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zrM~rPa+sC0Nn)Nym-jKfi6*?O&3tm#xo{??GY>=Ir3(*3K?*6PSArJt&YMTh$UO$^ z$bS5i=Flp9P-V|wMk7ppAx^}Zj{L{^{D&PtLrn9o{_7_3IsjaQ{RTg@WwA_;8HOT) QXXWJLfTN_8P?CiEKczJv7ytkO diff --git a/documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex b/documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex index 73e2358..954c9e7 100644 --- a/documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex +++ b/documents/math-minimal-distance-to-cubic-function/quadratic-functions.tex @@ -231,9 +231,16 @@ Now simplify the summands: &= \frac{a(1-3+4\sqrt{3}i + 6)}{4t\sqrt[3]{6}}\\ &= \frac{a(4+4\sqrt{3}i)}{4t \sqrt[3]{6}}\\ &= \frac{a(1+\sqrt{3}i)}{t \sqrt[3]{6}}\\ - \raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}} &= \frac{3(1+i\sqrt{3})a (1-2i\sqrt{3} - 3)t}{\sqrt[3]{12 \cdot 18^2}}\\ + \raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {3}}} &= 3 \left(\frac{(1+i\sqrt{3})a}{\sqrt[3]{12} \cdot t} \right) \left(\frac{(1-i\sqrt{3})t}{2 \sqrt[3]{18}}\right)^2\\ + &= \frac{3(1+i\sqrt{3})a (1-2i\sqrt{3} - 3)t}{\sqrt[3]{12 \cdot 18^2}}\\ &= \frac{3at((1-2i\sqrt{3}-3)+(i\sqrt{3} + 2\cdot 3 - 3i\sqrt{3}))}{\sqrt[3]{2^2 \cdot 3 \cdot (2 \cdot 3^2)^2}}\\ - &= + &= \frac{3at(4+i(-4\sqrt{3})}{4\cdot3 \sqrt[3]{9}}\\ + &= \frac{at(1-\sqrt{3}i}{\sqrt[3]{9}}\\ + \raisebox{.5pt}{\textcircled{\raisebox{-.9pt} {4}}} &= - \left(\frac{(1-i\sqrt{3})t}{2 \sqrt[3]{18}}\right)^3\\ + &= -\frac{(1-3i\sqrt{3} - 3 \cdot 3 + i \sqrt{27}) t^3}{8 \cdot 18}\\ + &=- \frac{t^3 (-8)}{2^4 \cdot 3^2}\\ + &= \frac{8t^3}{2^4 \cdot 3^2}\\ + &= \frac{t^3}{18} \end{align}