What is claimed is:1. A method of manufacturing a solid electrolyte for a lithium-ion battery, the method comprising:providing a starting material in which lanthanum nitrate [La(NO3)3.6H2O] and zirconium nitrate [ZrO(NO3)2.6H2O] are mixed at a molar ratio of 3:2;forming an aqueous solution by dissolving the starting material;forming a precipitate by adding a complex agent including ammonium hydroxide (NH4OH) to the aqueous solution;adding sodium hydroxide (NaOH) to the aqueous solution to adjust pH in a reactor, followed by mixing and co-precipitation of the aqueous solution;forming a primary precursor powder by washing, drying and pulverizing the precipitate; andforming a secondary precursor powder by mixing and ball-milling the primary precursor powder with lithium powder [LiOH.H2O], wherein the secondary precursor powder undergoes a heat treatment to form a heat-treated solid electrolyte powder having a composition of LixLayZrzO12, where x is 6 moles to 9 moles, y is 2 moles to 4 moles, and z is 1 mole to 3 moles.2. The method of claim 1, further comprising:forming the heat-treated solid electrolyte powder into a pellet sheet; andheat-treating the pellet sheet to change the heat-treated solid electrolyte powder to a crystal structure in which a cubic structure predominates.3. The method of claim 1,wherein the heat-treated solid electrolyte powder is changed to a crystalline structure in which one of a cubic structure and a tetragonal structure predominates depending on a temperature of the heat treatment.4. The method of claim 1,wherein a temperature of the heat treatment is 600° C. to 1200° C., and the heat-treated solid electrolyte powder has a structure in which a cubic structure coexists with a tetragonal structure or one of the cubic structure and the tetragonal structure predominates.5. The method of claim 1,wherein a temperature of the heat treatment is 700° C. to 800° C., and the heat-treated solid electrolyte powder is changed into a material in which a complete cubic structure or tetragonal structure predominates by calcination at 1200° C. for 2 hours to 8 hours.6. The method of claim 5, wherein the heat-treated solid electrolyte powder is changed into a material having a predominating cubic structure and a fine structure with a high density of 90% or higher by calcination at 1200° C. for 5 hours.7. The method of claim 1,wherein a temperature of the heat treatment is approximately 900° C., and the heat-treated solid electrolyte powder is manufactured by calcination at approximately 900° C. for 10 hours or longer.8. The method of claim 1,wherein a temperature of the heat treatment is approximately 900° C., and the heat-treated solid electrolyte powder is manufactured by calcination at approximately 900° C. for 2 hours to 10 hours.9. The method of claim 8, wherein the heat-treated solid electrolyte powder has a predominating tetragonal structure and a high-density fine structure with a relative density of 60% or higher by calcination at approximately 900° C. for 5 hours.10. The method of claim 1, wherein the complex agent is a 5 N ammonium hydroxide solution, and the sodium hydroxide is added to the aqueous solution such that the aqueous solution has a pH of 10 to 11.11. The method of claim 10, wherein the mixing of the aqueous solution is performed by titration with the starting material added at 4 mL/min simultaneously with titration of the complex agent added at 4 mL/min.12. The method of claim 10, wherein the sodium hydroxide is a 1 M solution and is automatically titrated based on a pH change, wherein the reactor is a co-precipitation reactor with co-precipitation proceeding.13. The method of claim 1, further comprising:putting the heat-treated solid electrolyte powder into a uniaxial compression mold for molding the heat-treated solid electrolyte powder;compressing the uniaxial compression mold; andheat-treating a pellet formed in the compressed uniaxial compression mold at a temperature of the heat treatment to form a high-density fine structure.14. The method of claim 13, wherein the heat-treated pellet changes to a cubic structure or a tetragonal structure at a calcination temperature that is the same as or higher than the heat treatment temperature.