Definition of complex numbers

1. $3x^2+27=0 \ \Rightarrow \ 3x^2=-27 \ \Rightarrow \ x=\pm \sqrt{-\dfrac{27}{3}}=\pm 3i$
2. $8x^2+4x-2=0 \ \Rightarrow$ $ \displaystyle \begin{array}{rl} x &=\frac{-4\pm\sqrt{16-4\cdot8\cdot2}}{16}= \frac{-4\pm\sqrt{-48}}{16}=\frac{-1}{4}\pm \frac{i\sqrt{48}}{16}\ &=\frac{-1}{4}\pm \frac{i\sqrt{2^4\cdot3}}{16} = \frac{-1}{4}\pm \frac{i\cdot 2^2\sqrt{3}}{16}= \frac{-1}{4}\pm \frac{i\sqrt{3}}{4} \end{array} $

If they must have a multiple of the imaginary unit, the easiest thing is to take an imaginary number, equate it to $x$ and then square it: $$12i=x \ \Rightarrow \ (12i)^2=x^2 \ \Rightarrow \ 144i^2=x^2 \ \Rightarrow \ -144=x^2 \ \Rightarrow \ x^2+144=0$$ it has $12i$ as solution, and $12i$ is a multiple of the imaginary unit $i$. Similarly we can obtain $x^2+169=0$ has a multiple of $i$. The solution would be $13i$.

The first one is a complex number, but it is not a pure imaginary since it has a real part. The pure imaginary number is $\sqrt{-27}$. This is because it is a multiple of the imaginary unit.