对于一元四次方程： a x 4 + b x 3 + c x 2 + d x + e = 0 ( a ≠ 0 ) \Large a x^{4}+b x^{3}+c x^{2}+d x+e=0(a\ne0) ax4+bx3+cx2+dx+e=0(a​=0) 记： { Δ 1 = c 2 − 3 b d + 12 a e Δ 2 = 2 c 3 − 9 b c d + 27 a d 2 + 27 b 2 e − 72 a c e \Large \begin{cases} \Delta_{1}=c^{2}-3 b d+12 a e \\ \Delta_{2}=2 c^{3}-9 b c d+27 a d^{2}+27 b^{2} e-72 a c e \end{cases} ⎩⎪⎨⎪⎧​Δ1​=c2−3bd+12aeΔ2​=2c3−9bcd+27ad2+27b2e−72ace​ 并记: Δ = 2 3 Δ 1 3 a Δ 2 + − 4 Δ 1 3 + Δ 2 2 3 + Δ 2 + − 4 Δ 1 3 + Δ 2 2 3 3 2 3 a \Large\Delta=\frac{\sqrt[3]{2} \Delta_{1}}{3 a \sqrt[3]{\Delta_{2}+\sqrt{-4 \Delta_{1}^{3}+\Delta_{2}^{2}}}}+\frac{\sqrt[3]{\Delta_{2}+\sqrt{-4 \Delta_{1}^{3}+\Delta_{2}^{2}}}}{3 \sqrt[3]{2} a} Δ=3a3Δ2​+−4Δ13​+Δ22​ ​ ​32 ​Δ1​​+332 ​a3Δ2​+−4Δ13​&#