Answer
The equation has no real solutions.
(Complex solutions: $\frac{1\pm\sqrt{11}i}{6}$)
Work Step by Step
$3x^{2}-x+1=0$
We solve using the quadratic formula ($a=3,\ b=-1,\ c=1$):
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\displaystyle \frac{-(-1)\pm\sqrt{(-1)^{2}-4*3*1}}{2*3}
=\displaystyle \frac{1\pm\sqrt{-11}}{6}=\frac{1\pm\sqrt{11}i}{6}$
However, we note that the discriminant is negative. Therefore, the equation has no real solutions.