Answer
$t=\left\{ -6-3i,-6+3i \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To find the non real complex solutions of the given equation, $
(t+6)^2=-9
,$ take the square root of both sides (Square Root Property). Then use the properties of radicals and use $i=\sqrt{-1}.$ Finally, simplify the radical and isolate the variable.
$\bf{\text{Solution Details:}}$
Taking the square root of both sides (Square Root Property), the equation above is equivalent to
\begin{array}{l}\require{cancel}
t+6=\pm\sqrt{-9}
.\end{array}
Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the equation above is equivalent to\begin{array}{l}\require{cancel}
t+6=\pm\sqrt{-1}\cdot\sqrt{9}
.\end{array}
Using $i=\sqrt{-1},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
t+6=\pm i\sqrt{9}
.\end{array}
Writing the radicand as an expression that contains a factor that is a perfect power of the given index and then extracting the root of that factor, the equation above is equivalent to
\begin{array}{l}\require{cancel}
t+6=\pm i\sqrt{(3)^2}
\\\\
t+6=\pm i(3)
\\\\
t+6=\pm 3i
.\end{array}
Using the properties of equality to isolate the variable, the equation above is equivalent to
\begin{array}{l}\require{cancel}
t=-6\pm 3i
.\end{array}
Hence, $
t=\left\{ -6-3i,-6+3i \right\}
.$
