4(3)(1)}}{2(3)}
\\\\
x=\dfrac{1\pm\sqrt{1-12}}{6}
\\\\
x=\dfrac{1\pm\sqrt{-11}}{6}
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy}$ and that $i=\sqrt{-1},$ the expression above is equivalent to\begin{array}{l}\require{cancel}
x=\dfrac{1\pm\sqrt{-1}\cdot\sqrt{11}}{6}
\\\\
x=\dfrac{1\pm i\sqrt{11}}{6}
.\end{array}
The solutions are $
x=\left\{ \dfrac{1-i\sqrt{11}}{6},\dfrac{1+i\sqrt{11}}{6} \right\}
.$
