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4*x^3-16*x=0

4*x^3-16*x=0 equation

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Numerical solution:

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The solution

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4*x  - 16*x = 0
4x3−16x=04 x^{3} - 16 x = 0
Simplify
Detail solution
Given the equation:
4x3−16x=04 x^{3} - 16 x = 0
transform
Take common factor x from the equation
we get:
x(4x2−16)=0x \left(4 x^{2} - 16\right) = 0
then:
x1=0x_{1} = 0
and also
we get the equation
4x2−16=04 x^{2} - 16 = 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=4a = 4
b=0b = 0
c=−16c = -16
, then
D = b^2 - 4 * a * c = 

(0)^2 - 4 * (4) * (-16) = 256

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

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

or
x2=2x_{2} = 2
x3=−2x_{3} = -2
The final answer for 4*x^3 - 16*x = 0:
x1=0x_{1} = 0
x2=2x_{2} = 2
x3=−2x_{3} = -2
Vieta's Theorem
rewrite the equation
4x3−16x=04 x^{3} - 16 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
x3−4x=0x^{3} - 4 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=−4q = -4
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=−4x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -4
x1x2x3=0x_{1} x_{2} x_{3} = 0
The graph
02468-21810121416-500010000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = -2
x1=−2x_{1} = -2 
x2 = 0
x2=0x_{2} = 0 
x3 = 2
x3=2x_{3} = 2 
x3 = 2
Sum and product of roots [src] 
sum
-2 + 2
−2+2-2 + 2 
=
0
00 
product
-2*0*2
2(−0)2 \left(- 0\right) 
=
0
00 
0
Numerical answer [src] 
x1 = 2.0
x2 = 0.0
x3 = -2.0
x3 = -2.0
The graph
4*x^3-16*x=0 equation
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