ANSWER
There are two solutions and they are both complex solutions. The solutions are:
[tex]a=2i-8;a=-2i-8[/tex]
EXPLANATION
We want to determine the number and nature of solutions to the equation:
[tex](3a+24)^2=-36[/tex]
To do this, solve the equation by first, finding the square root of both sides of the equation:
[tex]\begin{gathered} \sqrt[]{(3a+24)^2}=\pm\sqrt[]{-36}=\pm\sqrt[]{-1\cdot36} \\ \Rightarrow3a+24=\pm\mleft\lbrace\sqrt[]{36}\cdot\sqrt[]{-1}\mright\rbrace \\ 3a+24=\pm6i \end{gathered}[/tex]
Now, solve the equation for a:
[tex]\begin{gathered} 3a=\pm6i-24 \\ \Rightarrow a=\pm\frac{6i}{3}-\frac{24}{3} \\ \Rightarrow a=2i-8;a=-2i-8 \end{gathered}[/tex]
Hence, there are two solutions and they are complex solutions.
