Given the equation:
$$5 x^{3} - 5 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(5 x^{2} - 5\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$5 x^{2} - 5 = 0$$
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
$$a = 5$$
$$b = 0$$
$$c = -5$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (5) * (-5) = 100
Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = 1$$
Simplify$$x_{3} = -1$$
SimplifyThe final answer for (5*x^3 - 5*x) + 0 = 0:
$$x_{1} = 0$$
$$x_{2} = 1$$
$$x_{3} = -1$$