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Aus Online Mathematik Brückenkurs 2

When we complete the square of the second degree expression, z2+2z+3 we start with the squaring rule

z+a2=z2+2az+a2 

which we write as

z+a2−a2=z2+2az 

and adapt the constant to be a=1 so that the terms z2+2z are equal to z2+2az,, and therefore can be written as z+12−12 . The whole calculation becomes

z2+2z+3=z+12−12+3=z+12+2 

The underline terms show the actual completion of the square.