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Aus Online Mathematik Brückenkurs 2

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

z2+az=z+2a2−2a2.

In our case, we set a=3i in order to complete the square,

z2+3iz−41=z+23i2−23i2−41=z+23i2−49(−1)−41=z+23i2+2.