Answer
$x^2+81=0$
Work Step by Step
We are given the solutions of a quadratic equation:
$$\begin{cases}
x_1=-9i\\
x_2=9i.
\end{cases}$$
Because $x_1$ and $x_2$ are the solutions of the equation, it means that
$$x-x_1=0\text{ and }x-x_2=0,$$
so the quadratic equations can be written:
$$a(x-x_1)(x-x_2)=0,\text{ where } a\text{ is any real number}.$$
In our case we have:
$$\begin{align*}
a\left(x-\left(-9i\right)\right)\left(x-9i\right)&=0\\
a\left(x+9i\right)\left(x-9i\right)&=0.
\end{align*}$$
For example take $a=1$:
$$\begin{align*}
\left(x+9i\right)\left(x-9i\right)&=0\\
x^2-(9i)^2&=0\\
x^2+81&=0.
\end{align*}$$
The equation is:
$$x^2+81=0.$$
Note: Any equation in the form $k(x^2+81)=0$, where $k$ is a real number, fits.
