x^3 − 512 = 0

(x − 8)(x^2 + 8x + 64) = 0

Real solution:

x = 8

Complex solutions:

x = -4 - 4 i sqrt(3)

x = -4 + 4 i sqrt(3)

x- 8 is a factor of x³ - 512

Find other factors by synthetic division.

8 | 1_______0________0____ - 512

_ | ________ 8_______64______ 512

_ | 1 _______8_______64_______0

(x³ - 512) = (x - 8) ( x² + 8x + 64 )

(x³ - 512) = (x - 8) (x + 8) (x + 8)

x^3-512=0 has 1 real root & 2 non-real roots, which are

x=8

x=-4+/-6.928203i =-4+/-4sqr(3)i

Knowing 8 (or 8+0i) is a solution, the three solutions are evenly spaced about the circle radius 8 on the complex plane.

8+0i is at 0°, so the others are at 120° and 240°

These are 8cis(120°) and 8cis(240°), which are -4-4√3 i and -4+4√3 i.

x^3 - 512 = 0

x^3 - 8^3 = 0

(x - 8)(x^2 + 8x + 64) = 0

Quadratic formula for the trinomial factor...

(-8 +/- sqrt(8^2 - 4(1)(64))) / 2(1)

(-8 +/- sqrt(64 - 256)) / 2

(-8 +/- sqrt(-192)) / 2

(-8 +/- 8i sqrt(3)) / 2

-4 +/- 4i sqrt(3).

Other two solutions: x = -4 + 4i sqrt(3) and -4 - 4i sqrt(3).

Factor out the root 8, as in, factor out x-8. U will be left with a quadratic eqs, which you know how to solve.

Done!

Note that you can immediately jump into complex numbers/exponential form and readily get all three roots.