三次四次方程的公式通解

$$x_1=-\frac{b}{3 a}+ \sqrt[3]{\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a}+\sqrt{\left(\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a}\right)^2+ \left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^3}}+\sqrt[3]{\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a}-\sqrt{\left(\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a}\right)^2+ \left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^3}}$$

$$x_2=-\frac{b}{3 a}+\frac{-1+ \sqrt{3}\,\rm{{i}}}{2} \sqrt[3]{\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a} +\sqrt{\left(\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a}\right)^2+ \left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^3}}+\frac{-1- \sqrt{3}\,\rm{{i}}}{2} \sqrt[3]{\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a} -\sqrt{\left(\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a}\right)^2+ \left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^3}}$$

$$x_3=-\frac{b}{3 a}+\frac{-1- \sqrt{3}\,\rm{{i}}}{2} \sqrt[3]{\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a} +\sqrt{\left(\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a}\right)^2+ \left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^3}}+\frac{-1+ \sqrt{3}\,\rm{{i}}}{2} \sqrt[3]{\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a} -\sqrt{\left(\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a}\right)^2+ \left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^3}}$$

$$\Delta \ = \ \sqrt{\left(\frac{bc}{6a^2}-\frac{b^3}{27a^3}-\frac{d}{2a}\right)^2+ \left(\frac{c}{3a}-\frac{b^2}{9a^2}\right)^3}$$

$${x_1=-\frac{b}{4a}+\frac12\sqrt{\left(\frac{b}{2a}\right)^2-\frac{2c}{3a}+\frac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}}-\frac12\sqrt{\frac{b^2}{2a^2}-\frac{4c}{3a}-\frac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}-\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}+\frac{-b^3+4abc-8a^2d}{4a^3\sqrt{\left(\dfrac{b}{2a}\right)^2-\dfrac{2c}{3a}+\dfrac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\dfrac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}}}}}$$

$${x_2=-\frac{b}{4a}+\frac12\sqrt{\left(\frac{b}{2a}\right)^2-\frac{2c}{3a}+\frac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}}+\frac12\sqrt{\frac{b^2}{2a^2}-\frac{4c}{3a}-\frac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}-\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}-\frac{b^3-4abc+8a^2d}{4a^3\sqrt{\left(\dfrac{b}{2a}\right)^2-\dfrac{2c}{3a}+\dfrac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\dfrac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}}}}}$$

$${x_3=-\frac{b}{4a}-\frac12\sqrt{\left(\frac{b}{2a}\right)^2-\frac{2c}{3a}+\frac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}}-\frac12\sqrt{\frac{b^2}{2a^2}-\frac{4c}{3a}-\frac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}-\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}+\frac{b^3-4abc+8a^2d}{4a^3\sqrt{\left(\dfrac{b}{2a}\right)^2-\dfrac{2c}{3a}+\dfrac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\dfrac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}}}}}$$

$${x_4=-\frac{b}{4a}-\frac12\sqrt{\left(\frac{b}{2a}\right)^2-\frac{2c}{3a}+\frac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}}+\frac12\sqrt{\frac{b^2}{2a^2}-\frac{4c}{3a}-\frac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}-\frac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}+\frac{b^3-4abc+8a^2d}{4a^3\sqrt{\left(\dfrac{b}{2a}\right)^2-\dfrac{2c}{3a}+\dfrac{\sqrt[3]2(c^2-3bd+12ae)}{3a\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}+\dfrac{\sqrt[3]{2c^3-9bcd+27ad^2+27b^2e-72ace+\sqrt{(2c^3-9bcd+27ad^2+27b^2e-72ace)^2-4(c^2-3bd+12ae)^3}}}{3\sqrt[3]2a}}}}}$$

$$\Delta\ =\ 256a^3e^3-192a^2bde^2-128a^2c^2e^2+144a^2cd^2e-27a^2d^4+144ab^2ce^2-6ab^2d^2e-80abc^2de+18abcd^3+16ac^4e-4ac^3d^2-27b^4e^2+18b^3cde-4b^3d^3-4b^2c^3e+b^2c^2d^2$$