What are the solutions to the equation (z-2)^3=-64 ?

The solutions to the equation (z-2)^3=-64 are z=-2,z=4+2sqrt(3)i,z=4-2sqrt(3)i

Find the zeros of (z-2)^3=-64

The zeros of (z-2)^3=-64 are z=-2,z=4+2sqrt(3)i,z=4-2sqrt(3)i

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(z - 2)^{3} = - 64

z = - 2, z = 4 + 23 i, z = 4 - 23 i

Solution steps

(z - 2)^{3} = - 64

Expand

(z - 2)^{3} : z^{3} - 6 z^{2} + 12 z - 8

z^{3} - 6 z^{2} + 12 z - 8 = - 64

Move

64

to the left side

z^{3} - 6 z^{2} + 12 z + 56 = 0

Factor

z^{3} - 6 z^{2} + 12 z + 56 : (z + 2) (z^{2} - 8 z + 28)

(z + 2) (z^{2} - 8 z + 28) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

z + 2 = 0 or z^{2} - 8 z + 28 = 0

Solve

z + 2 = 0 : z = - 2

Solve

z^{2} - 8 z + 28 = 0 : z = 4 + 23 i, z = 4 - 23 i

The solutions are

z = - 2, z = 4 + 23 i, z = 4 - 23 i

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

4 x^{3} + 2 x - 18 = 0

y, x^{3} + y^{3} = 8

12 n^{2} + 48 n = - n^{3} - 64

x^{4} + 12 x^{3} + 54 x^{2} + 108 x + 81

What are the solutions to the equation (z-2)^3=-64 ?
The solutions to the equation (z-2)^3=-64 are z=-2,z=4+2sqrt(3)i,z=4-2sqrt(3)i
Find the zeros of (z-2)^3=-64
The zeros of (z-2)^3=-64 are z=-2,z=4+2sqrt(3)i,z=4-2sqrt(3)i