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got case 2.2 correct

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Martin Thoma 2013-12-25 10:30:35 +01:00
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documents/math-minimal-distance-to-cubic-function/quadratic-case-2.2.tex
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@ -40,8 +40,23 @@ Now get back to the original equation:
&\hphantom{{}=} + \alpha \left (\color{red}\frac{(1+i \sqrt{3})\alpha}{\sqrt[3]{12} \cdot t} \color{black}
\color{blue}-\frac{(1-i\sqrt{3}) t}{2\sqrt[3]{18}} \color{black} \right ) + \beta\\
&= \frac{-2 \alpha^3}{3 t^3}
+ \frac{\alpha^2(2(1+i\sqrt{3})-(2+\sqrt{3}i))}{2t\sqrt[3]{12}}
+ \frac{t^3}{18}
+ \beta\\
&= \frac{-24 \alpha^3 + (3\sqrt[3]{18}t^2)(\alpha^2\sqrt{3}i) + 2t^3+36 t^3 \beta}{36t^3}
&= \frac{-12 \alpha^3 + t^6+18 t^3 \beta}{18t^3}
\end{align}
Now continue with only the numerator
\begin{align}
0 &\stackrel{!}{=}
- 12 \alpha^3
+ (\sqrt{3(4 \alpha^3 + 27 \beta^2)}-9\beta)^2
+ 18 (\sqrt{3(4 \alpha^3 + 27 \beta^2)} - 9 \beta) \beta\\
&=
\color{red}- 12 \alpha^3 \color{black}+
\left (
3(\color{red}4 \alpha^3\color{black} + \color{blue}27 \beta^2 \color{black})
\color{orange}- 2 \cdot \sqrt{3(4 \alpha^3 + 27 \beta^2)} \cdot 9\beta\color{black}
+ \color{blue}81 \beta^2\color{black}
\right )\\
&\hphantom{{}=}+ 18 \beta (\color{orange}\sqrt{3(4 \alpha^3 + 27 \beta^2)}\color{black} \color{blue}- 9 \beta\color{black})
\end{align}