From Förberedande kurs i matematik 2

When we complete the square, we replace all \displaystyle z -terms in the second-degree expression with a quadratic term which contains \displaystyle z, according to the formula

\displaystyle z^{2}+az=\left( z+\frac{a}{2} \right)^{2}-\left( \frac{a}{2} \right)^{2}

In our case, we set \displaystyle a=\text{3}i\text{ } in order to complete the square:

\displaystyle \begin{align} & z^{2}+3iz-\frac{1}{4}=\left( z+\frac{3}{2}i \right)^{2}-\left( \frac{3}{2}i \right)^{2}-\frac{1}{4} \\ & =\left( z+\frac{3}{2}i \right)^{2}-\frac{9}{4}\left( -1 \right)-\frac{1}{4} \\ & =\left( z+\frac{3}{2}i \right)^{2}+2 \\ \end{align}