Solve the binomial equation x^3 + 8 = 0

To solve the binomial equation, we'll apply the formula of
the sum of cubes:

a^3 + b^3 = (a+b)(a^2 - ab +
b^2)

a^3 = x^3

a =
x

b^3 = 2^3 = 8

b =
2

x^3 + 8 = (x+2)(x^2 - 2x +
4)

If x^3 + 8 = 0, then (x+2)(x^2 - 2x + 4) =
0

If a product is zero, then each factor could be
zero.

x + 2 = 0

We'll subtract
2 both sides:

x1 =
-2

x^2 - 2x + 4 =
0

We'll apply the quadratic
formula:

x2 = [2 +
sqrt(4-16)]/2

x2 =
(2+2isqrt3)/2

We'll factorize by
2:

x2 =
2(1+isqrt3)/2

x2 =
1+isqrt3

x3 = 1-
isqrt3

The roots of the
equation are: {-2 , 1+isqrt3, 1- isqrt3 }.