t \in [0,1]


\begin{flalign*}
& ax^3 + bx^2 + cx + d = 0 \\
& a \neq 0 \\
\end{flalign*}


$ (x + y)^3 = (x^2 + 2xy + y^2)(x + y) = x^3 + 3x^2y + 3xy^2 + y^3 $

$ y $$ b = 3y $$ c = 3y^2 $$ s $$ t $$ x $
$ x = (t + s) $


\begin{align*}
a(t + s)^3 + b(t + s)^2 + c(t + s) + d &= 0 \\
a(t^3 + 3st^2 + 3s^2t + s^3) + b(t^2 + 2st + s^2) + c(t + s) + d &= 0 \\
at^3 + (3as + b)t^2 + (3as^2 + 2bs + c)t + as^3 + bs^2 + cs + d &= 0 \\
\end{align*}

$ s $$ t $$ s $$ x $$ t^2 $$ s $$ 3as + b = 0 $
\begin{flalign*}
& 3as + b = 0 \\
& s = -\frac{b}{3a} \\
\end{flalign*}


\begin{align*}
at^3 + (3as + b)t^2 + (3as^2 + 2bs + c)t + as^3 + bs^2 + cs + d &= 0 \\
at^3 + (-b + b)t^2 + (\frac{b^2}{3a} - \frac{2b^2}{3a} + c)t - \frac{b^3}{27a^2} + \frac{b^3}{9a^2} - \frac{bc}{3a} + d &= 0 \\
at^3 + \frac{3ac - b^2}{3a}t + \frac{2b^3 - 9abc + 27a^2d}{27a^2} &= 0 \\
t^3 + \frac{3ac - b^2}{3a^2}t + \frac{2b^3 - 9abc + 27a^2d}{27a^3} &= 0 \\
\end{align*}

$ p $$ q $
\begin{flalign*}
& t^3 + pt + q = 0 \\
& p = \frac{3ac - b^2}{3a^2} \\
& q = \frac{2b^3 - 9abc + 27a^2d}{27a^3} \\
\end{flalign*}

$ pt $$ u $$ v $$ t = u + v $
\begin{align*}
(v + u)^3 + p(v + u) + q &= 0 \\
v^3 + 3uv^2 + 3u^2v + u^3 + pv + pu + q &= 0 \\
v^3 + (3uv + p)(u + v) + u^3 + q &= 0 \\
\end{align*}

$ u $$ v $$ t $$ 3uv + p = 0 $Amoxicillin 875 mg tablet ingredients
$ v^3 + u^3 + q = 0 $

$ u $$ v $
\begin{flalign*}
& v = -\frac{p}{3u} \\
& u^3 + v^3 + q = 0 \\
\end{flalign*}


\begin{align*}
u^3 + v^3 + q &= 0 \\
u^3 + (-\frac{p}{3u})^3 + q &= 0 \\
u^3 -\frac{p^3}{27u^3} + q &= 0 \\
u^6 + qu^3 - \frac{p^3}{27} &= 0 \\
\end{align*}

$ z $$ z = u^3 $
\begin{flalign*}
& z^2 + qz - \frac{p^3}{27} = 0 \\
& z^2 + \frac{2qz}{2} + \frac{q^2}{4} = \frac{p^3}{27} + \frac{q^2}{4} \\
& (z + \frac{q}{2})^2  = \frac{p^3}{27} + \frac{q^2}{4} \\
& z + \frac{q}{2} = \pm\sqrt{\frac{p^3}{27} + \frac{q^2}{4}} \\
& u^3 = - \frac{q}{2} \pm \sqrt{\frac{p^3}{27} + \frac{q^2}{4}} \\
& u = \sqrt[3]{ - \frac{q}{2} \pm \sqrt{\frac{p^3}{27} + \frac{q^2}{4}}} \\
\end{flalign*}

