I Tried to Solve a Quartic

("tried" doing all the work here)

Solving a quartic is quite a bit harder than solving an equation for 4 times! (pun in Chinese doesn't work here) I saw other people playing with algebraic equations and they seemed to like it, so I thought I should try solving one too! I already read about solving cubics in Artin's book, so the only remaining mountain is the quartic. By knowing about resolvent cubics, I managed to derive a formula in an afternoon. It's a bit different from those I could find on the Internet, but I did manage to solve a quartic using that, so I guess it works sometimes?? Didn't investigate further though.

We'll setup some notation for the equation. The equation we'll solve shall be $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}f(x)=x^4+s_2x^2-s_3x+s_4=0,$$ and we'll name the solutions $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}x_i$. This makes $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}s_n$ coincide with $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}\sum x_{i_1}\cdots x_{i_n}$, which brings some convenience. I set $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}s_1$ to be zero because depressed quartics are easier to deal with in general, and giving a quartic depression is a relatively simple process.

The next step is to solve the resolvent. Let $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}b_1=x_1x_2+x_3x_4,\\ b_2=x_1x_3+x_2x_4,\\ b_3=x_1x_4+x_2x_3,$$ $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}b$ are solutions to a cubic whose coefficients we can express in terms of $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}s$: $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}g(x)=(x-b_1)(x-b_2)(x-b_3)=x^3-s_2x^2-4s_4x-s_3^2+4s_2s_4.$$

Then let $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}c_1=x_1x_2-x_3x_4,\\ c_2=x_1x_3-x_2x_4,\\ c_3=x_1x_4-x_2x_3,$$ we have $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}b_i^2-c_i^2=4s_4,$$ so we can solve for $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}c_i$.

Considering $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}(x_2+x_3+x_4)^2=x_2^2+x_3^2+x_4^2+2x_2x_3+2x_2x_4+2x_3x_4,$$ and $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}x_2x_3=\frac{b_3-c_3}{2},\\ x_2x_4=\frac{b_2-c_2}{2},\\ x_3x_4=\frac{b_1-c_1}{2},\\ x_1^2+x_2^2+x_3^2+x_4^2=s_1^2-2s_2,$$ we get $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}x_1^2=(s_1-x_1)^2=-x_1^2+s_1^2-2s_2+s_2-c_1-c_2-c_3,\\ x_1=\sqrt{\frac{-s_2-\sqrt{b_1^2-4s_4}-\sqrt{b_2^2-4s_4}-\sqrt{b_3^2-4s_4}}{2}}.$$

The radicals here don't give any information regarding which branch it is, so applying this procedure would involve a lot of verification of previous equations, even after you arrive at a result. I tried to solve $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}x^4-15 x^2-10 x+24$, and it worked. Yay?

I also tried expanding out everything, plugging in solutions to the resolvent from Mathematica, and the solution of $\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}x^4+bx^2-cx+d=0$ is: (drumroll) $$\providecommand\Gal{}\renewcommand\Gal[0]{\operatorname{Gal}}\providecommand\tr{}\renewcommand\tr[0]{\operatorname{tr}}\providecommand\GL{}\renewcommand\GL[0]{\operatorname{GL}}\providecommand\SL{}\renewcommand\SL[0]{\operatorname{SL}}\providecommand\PSL{}\renewcommand\PSL[0]{\operatorname{PSL}}\providecommand\SO{}\renewcommand\SO[0]{\operatorname{SO}}\providecommand\SU{}\renewcommand\SU[0]{\operatorname{SU}}\providecommand\im{}\renewcommand\im[0]{\operatorname{im}}\providecommand\cof{}\renewcommand\cof[0]{\operatorname{cof}}\providecommand\End{}\renewcommand\End[0]{\operatorname{End}}\providecommand\Tor{}\renewcommand\Tor[0]{\operatorname{Tor}}\providecommand\rk{}\renewcommand\rk[0]{\operatorname{rk}}\providecommand\Hom{}\renewcommand\Hom[0]{\operatorname{Hom}}\providecommand\diag{}\renewcommand\diag[0]{\operatorname{diag}}\providecommand\vspan{}\renewcommand\vspan[0]{\operatorname{span}}\providecommand\lcm{}\renewcommand\lcm[0]{\operatorname{lcm}}\providecommand\id{}\renewcommand\id[0]{\operatorname{id}}\providecommand\Ab{}\renewcommand\Ab[0]{\textsf{Ab}}\providecommand\Fld{}\renewcommand\Fld[0]{\textsf{Fld}}\providecommand\Mod{}\renewcommand\Mod[1]{#1\textsf{-Mod}}\providecommand\Grp{}\renewcommand\Grp[0]{\textsf{Grp}}\providecommand\dSet{}\renewcommand\dSet[1]{#1\textsf{-Set}}\providecommand\Set{}\renewcommand\Set[0]{\textsf{Set}}\providecommand\SetStar{}\renewcommand\SetStar[0]{\textsf{Set*}}\providecommand\Vect{}\renewcommand\Vect[1]{#1\textsf{-Vect}}\providecommand\Alg{}\renewcommand\Alg[1]{#1\textsf{-Alg}}\providecommand\Ring{}\renewcommand\Ring[0]{\textsf{Ring}}\providecommand\R{}\renewcommand\R[0]{\mathbb{R}}\providecommand\C{}\renewcommand\C[0]{\mathbb{C}}\providecommand\N{}\renewcommand\N[0]{\mathbb{N}}\providecommand\Z{}\renewcommand\Z[0]{\mathbb{Z}}\providecommand\Q{}\renewcommand\Q[0]{\mathbb{Q}}\providecommand\F{}\renewcommand\F[0]{\mathbb{F}}\providecommand\sfC{}\renewcommand\sfC[0]{\mathsf{C}}\providecommand\vphi{}\renewcommand\vphi[0]{\varphi}x=\sqrt{\frac{-b-\sqrt{\left(\frac{\sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}{3 \sqrt[3]{2}}-\frac{\sqrt[3]{2} \left(-b^2-12 d\right)}{3 \sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}+\frac{b}{3}\right)^2-4 d}-\sqrt{\left(\frac{\sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}{3 \sqrt[3]{2}}-\frac{\sqrt[3]{2} \left(-b^2-12 d\right)}{3 \sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}+\frac{b}{3}\right)^2-4 d}-\sqrt{\left(\frac{\sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}{3 \sqrt[3]{2}}-\frac{\sqrt[3]{2} \left(-b^2-12 d\right)}{3 \sqrt[3]{2 b^3+\sqrt{\left(2 b^3-72 b d+27 c^2\right)^2+4 \left(-b^2-12 d\right)^3}-72 b d+27 c^2}}+\frac{b}{3}\right)^2-4 d}}{2}}.$$ That is a lot of overflow.