diff --git a/documents/math-minimal-distance-to-cubic-function/math-minimal-distance-to-cubic-function.pdf b/documents/math-minimal-distance-to-cubic-function/math-minimal-distance-to-cubic-function.pdf
index d83ecc9..e1a0d00 100644
Binary files a/documents/math-minimal-distance-to-cubic-function/math-minimal-distance-to-cubic-function.pdf and b/documents/math-minimal-distance-to-cubic-function/math-minimal-distance-to-cubic-function.pdf differ
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}