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diff --git a/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-9.tex b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-9.tex new file mode 100644 index 00000000000..3a104cd3ba0 --- /dev/null +++ b/Master/texmf-dist/doc/latex/guide-to-latex/exercises/chap7/exer7-9.tex @@ -0,0 +1,12 @@ +\documentclass{article} +\begin{document} +The reduced cubic equation $y^3 + 3py +2q = 0$ has one real and two complex +solutions when $D = q^2 + p^3 > 0$. These are given by Cardan's formula as +\begin{equation} y_1 = u + v \end{equation} +\begin{equation} y_2 = -\frac{u+v}{2} + \frac{i}{2}\sqrt{3}(u-v) \end{equation} +\begin{equation} y_3 = -\frac{u+v}{2} - \frac{i}{2}\sqrt{3}(u-v) \end{equation} +where +\[ u = \sqrt[3]{-q + \sqrt{q^2+p^3}},\qquad v = \sqrt[3]{-q - \sqrt{q^2+p^3}} \] +\end{document} + + |