Answer
$0; 7-4i; 7+ 4i$
Work Step by Step
Given the cubic equation:
\begin{equation}
0.25 x^3-3.5 x^2+16.25 x=0.
\end{equation} First we factor out $0.25x$: $$0.25x(x^2-14x+65)=0.$$ We have: $$x=0\text{ or }x^2-14x+65=0.$$ 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 & =1, b=-14, c=65 \\
x & =\frac{-b \pm \sqrt{b^2-4 a c}}{2 a} \\
x & =\frac{14 \pm \sqrt{(-14)^2-4 \cdot 1\cdot(65)}}{2 \cdot 1}\\
& =\frac{14\pm \sqrt{-64}}{2} \\
& =\frac{14\pm 8i}{2} \\
&= 7\pm 4i.
\end{aligned}
\end{equation} The solution of the given equation is \begin{equation}
\begin{aligned}
x& =0\\
x&= 7-4i\\
x& =7+ 4i.
\end{aligned}
\end{equation}
