x^4-4*x^3+20=0

$$x^{4} – 4 x^{3} + 20 = 0$$

___________________________________________________________________________________________________________________________
/ ________________
/ / ____
/ / 20*/ 21 16 40
/ 8 – 2*3 / 20 + ——— + —————————————————————– – ———————–
_______________________________________________________ / / 9 _______________________________________________________ ________________
/ ________________ / / ________________ / ____
/ / ____ / / / ____ / 20*/ 21
/ / 20*/ 21 40 / / / 20*/ 21 40 3*3 / 20 + ———
/ 4 + 2*3 / 20 + ——— + ———————– / / 4 + 2*3 / 20 + ——— + ———————– / 9
/ / 9 ________________ / / / 9 ________________
/ / ____ / / / ____
/ / 20*/ 21 / / / 20*/ 21
/ 3*3 / 20 + ——— / / 3*3 / 20 + ———
/ / 9 / / / 9
x1 = 1 + —————————————————————– – ——————————————————————————————————————————————
2 2

$$x_{1} = – frac{1}{2} sqrt{- 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20} – frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + frac{16}{sqrt{4 + frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20}}} + 8} + 1 + frac{1}{2} sqrt{4 + frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20}}$$

___________________________________________________________________________________________________________________________
/ ________________
/ / ____
/ / 20*/ 21 16 40
/ 8 – 2*3 / 20 + ——— + —————————————————————– – ———————–
_______________________________________________________ / / 9 _______________________________________________________ ________________
/ ________________ / / ________________ / ____
/ / ____ / / / ____ / 20*/ 21
/ / 20*/ 21 40 / / / 20*/ 21 40 3*3 / 20 + ———
/ 4 + 2*3 / 20 + ——— + ———————– / / 4 + 2*3 / 20 + ——— + ———————– / 9
/ / 9 ________________ / / / 9 ________________
/ / ____ / / / ____
/ / 20*/ 21 / / / 20*/ 21
/ 3*3 / 20 + ——— / / 3*3 / 20 + ———
/ / 9 / / / 9
x2 = 1 + —————————————————————– + ——————————————————————————————————————————————
2 2

$$x_{2} = frac{1}{2} sqrt{- 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20} – frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + frac{16}{sqrt{4 + frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20}}} + 8} + 1 + frac{1}{2} sqrt{4 + frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20}}$$

___________________________________________________________________________________________________________________________
/ ________________
/ / ____
/ 16 / 20*/ 21 40
/ 8 – —————————————————————– – 2*3 / 20 + ——— – ———————–
_______________________________________________________ / _______________________________________________________ / 9 ________________
/ ________________ / / ________________ / ____
/ / ____ / / / ____ / 20*/ 21
/ / 20*/ 21 40 / / / 20*/ 21 40 3*3 / 20 + ———
/ 4 + 2*3 / 20 + ——— + ———————– / / 4 + 2*3 / 20 + ——— + ———————– / 9
/ / 9 ________________ / / / 9 ________________
/ / ____ / / / ____
/ / 20*/ 21 / / / 20*/ 21
/ 3*3 / 20 + ——— / / 3*3 / 20 + ———
/ / 9 / / / 9
x3 = 1 – —————————————————————– – ——————————————————————————————————————————————
2 2

$$x_{3} = – frac{1}{2} sqrt{4 + frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 1 – frac{1}{2} sqrt{- 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20} – frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} – frac{16}{sqrt{4 + frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20}}} + 8}$$

___________________________________________________________________________________________________________________________
/ ________________
/ / ____
/ 16 / 20*/ 21 40
/ 8 – —————————————————————– – 2*3 / 20 + ——— – ———————–
/ _______________________________________________________ / 9 ________________ _______________________________________________________
/ / ________________ / ____ / ________________
/ / / ____ / 20*/ 21 / / ____
/ / / 20*/ 21 40 3*3 / 20 + ——— / / 20*/ 21 40
/ / 4 + 2*3 / 20 + ——— + ———————– / 9 / 4 + 2*3 / 20 + ——— + ———————–
/ / / 9 ________________ / / 9 ________________
/ / / ____ / / ____
/ / / 20*/ 21 / / 20*/ 21
/ / 3*3 / 20 + ——— / 3*3 / 20 + ———
/ / / 9 / / 9
x4 = 1 + —————————————————————————————————————————————— – —————————————————————–
2 2

$$x_{4} = – frac{1}{2} sqrt{4 + frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 1 + frac{1}{2} sqrt{- 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20} – frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} – frac{16}{sqrt{4 + frac{40}{3 sqrt[3]{frac{20 sqrt{21}}{9} + 20}} + 2 sqrt[3]{frac{20 sqrt{21}}{9} + 20}}} + 8}$$

x1 = -0.904564276865 – 1.29517336229*i

x2 = 3.55475184132000

x3 = 2.25437671241000

x4 = -0.904564276865 + 1.29517336229*i