Solution 3.3:3c

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Current revision (13:45, 30 October 2008) (edit) (undo)
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If we take the minus sign out in front of the whole expression,
If we take the minus sign out in front of the whole expression,
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{{Displayed math||<math>-\bigl(z^2+2iz-4z-1\bigr)\,,</math>}}
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<math>-\left( z^{2}+2iz-4z-1 \right)</math>
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and collect together the first-degree terms,
and collect together the first-degree terms,
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{{Displayed math||<math>-\bigl(z^2+(-4+2i)z-1\bigr)\,,</math>}}
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<math>-\left( z^{2}+\left( -4+2i \right)z-1 \right)</math>
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we can then complete the square of the expression inside the outer bracket,
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we can then complete the square of the expression inside the outer bracket
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<math>\begin{align}
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{{Displayed math||<math>\begin{align}
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& -\left( z^{2}+\left( -4+2i \right)z-1 \right)=-\left( \left( z+\frac{-4+2i}{2} \right)^{2}-\left( \frac{-4+2i}{2} \right)^{2}-1 \right) \\
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-\bigl(z^2+(-4+2i)z-1\bigr)
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& =-\left( \left( z-2+i \right)^{2}-\left( -2+i \right)^{2}-1 \right) \\
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&= -\Bigl(\Bigl(z+\frac{-4+2i}{2}\Bigr)^2-\Bigl(\frac{-4+2i}{2}\Bigr)^2-1\Bigr)\\[5pt]
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& =-\left( \left( z-2+i \right)^{2}-\left( -2 \right)^{2}+4i-i^{2}-1 \right) \\
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&= -\bigl((z-2+i)^2-(-2+i)^2-1\bigr)\\[5pt]
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& =-\left( \left( z-2+i \right)^{2}-4+4i+1-1 \right) \\
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&= -\bigl((z-2+i)^2-(-2)^2+4i-i^2-1\bigr)\\[5pt]
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& =-\left( \left( z-2+i \right)^{2}-4+4i \right) \\
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&= -\bigl((z-2+i)^2-4+4i+1-1\bigr)\\[5pt]
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& =-\left( z-2+i \right)^{2}+4-4i. \\
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&= -\bigl((z-2+i)^2-4+4i\bigr)\\[5pt]
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\end{align}</math>
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&= -(z-2+i)^2+4-4i\,\textrm{.}
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\end{align}</math>}}

Current revision

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

\displaystyle -\bigl(z^2+2iz-4z-1\bigr)\,,

and collect together the first-degree terms,

\displaystyle -\bigl(z^2+(-4+2i)z-1\bigr)\,,

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

\displaystyle \begin{align}

-\bigl(z^2+(-4+2i)z-1\bigr) &= -\Bigl(\Bigl(z+\frac{-4+2i}{2}\Bigr)^2-\Bigl(\frac{-4+2i}{2}\Bigr)^2-1\Bigr)\\[5pt] &= -\bigl((z-2+i)^2-(-2+i)^2-1\bigr)\\[5pt] &= -\bigl((z-2+i)^2-(-2)^2+4i-i^2-1\bigr)\\[5pt] &= -\bigl((z-2+i)^2-4+4i+1-1\bigr)\\[5pt] &= -\bigl((z-2+i)^2-4+4i\bigr)\\[5pt] &= -(z-2+i)^2+4-4i\,\textrm{.} \end{align}

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