solvefor r,x= 2/3 pir^3

Lösung

löse nach

r, x = \frac{2}{3} π r^{3}

r = 3 \frac{3 x}{2 π}, r = - \frac{2 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{4 π} + i \frac{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{5}{6}} π ^{\frac{2}{3}} 3 x}{4 π}, r = - \frac{2 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{4 π} - i \frac{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{5}{6}} π ^{\frac{2}{3}} 3 x}{4 π}

Schritte zur Lösung

x = \frac{2}{3} π r^{3}

Tausche die Seiten

\frac{2}{3} π r^{3} = x

Vereinfache

\frac{2}{3} π : \frac{2 π}{3}

\frac{2 π}{3} r^{3} = x

Multipliziere beide Seiten mit

3

2 π r^{3} = 3 x

Teile beide Seiten durch

2 π

r^{3} = \frac{3 x}{2 π}

Für

x^{3} = f (a)

sind die Lösungen

x = 3 f (a), 3 f (a) \frac{- 1 - 3 i}{2}, 3 f (a) \frac{- 1 + 3 i}{2}

r = 3 \frac{3 x}{2 π}, r = 3 \frac{3 x}{2 π} \frac{- 1 + 3 i}{2}, r = 3 \frac{3 x}{2 π} \frac{- 1 - 3 i}{2}

Vereinfache

3 \frac{3 x}{2 π} \frac{- 1 + 3 i}{2} : - \frac{2 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{4 π} + i \frac{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{5}{6}} π ^{\frac{2}{3}} 3 x}{4 π}

Vereinfache

3 \frac{3 x}{2 π} \frac{- 1 - 3 i}{2} : - \frac{2 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{4 π} - i \frac{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{5}{6}} π ^{\frac{2}{3}} 3 x}{4 π}

r = 3 \frac{3 x}{2 π}, r = - \frac{2 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{4 π} + i \frac{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{5}{6}} π ^{\frac{2}{3}} 3 x}{4 π}, r = - \frac{2 ^{\frac{2}{3}} 3 3 π ^{\frac{2}{3}} 3 x}{4 π} - i \frac{2 ^{\frac{2}{3}} \cdot 3 ^{\frac{5}{6}} π ^{\frac{2}{3}} 3 x}{4 π}

x^{3} - 2 x^{2} + 5 x - 10 = 0

x^{3} + 3 x^{2} = 81 x + 243

12 x^{3} - 30 x^{2} - 24 x + 10 = 0

6 (x + 8)^{5} = 64

2 x (2 x + 9) (16 x^{2} - 9) = 0