Polynomial Roots
This is just me practicing $\LaTeX$ by typing long complicated formulas also testing my speed :P
But you can see these formulas (if interested)
Solution of $ax+b$ is: \begin{equation} r=-\frac{b}{a} \end{equation} Solution of $ax^2+bx+c$ is: \begin{equation} r_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\equiv\frac{-2c}{b\pm\sqrt{b^2-4ac}} \end{equation} Solution of $ax^3+bx^2+cx+d$ is:
\begin{aligned}
r_1&=-\frac{1}{3a}\left[b+
\sqrt[3]{\frac{2b^3-9abc+27a^2d\pm\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}}{2}}
+\frac{b^2-3ac}{\sqrt[3]{\frac{2b^3-9abc+27a^2d\pm\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}}{2}}
}\right]\\
r_2&=-\frac{1}{3a}\left[b+
\left(\frac{-1+\sqrt{-3}}{2}\right)\sqrt[3]{\frac{2b^3-9abc+27a^2d\pm\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}}{2}}
+\frac{b^2-3ac}{\left(\dfrac{-1+\sqrt{-3}}{2}\right)\sqrt[3]{\frac{2b^3-9abc+27a^2d\pm\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}}{2}}
}\right]\\
r_3&=-\frac{1}{3a}\left[b+
\left(\frac{-1-\sqrt{-3}}{2}\right)\sqrt[3]{\frac{2b^3-9abc+27a^2d\pm\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}}{2}}
+\frac{b^2-3ac}{\left(\dfrac{-1-\sqrt{-3}}{2}\right)\sqrt[3]{\frac{2b^3-9abc+27a^2d\pm\sqrt{\left(2b^3-9abc+27a^2d\right)^2-4\left(b^2-3ac\right)^3}}{2}}
}\right]\\
\end{aligned}
Solution of $x^4+ax^3+bx^2+cx+d$ is:
The last one is way too long to fit in page. Check here!
The PDF for same can be found here