Quartic Roots
Solution of $x^4+ax^3+bx^2+cx+d$ is:
\begin{align} r_1&=-\frac{a}{4} -\frac{1}{2}\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}+\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}} -\frac{1}{2}\sqrt{\frac{a^2}{2}-\frac{4b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{3\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}-\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}-\frac{-a^3+4ab-8c}{4\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{3\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}+\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}}}}\\ r_2&=-\frac{a}{4} -\frac{1}{2}\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}+\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}} +\frac{1}{2}\sqrt{\frac{a^2}{2}-\frac{4b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{3\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}-\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}-\frac{-a^3+4ab-8c}{4\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{3\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}+\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}}}}\\ r_3&=-\frac{a}{4} -\frac{1}{2}\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}+\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}} -\frac{1}{2}\sqrt{\frac{a^2}{2}-\frac{4b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{3\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}-\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}+\frac{-a^3+4ab-8c}{4\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{3\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}+\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}}}}\\ r_4&=-\frac{a}{4} -\frac{1}{2}\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}+\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}} +\frac{1}{2}\sqrt{\frac{a^2}{2}-\frac{4b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{3\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}-\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}+\frac{-a^3+4ab-8c}{4\sqrt{\frac{a^2}{4}-\frac{2b}{3}+\frac{\sqrt[3]{2}\left(b^2-3ac+12d\right)}{3\sqrt[3]{\left(2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}\right)}}+\sqrt[3]{\left(\frac{2b^3-9abc+27c^2+27a^2d-72bd+\sqrt{-4\left(b^2-3ac+12d\right)^3+\left(2b^3-9abc+27c^2+27a^2d-72bd\right)^2}}{54}\right)}}}} \end{align}