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5x^3-5x=0

5x^3-5x=0 equation

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Numerical solution:

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The solution

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5*x  - 5*x = 0
5x3−5x=05 x^{3} - 5 x = 0
Simplify
Detail solution
Given the equation:
5x3−5x=05 x^{3} - 5 x = 0
transform
Take common factor x from the equation
we get:
x(5x2−5)=0x \left(5 x^{2} - 5\right) = 0
then:
x1=0x_{1} = 0
and also
we get the equation
5x2−5=05 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:
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=5a = 5
b=0b = 0
c=−5c = -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
x2=1x_{2} = 1
Simplify
x3=−1x_{3} = -1
Simplify
The final answer for (5*x^3 - 5*x) + 0 = 0:
x1=0x_{1} = 0
x2=1x_{2} = 1
x3=−1x_{3} = -1
Vieta's Theorem
rewrite the equation
5x3−5x=05 x^{3} - 5 x = 0
of
ax3+bx2+cx+d=0a x^{3} + b x^{2} + c x + d = 0
as reduced cubic equation
x3+bx2a+cxa+da=0x^{3} + \frac{b x^{2}}{a} + \frac{c x}{a} + \frac{d}{a} = 0
x3−x=0x^{3} - x = 0
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=−1q = -1
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=−1x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -1
x1x2x3=0x_{1} x_{2} x_{3} = 0
The graph
05-15-10-51015-1000010000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = -1
x1=−1x_{1} = -1
Simplify 
x2 = 0
x2=0x_{2} = 0
Simplify 
x3 = 1
x3=1x_{3} = 1
Simplify
Sum and product of roots [src] 
sum
0 - 1 + 0 + 1
((−1+0)+0)+1\left(\left(-1 + 0\right) + 0\right) + 1 
=
0
00 
product
1*-1*0*1
1(−1)0⋅11 \left(-1\right) 0 \cdot 1 
=
0
00 
0
Numerical answer [src] 
x1 = 0.0
x2 = 1.0
x3 = -1.0
x3 = -1.0
The graph
5x^3-5x=0 equation
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