Answer
$2 - \frac{1}{3}\sqrt{42}\ i$; $2 + \frac{1}{3}\sqrt{42}\ i$
Work Step by Step
Given the quadratic equation \begin{equation}
-3 w^2+12 w-26=0.
\end{equation} Factor out $-1$: $$3 w^2-12 w+26=0.$$ The quadratic equation can be best solved by the quadratic formula. We identify the constants $a$, $b$, $c$ from the general form of the equation and solve it using the quadratic formula: \begin{equation}
\begin{aligned}
a&x^2+bx+c=0\\
a & =3, b=-12, c=26 \\
x & =\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} \\
x & =\frac{12 \pm \sqrt{(-12)^2-4 \cdot 3\cdot(26)}}{2 \cdot 3}\\
& =\frac{12\pm \sqrt{-168}}{6} \\
& =\frac{12\pm 2\sqrt{42} i}{6} \\
& =\frac{6\pm \sqrt{42} i}{3}.
\end{aligned}
\end{equation} The solution is \begin{equation}
\begin{aligned}
x&= 2 - \frac{1}{3}\sqrt{42}\ i\\
x& = 2 + \frac{1}{3}\sqrt{42}\ i.
\end{aligned}
\end{equation}
