Lösung 3.3:4c

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K (Lösning 3.3:4c moved to Solution 3.3:4c: Robot: moved page)
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{{NAVCONTENT_START}}
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We complete the square on the left-hand side:
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<center> [[Image:3_3_4c.gif]] </center>
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{{NAVCONTENT_STOP}}
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<math>\begin{align}
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& \left( z+1 \right)^{\text{2}}-1^{2}+3=0 \\
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& \left( z+1 \right)^{\text{2}}+2=0 \\
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\end{align}</math>
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Taking the root now gives
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<math>z+1=\pm i\sqrt{2}</math>
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i.e.
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<math>z=-1+i\sqrt{2}</math>
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and
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<math>z=-1-i\sqrt{2}</math>.
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We test the solutions in the equation to ascertain that we have calculated correctly.
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<math>\begin{align}
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& z=-1+i\sqrt{2}:\quad z^{2}+2z+3=\left( -1+i\sqrt{2} \right)^{2}+2\left( -1+i\sqrt{2} \right)+3 \\
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& =\left( -1 \right)^{2}-2\centerdot i\sqrt{2}+i^{2}\left( \sqrt{2} \right)^{2}-2+2i\sqrt{2}+3 \\
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& =1-2\centerdot i\sqrt{2}-2-2+2i\sqrt{2}+3=0, \\
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\end{align}</math>
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<math>\begin{align}
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& z=-1-i\sqrt{2}:\quad z^{2}+2z+3=\left( -1-i\sqrt{2} \right)^{2}+2\left( -1-i\sqrt{2} \right)+3 \\
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& =\left( -1 \right)^{2}+2\centerdot i\sqrt{2}+i^{2}\left( \sqrt{2} \right)^{2}-2-2i\sqrt{2}+3 \\
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& =1+2\centerdot i\sqrt{2}-2-2-2\sqrt{2}i+3=0, \\
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\end{align}</math>

Version vom 10:32, 25. Okt. 2008

We complete the square on the left-hand side:


\displaystyle \begin{align} & \left( z+1 \right)^{\text{2}}-1^{2}+3=0 \\ & \left( z+1 \right)^{\text{2}}+2=0 \\ \end{align}


Taking the root now gives \displaystyle z+1=\pm i\sqrt{2} i.e. \displaystyle z=-1+i\sqrt{2} and \displaystyle z=-1-i\sqrt{2}.

We test the solutions in the equation to ascertain that we have calculated correctly.


\displaystyle \begin{align} & z=-1+i\sqrt{2}:\quad z^{2}+2z+3=\left( -1+i\sqrt{2} \right)^{2}+2\left( -1+i\sqrt{2} \right)+3 \\ & =\left( -1 \right)^{2}-2\centerdot i\sqrt{2}+i^{2}\left( \sqrt{2} \right)^{2}-2+2i\sqrt{2}+3 \\ & =1-2\centerdot i\sqrt{2}-2-2+2i\sqrt{2}+3=0, \\ \end{align}


\displaystyle \begin{align} & z=-1-i\sqrt{2}:\quad z^{2}+2z+3=\left( -1-i\sqrt{2} \right)^{2}+2\left( -1-i\sqrt{2} \right)+3 \\ & =\left( -1 \right)^{2}+2\centerdot i\sqrt{2}+i^{2}\left( \sqrt{2} \right)^{2}-2-2i\sqrt{2}+3 \\ & =1+2\centerdot i\sqrt{2}-2-2-2\sqrt{2}i+3=0, \\ \end{align}

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.3:4c“