Solving cubic equation of one variable x^3+x^2=1

Solving cubic equation of one variable x^3+x^2=1

Answer: between 0.7548776662 and 0.7548776663
The other two solutions are complex solutions
x₂= -0.8774388331 + 0.7448617666i
x₃= -0.8774388331 - 0.7448617666i.
There are many kinds of solutions, the trial method is not applicable, the Newton method can be applied, the following iterative method
Any number, as long as it is not 1, can be solved, now choose X. = 0.5
x³ + x² = 1
x₁= (1 - x².)^(⅓) = 0.9085602964
x₂= (1 - x²₁)^(⅓) = 0.5588306678
x₃= (1 - x²₂)^(⅓) = 0.8826762043
x₄= (1 - x²₃)^(⅓) = 0.6044873908
x5 = (1 - x²₄)^(⅓) = 0.8593410286
x6 = (1 - x²5 )^(⅓) = 0.6395023773
x7 = (1 - x²6 )^(⅓) = 0.8392116136
x8 = (1 - x²7 )^(⅓) = 0.6662370684
x9 = (1 - x²8 )^(⅓) = 0.8223530315
x10= (1 - x²9 )^(⅓) = 0.6866415893
x11= (1 - x²10)^(⅓) = 0.8085149479
x12= (1 - x²11)^(⅓) = 0.7022401560
x13= (1 - x²12)^(⅓) = 0.7973132595
x14= (1 - x²13)^(⅓) = 0.7141942878
x15= (1 - x²14)^(⅓) = 0.7883341060
x16= (1 - x²15)^(⅓) = 0.7233800293
x17= (1 - x²16)^(⅓) = 0.7811867390
x18= (1 - x²17)^(⅓) = 0.7304565099
x19= (1 - x²18)^(⅓) = 0.7755262674
x20= (1 - x²19)^(⅓) = 0.7359204588
x21= (1 - x²20)^(⅓) = 0.7710600334
x22= (1 - x²21)^(⅓) = 0.7401475387
x23= (1 - x²22)^(⅓) = 0.7675458045
x24= (1 - x²23)^(⅓) = 0.7434230494
x25= (1 - x²24)^(⅓) = 0.7647863727
x26= (1 - x²25)^(⅓) = 0.7459645804
x27= (1 - x²26)^(⅓) = 0.7626230032
x28= (1 - x²27)^(⅓) = 0.7479387282
x29= (1 - x²28)^(⅓) = 0.7609289534
x30= (1 - x²29)^(⅓) = 0.7494734840
x31= (1 - x²30)^(⅓) = 0.7596036119
x32= (1 - x²31)^(⅓) = 0.7506674668
x33= (1 - x²32)^(⅓) = 0.7585674505
x34= (1 - x²33)^(⅓) = 0.7515968483
x35= (1 - x²34)^(⅓) = 0.7577578060
x36= (1 - x²35)^(⅓) = 0.7523205798
x37= (1 - x²36)^(⅓) = 0.7571254206
x38= (1 - x²37)^(⅓) = 0.7528843583
x39= (1 - x²38)^(⅓) = 0.7566316441
x40= (1 - x²39)^(⅓) = 0.7533236524
x41= (1 - x²40)^(⅓) = 0.7562461911
x42= (1 - x²41)^(⅓) = 0.7536660207
x43= (1 - x²42)^(⅓) = 0.7559453557
x44= (1 - x²43)^(⅓) = 0.7539328925
x45= (1 - x²44)^(⅓) = 0.7557105972
x46= (1 - x²45)^(⅓) = 0.7541409426
x47= (1 - x²46)^(⅓) = 0.7555274234
x48= (1 - x²47)^(⅓) = 0.7543031522
x49= (1 - x²48)^(⅓) = 0.7553845122
x50= (1 - x²49)^(⅓) = 0.7544296315
x51= (1 - x²50)^(⅓) = 0.7552730215
x52= (1 - x²51)^(⅓) = 0.7545282570
x53= (1 - x²52)^(⅓) = 0.7551860480
