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z^3+6z+20=0

z^3+6z+20=0 equation

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Numerical solution:

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The solution

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 3               
z  + 6*z + 20 = 0
z3+6z+20=0z^{3} + 6 z + 20 = 0
Simplify
Detail solution
Given the equation:
z3+6z+20=0z^{3} + 6 z + 20 = 0
transform
z3+6z+20=0z^{3} + 6 z + 20 = 0
or
z3+6z+20=0z^{3} + 6 z + 20 = 0
z3+6z+20=0z^{3} + 6 z + 20 = 0
(z+2)(z2−2z+4)+6z+12=0\left(z + 2\right) \left(z^{2} - 2 z + 4\right) + 6 z + 12 = 0
Take common factor z+2z + 2 from the equation
we get:
(z+2)(z2−2z+10)=0\left(z + 2\right) \left(z^{2} - 2 z + 10\right) = 0
or
(z+2)(z2−2z+10)=0\left(z + 2\right) \left(z^{2} - 2 z + 10\right) = 0
then:
z1=−2z_{1} = -2
and also
we get the equation
z2−2z+10=0z^{2} - 2 z + 10 = 0
This equation is of the form
a∗z2+b∗z+c=0a*z^2 + b*z + c = 0
A quadratic equation can be solved using the discriminant
The roots of the quadratic equation:
z2=D−b2az_{2} = \frac{\sqrt{D} - b}{2 a}
z3=−D−b2az_{3} = \frac{- \sqrt{D} - b}{2 a}
where D=b2−4acD = b^2 - 4 a c is the discriminant.
Because
a=1a = 1
b=−2b = -2
c=10c = 10
, then
D=b2−4∗a∗c=D = b^2 - 4 * a * c =
(−1)1⋅4⋅10+(−2)2=−36\left(-1\right) 1 \cdot 4 \cdot 10 + \left(-2\right)^{2} = -36
Because D<0, then the equation
has no real roots,
but complex roots is exists.
z2=(−b+D)2az_2 = \frac{(-b + \sqrt{D})}{2 a}
z3=(−b−D)2az_3 = \frac{(-b - \sqrt{D})}{2 a}
or
z2=1+3iz_{2} = 1 + 3 i
Simplify
z3=1−3iz_{3} = 1 - 3 i
Simplify
The final answer for (z^3 + 6*z + 20) + 0 = 0:
z1=−2z_{1} = -2
z2=1+3iz_{2} = 1 + 3 i
z3=1−3iz_{3} = 1 - 3 i
Vieta's Theorem
it is reduced cubic equation
pz2+z3+qz+v=0p z^{2} + z^{3} + q z + v = 0
where
p=bap = \frac{b}{a}
p=0p = 0
q=caq = \frac{c}{a}
q=6q = 6
v=dav = \frac{d}{a}
v=20v = 20
Vieta Formulas
z1+z2+z3=−pz_{1} + z_{2} + z_{3} = - p
z1z2+z1z3+z2z3=qz_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = q
z1z2z3=vz_{1} z_{2} z_{3} = v
z1+z2+z3=0z_{1} + z_{2} + z_{3} = 0
z1z2+z1z3+z2z3=6z_{1} z_{2} + z_{1} z_{3} + z_{2} z_{3} = 6
z1z2z3=20z_{1} z_{2} z_{3} = 20
The graph
-17.5-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.0-200200
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
-2 + 1 - 3*I + 1 + 3*I
(−2)+(1−3i)+(1+3i)\left(-2\right) + \left(1 - 3 i\right) + \left(1 + 3 i\right) 
=
0
00
Simplify 
product
-2 * 1 - 3*I * 1 + 3*I
(−2)∗(1−3i)∗(1+3i)\left(-2\right) * \left(1 - 3 i\right) * \left(1 + 3 i\right) 
=
-20
−20-20
Simplify
Rapid solution [src] 
z_1 = -2
z1=−2z_{1} = -2
Simplify 
z_2 = 1 - 3*I
z2=1−3iz_{2} = 1 - 3 i
Simplify 
z_3 = 1 + 3*I
z3=1+3iz_{3} = 1 + 3 i
Simplify
Numerical answer [src] 
z1 = 1.0 - 3.0*i
z2 = 1.0 + 3.0*i
z3 = -2.0
z3 = -2.0
The graph
z^3+6z+20=0 equation
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