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x³-2x²+2x-4=0

x³-2x²+2x-4=0 equation

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Numerical solution:

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The solution

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 3      2              
x  - 2*x  + 2*x - 4 = 0
x3−2x2+2x−4=0x^{3} - 2 x^{2} + 2 x - 4 = 0
Simplify
Detail solution
Given the equation:
x3−2x2+2x−4=0x^{3} - 2 x^{2} + 2 x - 4 = 0
transform
x3−2x2+2x−4=0x^{3} - 2 x^{2} + 2 x - 4 = 0
or
x3+2x−12=0x^{3} + 2 x - 12 = 0
x3−2x2+2x−4=0x^{3} - 2 x^{2} + 2 x - 4 = 0
(−2x+4)(x+2)+(x−2)(x2+2x+4)+2x−4=0\left(- 2 x + 4\right) \left(x + 2\right) + \left(x - 2\right) \left(x^{2} + 2 x + 4\right) + 2 x - 4 = 0
Take common factor x−2x - 2 from the equation
we get:
(x−2)(x2+2)=0\left(x - 2\right) \left(x^{2} + 2\right) = 0
or
(x−2)(x2+2)=0\left(x - 2\right) \left(x^{2} + 2\right) = 0
then:
x1=2x_{1} = 2
and also
we get the equation
x2+2=0x^{2} + 2 = 0
This equation is of the form
a∗x2+b∗x+c=0a*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=b2−4acD = b^2 - 4 a c is the discriminant.
Because
a=1a = 1
b=0b = 0
c=2c = 2
, then
D=b2−4∗a∗c=D = b^2 - 4 * a * c =
(−1)1⋅4⋅2+02=−8\left(-1\right) 1 \cdot 4 \cdot 2 + 0^{2} = -8
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x2=(−b+D)2ax_2 = \frac{(-b + \sqrt{D})}{2 a}
x3=(−b−D)2ax_3 = \frac{(-b - \sqrt{D})}{2 a}
or
x2=2ix_{2} = \sqrt{2} i
Simplify
x3=−2ix_{3} = - \sqrt{2} i
Simplify
The final answer for (x^3 - 2*x^2 + 2*x - 1*4) + 0 = 0:
x1=2x_{1} = 2
x2=2ix_{2} = \sqrt{2} i
x3=−2ix_{3} = - \sqrt{2} i
Vieta's Theorem
it is reduced cubic equation
px2+x3+qx+v=0p x^{2} + x^{3} + q x + v = 0
where
p=bap = \frac{b}{a}
p=−2p = -2
q=caq = \frac{c}{a}
q=2q = 2
v=dav = \frac{d}{a}
v=−4v = -4
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=2x_{1} + x_{2} + x_{3} = 2
x1x2+x1x3+x2x3=2x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = 2
x1x2x3=−4x_{1} x_{2} x_{3} = -4
The graph
-10.0-7.5-5.0-2.50.02.55.07.510.012.515.017.5-200200
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
         ___       ___
2 + -I*\/ 2  + I*\/ 2
(2)+(−2i)+(2i)\left(2\right) + \left(- \sqrt{2} i\right) + \left(\sqrt{2} i\right) 
=
2
22
Simplify 
product
         ___       ___
2 * -I*\/ 2  * I*\/ 2
(2)∗(−2i)∗(2i)\left(2\right) * \left(- \sqrt{2} i\right) * \left(\sqrt{2} i\right) 
=
4
44
Simplify
Rapid solution [src] 
x_1 = 2
x1=2x_{1} = 2
Simplify 
           ___
x_2 = -I*\/ 2
x2=−2ix_{2} = - \sqrt{2} i
Simplify 
          ___
x_3 = I*\/ 2
x3=2ix_{3} = \sqrt{2} i
Simplify
Numerical answer [src] 
x1 = -1.4142135623731*i
x2 = 2.0
x3 = 1.4142135623731*i
x3 = 1.4142135623731*i
The graph
x³-2x²+2x-4=0 equation
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