x54= (1 - x²53)^(⅓) = 0.7546051664
x55= (1 - x²54)^(⅓) = 0.7551182031
x56= (1 - x²55)^(⅓) = 0.7546651435
x57= (1 - x²56)^(⅓) = 0.7550652816
x58= (1 - x²57)^(⅓) = 0.7547119176
x59= (1 - x²58)^(⅓) = 0.7550240019
x60= (1 - x²59)^(⅓) = 0.7547483959
x61= (1 - x²60)^(⅓) = 0.7549918037
x61= (1 - x²60)^(⅓) = 0.7547768452
x62= (1 - x²61)^(⅓) = 0.7549666894
x63= (1 - x²62)^(⅓) = 0.7547990330
x64= (1 - x²63)^(⅓) = 0.7549471008
x65= (1 - x²64)^(⅓) = 0.7548163376
x66= (1 - x²65)^(⅓) = 0.7549318222
x67= (1 - x²66)^(⅓) = 0.7548298339
x68= (1 - x²67)^(⅓) = 0.7549199055
x69= (1 - x²68)^(⅓) = 0.7548403600
x70= (1 - x²69)^(⅓) = 0.7549106108
x71= (1 - x²70)^(⅓) = 0.7548485696
x72= (1 - x²71)^(⅓) = 0.7549033614
x73= (1 - x²72)^(⅓) = 0.7548549726
x74= (1 - x²73)^(⅓) = 0.7548977072
x75= (1 - x²74)^(⅓) = 0.7548599665
x76= (1 - x²75)^(⅓) = 0.7548932972
x77= (1 - x²76)^(⅓) = 0.7548638615
x78= (1 - x²77)^(⅓) = 0.7548898576
x79= (1 - x²78)^(⅓) = 0.7548668993
x80= (1 - x²79)^(⅓) = 0.7548871748
x81= (1 - x²80)^(⅓) = 0.7548692686
x82= (1 - x²81)^(⅓) = 0.7548850824
x83= (1 - x²82)^(⅓) = 0.7548711166
x84= (1 - x²83)^(⅓) = 0.7548834505
x85= (1 - x²84)^(⅓) = 0.7548725579
x86= (1 - x²85)^(⅓) = 0.7548821776
x87= (1 - x²86)^(⅓) = 0.7548736820
x88= (1 - x²87)^(⅓) = 0.7548811849
x89= (1 - x²88)^(⅓) = 0.7548745588
x90= (1 - x²89)^(⅓) = 0.7548804106
x91= (1 - x²90)^(⅓) = 0.7548752426
x92= (1 - x²91)^(⅓) = 0.7548798067
x93= (1 - x²92)^(⅓) = 0.7548757759
x94= (1 - x²93)^(⅓) = 0.7548793357
x95= (1 - x²94)^(⅓) = 0.7548761919
x96= (1 - x²95)^(⅓) = 0.7548789683
x97= (1 - x²96)^(⅓) = 0.7548765163
x98= (1 - x²97)^(⅓) = 0.7548786818
x99= (1 - x²98)^(⅓) = 0.7548767694
x100=(1 - x²99)^(⅓) = 0.7548784583
x101=(1 -x²100)^(⅓) = 0.7548769667
x102=(1 -x²101)^(⅓) = 0.7548782840
x103=(1 -x²102)^(⅓) = 0.7548771207
x104=(1 -x²103)^(⅓) = 0.7548781481
x105=(1 -x²104)^(⅓) = 0.7548772407
x106=(1 -x²105)^(⅓) = 0.7548780420
x107=(1 -x²106)^(⅓) = 0.7548773344
x108=(1 -x²107)^(⅓) = 0.7548779593
x109=(1 -x²108)^(⅓) = 0.7548774074
x110=(1 -x²109)^(⅓) = 0.7548778949
x111=(1 -x²110)^(⅓) = 0.7548774644
x112=(1 -x²111)^(⅓) = 0.7548778445
x113=(1 -x²112)^(⅓) = 0.7548775088
x114=(1 -x²113)^(⅓) = 0.7548778053
x115=(1 -x²114)^(⅓) = 0.7548775434
x116=(1 -x²115)^(⅓) = 0.7548777747
x117=(1 -x²116)^(⅓) = 0.7548775705
x118=(1 -x²117)^(⅓) = 0.7548777508
x119=(1 -x²118)^(⅓) = 0.7548775915
