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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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 3           
x  - 12*x = 0
x3−12x=0x^{3} - 12 x = 0
Simplify
Detail solution
Given the equation:
x3−12x=0x^{3} - 12 x = 0
transform
Take common factor x from the equation
we get:
x(x2−12)=0x \left(x^{2} - 12\right) = 0
then:
x1=0x_{1} = 0
and also
we get the equation
x2−12=0x^{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:
x2=D−b2ax_{2} = \frac{\sqrt{D} - b}{2 a}
x3=−D−b2ax_{3} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=1a = 1
b=0b = 0
c=−12c = -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
x2=23x_{2} = 2 \sqrt{3}
x3=−23x_{3} = - 2 \sqrt{3}
The final answer for x^3 - 12*x = 0:
x1=0x_{1} = 0
x2=23x_{2} = 2 \sqrt{3}
x3=−23x_{3} = - 2 \sqrt{3}
Vieta's Theorem
it is reduced cubic equation
px2+qx+v+x3=0p x^{2} + q x + v + x^{3} = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=−12q = -12
v=dav = \frac{d}{a}
v=0v = 0
Vieta Formulas
x1+x2+x3=−px_{1} + x_{2} + x_{3} = - p
x1x2+x1x3+x2x3=qx_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q
x1x2x3=vx_{1} x_{2} x_{3} = v
x1+x2+x3=0x_{1} + x_{2} + x_{3} = 0
x1x2+x1x3+x2x3=−12x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -12
x1x2x3=0x_{1} x_{2} x_{3} = 0
The graph
-2.50.02.55.07.510.012.515.017.520.0-25002500
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = 0
x1=0x_{1} = 0 
          ___
x2 = -2*\/ 3
x2=−23x_{2} = - 2 \sqrt{3} 
         ___
x3 = 2*\/ 3
x3=23x_{3} = 2 \sqrt{3} 
x3 = 2*sqrt(3)
Sum and product of roots [src] 
sum
      ___       ___
- 2*\/ 3  + 2*\/ 3
−23+23- 2 \sqrt{3} + 2 \sqrt{3} 
=
0
00 
product
       ___     ___
0*-2*\/ 3 *2*\/ 3
23⋅0(−23)2 \sqrt{3} \cdot 0 \left(- 2 \sqrt{3}\right) 
=
0
00 
0
Numerical answer [src] 
x1 = 0.0
x2 = 3.46410161513775
x3 = -3.46410161513775
x3 = -3.46410161513775
The graph
x^3-12*x=0 equation
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