谁展开$\Delta$谁傻逼

令

$$ \Delta _1=c^2-3bd+12ae $$

$$ \Delta _2=2c^3-9bcd+27ad^2+27b^2e-72ace $$

$$ \Delta = \frac {\sqrt [3] {2} \Delta_1} {3a \sqrt [3] {\Delta_2+ \sqrt {-4 \Delta^3_1+\Delta^2_2}}}+\frac {\sqrt [3] {\Delta_2+\sqrt {-4\Delta^3_1+\Delta^2_2} } } {3 \sqrt [3] {2} a} $$

即有

$$ \Delta = \frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}\\+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a} $$

若 $ ax^4+bx^3+cx^2+dx+e=0, a,b,c,d,e \in \mathbb R, a \ne 0 $ ，则有：

$$ x_1=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+\Delta}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{4c}{3a}-\Delta- \frac{-\frac{b^3}{a^3}+\frac{4bc}{a^2}-\frac{8d}{a}} {4\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+\Delta}}} $$

即

$$ x_1=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})}\\-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{4c}{3a}-(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})- \frac{-\frac{b^3}{a^3}+\frac{4bc}{a^2}-\frac{8d}{a}} {4\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})}}} $$

$$ x_2=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+\Delta}+\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{4c}{3a}-\Delta- \frac{-\frac{b^3}{a^3}+\frac{4bc}{a^2}-\frac{8d}{a}} {4\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+\Delta}}} $$

即

$$ x_2=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})}\\+\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{4c}{3a}-(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})- \frac{-\frac{b^3}{a^3}+\frac{4bc}{a^2}-\frac{8d}{a}} {4\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})}}} $$

$$ x_3=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+\Delta}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{4c}{3a}-\Delta+ \frac{-\frac{b^3}{a^3}+\frac{4bc}{a^2}-\frac{8d}{a}} {4\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+\Delta}}} $$

即

$$ x_3=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})}\\-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{4c}{3a}-(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})+ \frac{-\frac{b^3}{a^3}+\frac{4bc}{a^2}-\frac{8d}{a}} {4\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})}}} $$

$$ x_4=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+\Delta}+\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{4c}{3a}-\Delta+ \frac{-\frac{b^3}{a^3}+\frac{4bc}{a^2}-\frac{8d}{a}} {4\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+\Delta}}} $$

即

$$ x_4=\frac{-b}{4a}-\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})}\\+\frac{1}{2}\sqrt{\frac{b^2}{4a^2}-\frac{4c}{3a}-(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})+ \frac{-\frac{b^3}{a^3}+\frac{4bc}{a^2}-\frac{8d}{a}} {4\sqrt{\frac{b^2}{4a^2}-\frac{2c}{3a}+(\frac {\sqrt [3] {2} (c^2-3bd+12ae)} {3a \sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+ \sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2}}}+\frac {\sqrt [3] {(2c^3-9bcd+27ad^2+27b^2e-72ace)+\sqrt {-4(c^2-3bd+12ae)^3+(2c^3-9bcd+27ad^2+27b^2e-72ace)^2} } } {3 \sqrt [3] {2} a})}}} $$