Answer
$x=\left\{
-\dfrac{4}{7}-\dfrac{\sqrt{26}}{7}i, -\dfrac{4}{7}+\dfrac{\sqrt{26}}{7}i
\right\}
$
Work Step by Step
Using $ax^2+bx+c=0,$ the given equation,
\begin{align*}
7x^2+8x&=-6
\\
7x^2+8x+6&=0
,\end{align*} has $a=
7
,$ $b=
8
,$ and $c=
6
.$ Using the Quadratic Formula which is given by $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a},$ then
\begin{align*}\require{cancel}
x&=\dfrac{-8\pm\sqrt{8^2-4(7)(6)}}{2(7)}
\\\\&=
\dfrac{-8\pm\sqrt{64-168}}{14}
\\\\&=
\dfrac{-8\pm\sqrt{-104}}{14}
\\\\&=
\dfrac{-8\pm\sqrt{-1}\cdot\sqrt{4}\cdot\sqrt{26}}{14}
\\\\&=
\dfrac{-8\pm\sqrt{-1}\cdot2\cdot\sqrt{26}}{14}
\\\\&=
\dfrac{-8\pm i\cdot2\cdot\sqrt{26}}{14}
&\text{ (use $i=\sqrt{-1}$)}
\\\\&=
\dfrac{-8\pm 2i\sqrt{26}}{14}
\\\\&=
\dfrac{-\cancel8^4\pm \cancel2^1i\sqrt{26}}{\cancel{14}^7}
&\text{ (divide by $2$)}
\\\\&=
\dfrac{-4\pm i\sqrt{26}}{7}
\\\\&=
-\dfrac{4}{7}\pm \dfrac{\sqrt{26}}{7}i
.\end{align*}
The solutions are $
x=\left\{
-\dfrac{4}{7}-\dfrac{\sqrt{26}}{7}i, -\dfrac{4}{7}+\dfrac{\sqrt{26}}{7}i
\right\}
.$
