The fact that every polynomial equation of positive degree has solutions, possibly non-real, was asserted during the 17th century, but completely proved only at the beginning of the 19th century. This is the fundamental theorem of algebra, which does not provide any tool for computing exactly the solutions, although Newton's method allows approximating the solutions to any desired accuracy.
tHE fAct thAT every polynoMIal eqUatiOn of positIve DegReE hAs soLutiONs, poSsiBly Non-rEAL, wAs AsSerteD dUrinG thE 17th cenTury, but ComplEtely Proved oNLY At tHe bEGinNIng of The 19Th cEnTury. tHIS Is thE FuNdameNtal theorem of algebRa, which Does not PrOviDE any ToOl FOr cOmPUtIng eXactly the sOlutiOns, aLthougH nEwtOn'S methoD alLowS APprOximating ThE soluTioNS to any DesireD accuracy.