Answer
The solution for the equation ${{x}^{2}}+4x+6=0$ is $x=-2\pm \sqrt{2}i$
Work Step by Step
On comparing ${{x}^{2}}+4x+6=0$ with the standard form of the equation,
$\begin{align}
& a=1, \\
& b=4, \\
& c=6 \\
\end{align}$
Put these values in the formula:
$\begin{align}
& p=\frac{-4\pm \sqrt{{{4}^{2}}-4\cdot 1\cdot 6}}{2\cdot 1} \\
& =\frac{-4\pm \sqrt{16-24}}{2} \\
& =\frac{-4\pm \sqrt{-8}}{2} \\
& =\frac{-4\pm \sqrt{8}\cdot \sqrt{-1}}{2}
\end{align}$
And use the definition of the complex number i to obtain,
$\begin{align}
& p=\frac{-4\pm 2\sqrt{2}i}{2} \\
& =\frac{-4}{2}\pm \frac{2\sqrt{2}i}{2} \\
& =-2\pm \sqrt{2}i
\end{align}$
Thus, the two solutions of the provided equations are $x=-2+\sqrt{2}i$ and $x=-2-\sqrt{2}i$.