x120=(1 -x²119)^(⅓) = 0.7548776322
x121=(1 -x²120)^(⅓) = 0.7548776608
x122=(1 -x²121)^(⅓) = 0.7548776177
x123=(1 -x²122)^(⅓) = 0.7548776208
x124=(1 -x²123)^(⅓) = 0.7548776064
x125=(1 -x²124)^(⅓) = 0.7548776308
x126=(1 -x²125)^(⅓) = 0.7548776976
x127=(1 -x²126)^(⅓) = 0.7548776386
x128=(1 -x²127)^(⅓) = 0.7548776907
x129=(1 -x²128)^(⅓) = 0.7548776447
x130=(1 -x²129)^(⅓) = 0.7548776853
x131=(1 -x²130)^(⅓) = 0.7548776494
x132=(1 -x²131)^(⅓) = 0.7548776811
x133=(1 -x²132)^(⅓) = 0.7548776531
x134=(1 -x²133)^(⅓) = 0.7548776778
x135=(1 -x²134)^(⅓) = 0.7548776560
x136=(1 -x²135)^(⅓) = 0.7548776753
x137=(1 -x²136)^(⅓) = 0.7548776583
x138=(1 -x²137)^(⅓) = 0.7548776733
x139=(1 -x²136)^(⅓) = 0.7548776600
x140=(1 -x²139)^(⅓) = 0.7548776717
x141=(1 -x²140)^(⅓) = 0.7548776614
x142=(1 -x²141)^(⅓) = 0.7548776705
x143=(1 -x²142)^(⅓) = 0.7548776625
x144=(1 -x²143)^(⅓) = 0.7548776696
x145=(1 -x²144)^(⅓) = 0.7548776633
x146=(1 -x²145)^(⅓) = 0.7548776689
x147=(1 -x²146)^(⅓) = 0.7548776639
x148=(1 -x²147)^(⅓) = 0.7548776683
x149=(1 -x²148)^(⅓) = 0.7548776645
x150=(1 -x²149)^(⅓) = 0.7548776678
x151=(1 -x²150)^(⅓) = 0.7548776648
x152=(1 -x²151)^(⅓) = 0.7548776675
x153=(1 -x²152)^(⅓) = 0.7548776652
x154=(1 -x²153)^(⅓) = 0.7548776672
x155=(1 -x²154)^(⅓) = 0.7548776654
x156=(1 -x²155)^(⅓) = 0.7548776667
x157=(1 -x²156)^(⅓) = 0.7548776656
x158=(1 -x²157)^(⅓) = 0.7548776668
x159=(1 -x²158)^(⅓) = 0.7548776657
x160=(1 -x²159)^(⅓) = 0.7548776667
x161=(1 -x²160)^(⅓) = 0.7548776658
x162=(1 -x²161)^(⅓) = 0.7548776666
x163=(1 -x²162)^(⅓) = 0.7548776659
x164=(1 -x²163)^(⅓) = 0.7548776665
x165=(1 -x²164)^(⅓) = 0.7548776666
x166=(1 -x²165)^(⅓) = 0.7548776665
x167=(1 -x²166)^(⅓) = 0.7548776661
x168=(1 -x²167)^(⅓) = 0.7548776664
x169=(1 -x²168)^(⅓) = 0.7548776661
x170=(1 -x²169)^(⅓) = 0.7548776664
x171=(1 -x²170)^(⅓) = 0.7548776661
x172=(1 -x²171)^(⅓) = 0.7548776663
x173=(1 -x²172)^(⅓) = 0.7548776662
x174=(1 -x²173)^(⅓) = 0.7548776663
x175=(1 -x²174)^(⅓) = 0.7548776662
x176=(1 -x²175)^(⅓) = 0.7548776663
x177=(1 -x²176)^(⅓) = 0.7548776662
x178=(1 -x²177)^(⅓) = 0.7548776663
x179=(1 -x²178)^(⅓) = 0.7548776662
x180=(1 -x²179)^(⅓) = 0.7548776663
x181=(1 -x²180)^(⅓) = 0.7548776662
x182=(1 -x²181)^(⅓) = 0.7548776663
x183=(1 -x²182)^(⅓) = 0.7548776662
x184=(1 -x²183)^(⅓) = 0.7548776663
x185=(1 -x²184)^(⅓) = 0.7548776662
x186=(1 -x²185)^(⅓) = 0.7548776663
.