Answer
$t=-\dfrac{3}{2}\pm\dfrac{\sqrt{3}}{2}i$
Work Step by Step
$t+3+\dfrac{3}{t}=0$
Multiply the whole equation by $t$:
$t\Big(t+3+\dfrac{3}{t}=0\Big)$
$t^{2}+3t+3=0$
Use the quadratic formula to solve this equation. The formula is $t=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For this equation, $a=1$, $b=3$ and $c=3$.
Substitute the known values in the formula:
$t=\dfrac{-3\pm\sqrt{3^{2}-4(1)(3)}}{2(1)}=\dfrac{-3\pm\sqrt{9-12}}{2}=...$
$...=\dfrac{-3\pm\sqrt{-3}}{2}=\dfrac{-3\pm\sqrt{3}i}{2}=-\dfrac{3}{2}\pm\dfrac{\sqrt{3}}{2}i$
The answer is $t=-\dfrac{3}{2}\pm\dfrac{\sqrt{3}}{2}i$
