Answer
$x=-\frac{1}{3} +\frac{2}{3}\sqrt{11}\ i$
$x=-\frac{1}{3} -\frac{2}{3}\sqrt{11}\ i$
Work Step by Step
Given the quadratic equation
\begin{equation}
3 x^2+2 x+7=-8.
\end{equation} This equation can be best solved by the quadratic formula after rewriting it in the form
\begin{equation}
\begin{aligned}
ax^2+bx+c&=0\\
3 x^2+2 x+7&=-8 \\
3 x^2+2 x+15&=0.
\end{aligned}
\end{equation} Identify the constants $a$, $b$, $c$ and solve the equation: \begin{equation}
\begin{aligned}
a & =3, b=2, c=15 \\
x & =\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} \\
x & =\frac{-2 \pm \sqrt{2^2-4 \cdot 3\cdot(15)}}{2 \cdot 3}\\
& =\frac{-2\pm \sqrt{-176}}{6} \\
& =\frac{-2\pm \sqrt{-16\cdot 11}}{6} \\
&=\frac{-2\pm 4\sqrt{11}}{6}\ i \\
&=\frac{-1\pm 2\sqrt{11}}{3}\ i.
\end{aligned}
\end{equation} The solution is
\begin{equation}
\begin{aligned}
x& =-\frac{1}{3} +\frac{2}{3}\sqrt{11}\ i\\
x& =-\frac{1}{3} -\frac{2}{3}\sqrt{11}\ i.
\end{aligned}
\end{equation}
