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x^3-12*x=0

x^3-12*x=0 equation

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Numerical solution:

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The solution

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x  - 12*x = 0
$$x^{3} - 12 x = 0$$
Detail solution
Given the equation:
$$x^{3} - 12 x = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} - 12\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} - 12 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = -12$$
, then
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (1) * (-12) = 48

Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)

x3 = (-b - sqrt(D)) / (2*a)

or
$$x_{2} = 2 \sqrt{3}$$
$$x_{3} = - 2 \sqrt{3}$$
The final answer for x^3 - 12*x = 0:
$$x_{1} = 0$$
$$x_{2} = 2 \sqrt{3}$$
$$x_{3} = - 2 \sqrt{3}$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = -12$$
$$v = \frac{d}{a}$$
$$v = 0$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = 0$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -12$$
$$x_{1} x_{2} x_{3} = 0$$
The graph
Rapid solution [src]
x1 = 0
$$x_{1} = 0$$
          ___
x2 = -2*\/ 3 
$$x_{2} = - 2 \sqrt{3}$$
         ___
x3 = 2*\/ 3 
$$x_{3} = 2 \sqrt{3}$$
x3 = 2*sqrt(3)
Sum and product of roots [src]
sum
      ___       ___
- 2*\/ 3  + 2*\/ 3 
$$- 2 \sqrt{3} + 2 \sqrt{3}$$
=
0
$$0$$
product
       ___     ___
0*-2*\/ 3 *2*\/ 3 
$$2 \sqrt{3} \cdot 0 \left(- 2 \sqrt{3}\right)$$
=
0
$$0$$
0
Numerical answer [src]
x1 = 0.0
x2 = 3.46410161513775
x3 = -3.46410161513775
x3 = -3.46410161513775
The graph
x^3-12*x=0 equation