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Aus Online Mathematik Brückenkurs 2

If we take the minus sign out in front of the whole expression,

−z2+2iz−4z−1

and collect together the first-degree terms,

−z2+(−4+2i)z−1

we can then complete the square of the expression inside the outer bracket,

−z2+(−4+2i)z−1=−z+2−4+2i2−2−4+2i2−1=−(z−2+i)2−(−2+i)2−1=−(z−2+i)2−(−2)2+4i−i2−1=−(z−2+i)2−4+4i+1−1=−(z−2+i)2−4+4i=−(z−2+i)2+4−4i.