Answer
$7 - \frac{1}{2}\sqrt{2}\ i;7 + \frac{1}{2}\sqrt{2}\ i$
Work Step by Step
Given the quadratic equation
\begin{equation}
2 x^2-28 x+45=-54.
\end{equation} We bring the equation to the general form:
$$\begin{align*}
ax^2+bx+c&=0\\
2x^2-28x+45+54&=0\\
2x^2-28x+99&=0.
\end{align*}$$ 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 & =2, b=-28, c=99 \\
x & =\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} \\
x & =\frac{28 \pm \sqrt{(-28)^2-4 \cdot 2\cdot(99)}}{2 \cdot 2}\\
& =\frac{28\pm \sqrt{-8}}{4} \\
& =\frac{28\pm 2\sqrt{2} i}{4} \\
&=7\pm\frac{1}{2}\sqrt 2 \ i.
\end{aligned}
\end{equation} The solution is \begin{equation}
\begin{aligned}
x&= 7 - \frac{1}{2}\sqrt{2}\ i\\
x& = 7 + \frac{1}{2}\sqrt{2}\ i.
\end{aligned}
\end{equation}
