What are the solutions to the equation z^6+6z^3+10=0 ?

The solutions to the equation z^6+6z^3+10=0 are

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z^6+6z^3+10=0

z^{6} + 6 z^{3} + 10 = 0

z = 610 cos (\frac{- arctan ( \frac{1}{3} ) + π}{3}) + 610 i sin (\frac{- arctan ( \frac{1}{3} ) + π}{3}), z = 610 cos (\frac{- arctan ( \frac{1}{3} ) + 3 π}{3}) + 610 i sin (\frac{- arctan ( \frac{1}{3} ) + 3 π}{3}), z = 610 cos (\frac{- arctan ( \frac{1}{3} ) + 5 π}{3}) + 610 i sin (\frac{- arctan ( \frac{1}{3} ) + 5 π}{3}), z = 610 cos (\frac{arctan ( \frac{1}{3} ) - π}{3}) + 610 i sin (\frac{arctan ( \frac{1}{3} ) - π}{3}), z = 610 cos (\frac{arctan ( \frac{1}{3} ) + π}{3}) + 610 i sin (\frac{arctan ( \frac{1}{3} ) + π}{3}), z = 610 cos (\frac{arctan ( \frac{1}{3} ) + 3 π}{3}) + 610 i sin (\frac{arctan ( \frac{1}{3} ) + 3 π}{3})

Solution steps

z^{6} + 6 z^{3} + 10 = 0

Rewrite the equation with

u = z^{3}

and

u^{2} = z^{6}

u^{2} + 6 u + 10 = 0

Solve

u^{2} + 6 u + 10 = 0 : u = - 3 + i, u = - 3 - i

u = - 3 + i, u = - 3 - i

Substitute back

u = z^{3},

solve for

z

Solve

z^{3} = - 3 + i : z = 610 cos (\frac{- arctan ( \frac{1}{3} ) + π}{3}) + 610 i sin (\frac{- arctan ( \frac{1}{3} ) + π}{3}), z = 610 cos (\frac{- arctan ( \frac{1}{3} ) + 3 π}{3}) + 610 i sin (\frac{- arctan ( \frac{1}{3} ) + 3 π}{3}), z = 610 cos (\frac{- arctan ( \frac{1}{3} ) + 5 π}{3}) + 610 i sin (\frac{- arctan ( \frac{1}{3} ) + 5 π}{3})

Solve

z^{3} = - 3 - i : z = 610 cos (\frac{arctan ( \frac{1}{3} ) - π}{3}) + 610 i sin (\frac{arctan ( \frac{1}{3} ) - π}{3}), z = 610 cos (\frac{arctan ( \frac{1}{3} ) + π}{3}) + 610 i sin (\frac{arctan ( \frac{1}{3} ) + π}{3}), z = 610 cos (\frac{arctan ( \frac{1}{3} ) + 3 π}{3}) + 610 i sin (\frac{arctan ( \frac{1}{3} ) + 3 π}{3})

The solutions are

z = 610 cos (\frac{- arctan ( \frac{1}{3} ) + π}{3}) + 610 i sin (\frac{- arctan ( \frac{1}{3} ) + π}{3}), z = 610 cos (\frac{- arctan ( \frac{1}{3} ) + 3 π}{3}) + 610 i sin (\frac{- arctan ( \frac{1}{3} ) + 3 π}{3}), z = 610 cos (\frac{- arctan ( \frac{1}{3} ) + 5 π}{3}) + 610 i sin (\frac{- arctan ( \frac{1}{3} ) + 5 π}{3}), z = 610 cos (\frac{arctan ( \frac{1}{3} ) - π}{3}) + 610 i sin (\frac{arctan ( \frac{1}{3} ) - π}{3}), z = 610 cos (\frac{arctan ( \frac{1}{3} ) + π}{3}) + 610 i sin (\frac{arctan ( \frac{1}{3} ) + π}{3}), z = 610 cos (\frac{arctan ( \frac{1}{3} ) + 3 π}{3}) + 610 i sin (\frac{arctan ( \frac{1}{3} ) + 3 π}{3})

- 2 x^{4} + x^{3} - x^{2} + 4 x

- 3 x^{3} + 5 x^{2} - 27 x + 45 = 0

2 x^{4} - 9 x^{3} + 17 x^{2} - 36 x + 36 = 0

x^{4} + 7 x^{3} + 8 x^{2} - 7 x + 15 = 0

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

What are the solutions to the equation z^6+6z^3+10=0 ?
The solutions to the equation z^6+6z^3+10=0 are
Find the zeros of z^6+6z^3+10=0
The zeros of z^6+6z^3+10=0 are