Answer
$x^2+36=0$
Work Step by Step
The sum of the given roots, $
-6i
\text{ and }
6i
,$ is
\begin{align*}
&
-6i+6i
\\&=
0
.\end{align*}
Since the sum is given by the ratio $-\dfrac{b}{a},$ then
\begin{align*}
-\dfrac{b}{a}&=0
\\\\
-\dfrac{b}{1}&=0
&\text{ (given $a=1$)}
\\\\
-b&=0
\\
b&=0
.\end{align*}
The product of the given roots is
\begin{align*}\require{cancel}
&
-6i(6i)
\\&=
-36i^2
\\&=
-36(-1)
&\text{ (use $i^2=-1$)}
\\&=
36
.\end{align*}
Since the product is given by the ratio $\dfrac{c}{a},$ then
\begin{align*}
\dfrac{c}{a}&=36
\\\\
\dfrac{c}{1}&=36
&\text{ (given $a=1$)}
\\\\
c&=36
.\end{align*}
Hence, the quadratic equation $ax^2+bx+c=0$ with the given roots is
\begin{align*}
1x^2+0x+36&=0
\\
x^2+36&=0
.\end{align*}
