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x^3-3*x-2=0

x^3-3*x-2=0 equation

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Numerical solution:

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The solution

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

(-1)^2 - 4 * (1) * (-2) = 9

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=−1x_{3} = -1
The final answer for x^3 - 3*x - 2 = 0:
x1=−1x_{1} = -1
x2=2x_{2} = 2
x3=−1x_{3} = -1
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=−3q = -3
v=dav = \frac{d}{a}
v=−2v = -2
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=−3x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -3
x1x2x3=−2x_{1} x_{2} x_{3} = -2
The graph
05-15-10-51015-50005000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = -1
x1=−1x_{1} = -1 
x2 = 2
x2=2x_{2} = 2 
x2 = 2
Sum and product of roots [src] 
sum
-1 + 2
−1+2-1 + 2 
=
1
11 
product
-2
−2- 2 
=
-2
−2-2 
-2
Numerical answer [src] 
x1 = 2.0
x2 = -1.0
x2 = -1.0
The graph
x^3-3*x-2=0 equation
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