#### Answer

$x=-3\pm\sqrt{7}$

#### Work Step by Step

$\dfrac{1}{6}x^{2}+x+\dfrac{1}{3}=0$
Multiply the whole equation by $6$ to avoid working with fractions:
$6\Big(\dfrac{1}{6}x^{2}+x+\dfrac{1}{3}=0\Big)$
$x^{2}+6x+2=0$
Use the quadratic formula to solve this equation. The formula is $x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For this equation, $a=1$, $b=6$ and $c=2$
Substitute:
$x=\dfrac{-6\pm\sqrt{6^{2}-4(1)(2)}}{2(1)}=\dfrac{-6\pm\sqrt{36-8}}{2}=\dfrac{-6\pm\sqrt{28}}{2}=...$
$...=\dfrac{-6\pm2\sqrt{7}}{2}=-3\pm\sqrt{7}$