Given the equation:
$$\left(x^{3} + 2 x\right) - 3 = 0$$
transform
$$\left(2 x + \left(x^{3} - 1\right)\right) - 2 = 0$$
or
$$\left(2 x + \left(x^{3} - 1^{3}\right)\right) - 2 = 0$$
$$2 \left(x - 1\right) + \left(x^{3} - 1^{3}\right) = 0$$
$$\left(x - 1\right) \left(\left(x^{2} + x\right) + 1^{2}\right) + 2 \left(x - 1\right) = 0$$
Take common factor -1 + x from the equation
we get:
$$\left(x - 1\right) \left(\left(\left(x^{2} + x\right) + 1^{2}\right) + 2\right) = 0$$
or
$$\left(x - 1\right) \left(x^{2} + x + 3\right) = 0$$
then:
$$x_{1} = 1$$
and also
we get the equation
$$x^{2} + x + 3 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 1$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (1) * (3) = -11
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{11} i}{2}$$
The final answer for x^3 + 2*x - 3 = 0:
$$x_{1} = 1$$
$$x_{2} = - \frac{1}{2} + \frac{\sqrt{11} i}{2}$$
$$x_{3} = - \frac{1}{2} - \frac{\sqrt{11} i}{2}$$