$ u $$ v $$ v $
\begin{flalign*}
& u^3 + v^3 + q = 0 \\
&v^3 + q = \frac{q}{2} - \sqrt{\frac{p^3}{27} + \frac{q^2}{4}} \\
&v^3 = -\frac{q}{2} - \sqrt{\frac{p^3}{27} + \frac{q^2}{4}} \\
& v = \sqrt[3]{ - \frac{q}{2} - \sqrt{\frac{p^3}{27} + \frac{q^2}{4}}} \\
\end{flalign*}

\begin{flalign*}
& 3uv + p = 0 \\
& v = -\frac{p}{3u} \\
\end{flalign*}

$ u $$ v $
\begin{align*}
p &= \frac{3ac - b^2}{3a^2} \\
q &= \frac{2b^3 - 9abc + 27a^2d}{27a^3} \\
u &= \sqrt[3]{ -\frac{q}{2} + \sqrt{\frac{p^3}{27} + \frac{q^2}{4}}} \\
t &= u - \frac{p}{3u} \\
x &= t -\frac{b}{3a}
\end{align*}

$ \frac{p^3}{27} + \frac{q^2}{4} > 0 $$ 3uv + p = 0 $$ \frac{p^3}{27} + \frac{q^2}{4} < 0 $
\begin{align*}
u^3 &= -\frac{q}{2} + i\sqrt{\lvert\frac{p^3}{27} + \frac{q^2}{4}\rvert} \\
v^3 &= -\frac{q}{2} - i\sqrt{\lvert\frac{p^3}{27} + \frac{q^2}{4}\rvert} \\
u &= \sqrt[3]{ - \frac{q}{2} + i\sqrt{\frac{p^3}{27} + \frac{q^2}{4}}} \\
v &= \sqrt[3]{ - \frac{q}{2} - i\sqrt{\frac{p^3}{27} + \frac{q^2}{4}}}
\end{align*}
$ i = \sqrt{-1} $$ i^2 = -1 $

$ x $$ y $
\begin{align*}
u &= u_x + u_yi \\
v &= v_x + v_yi \\
\end{align*}

$ r $$ \theta $$ u_x = 1, u_yi = 1 $
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$ u_x = 1, u_yi = 1 $

$ u $
\begin{flalign*}
& u_x = r\cos(\theta) \\
& u_y = r\sin(\theta) \\
& \frac{u_y}{u_y} = \tan(\theta) \\
\end{flalign*}

\sqrt[3]{ - \frac{q}{2} + i\sqrt{\frac{p^3}{27} + \frac{q^2}{4}}}$ r $$ \theta $$ n $
$ \sqrt[3]{u_x + u_yi} = \sqrt[3]{r}\cos(\frac{\theta}{3}) + \sqrt[3]{r}\sin(\frac{\theta}{3})i $



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$ v $

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$ u_x = v_x $$ u_yi = -v_yi $$ t $$ u_yi $$ v_yi $$ t $$ 2u_x

$ \theta $$ n $$ n $$ u_1 $$ u_2 $$ u_0 $$ \pm120^\circ $


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$ u $$ v $$ y $
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$ u $$ v $
 \begin{flalign*}
& uv = -\frac{p}{3} \\
& (u_x + u_yi)(v_x + v_yi) = -\frac{p}{3} \\
& u_xv_x + u_xv_yi + v_xu_yi - u_yv_y = -\frac{p}{3} \\
\end{flalign*}

$ p $$ u_xv_yi + v_xu_yi = 0 $$ r $$ u $$ v $
 \begin{flalign*}
& u_xv_yi + v_xu_yi = 0 \\
& u_xv_y + v_xu_y = 0 \\
& r\cos(U) \cdot r\sin(V) + r\cos(V) \cdot r\sin(U) = 0 \\
& \cos(U) \cdot \sin(V) + \cos(V) \cdot \sin(U) = 0 \\
& \cos(U) \cdot \sin(V) = -\cos(V) \cdot \sin(U) \\
& \frac{\sin(V)}{\cos(V)} = -\frac{\sin(U)}{\cos(U)} \\
& \tan(V) = -\tan(U) \\
\end{flalign*}

$ u $$ v $$ \frac{p^3}{27} + \frac{q^2}{4} < 0 $$ v_yi = -u_yi $$ u_x = v_x $$ t = 2u_x $

$ \frac{p^3}{27} + \frac{q^2}{4} > 0 $$ u $$ v $$ u_yi = v_yi = 0 $$ u_x \neq v_x $$ u_r \neq v_r $$ r $$ u $$ v $

$ u $$ v $$ u_yi = v_yi = 0 $$ u_x = v_x $$ u_r = v_r $$ u $$ v $