Lösung 3.3:3a

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When we complete the square of the second degree expression,
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When we complete the square of the second degree expression, <math>z^2+2z+3</math>, we start with the forumla
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<math>z^{2}+2z+3</math>
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we start with the squaring rule
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<math>\left( z+a \right)^{2}=z^{2}+2az+a^{2}</math>
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{{Displayed math||<math>(z+a)^2 = z^2+2az+a^2</math>}}
which we write as
which we write as
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{{Displayed math||<math>(z+a)^2-a^2 = z^2+2az</math>}}
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<math>\left( z+a \right)^{2}-a^{2}=z^{2}+2az</math>
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and adapt the constant to be <math>a=1</math> so that the terms <math>z^2+2z</math> are equal to <math>z^2+2az</math>, and therefore can be written as <math>(z+1)^2-1^2</math>. The whole calculation becomes
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and adapt the constant to be
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<math>a=\text{1 }</math>
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so that the terms
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<math>z^{2}+2z</math>
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are equal to
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<math>z^{2}+2az</math>,, and therefore can be written as
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<math>\left( z+1 \right)^{2}-1^{2}</math>. The whole calculation becomes
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<math>\underline{z^{2}+2z}+3=\underline{\left( z+1 \right)^{2}-1^{2}}+3=\left( z+1 \right)^{2}+2</math>
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{{Displayed math||<math>\underline{z^2+2z\vphantom{()}}+3 = \underline{(z+1)^2-1^2}+3 = (z+1)^2 + 2\,\textrm{.}</math>}}
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The underline terms show the actual completion of the square.
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The underlined terms show the actual completion of the square.

Version vom 13:27, 30. Okt. 2008

When we complete the square of the second degree expression, \displaystyle z^2+2z+3, we start with the forumla

\displaystyle (z+a)^2 = z^2+2az+a^2

which we write as

\displaystyle (z+a)^2-a^2 = z^2+2az

and adapt the constant to be \displaystyle a=1 so that the terms \displaystyle z^2+2z are equal to \displaystyle z^2+2az, and therefore can be written as \displaystyle (z+1)^2-1^2. The whole calculation becomes

\displaystyle \underline{z^2+2z\vphantom{()}}+3 = \underline{(z+1)^2-1^2}+3 = (z+1)^2 + 2\,\textrm{.}

The underlined terms show the actual completion of the square.

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.3:3a“