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z^2+6*z+13=0

z^2+6*z+13=0 equation

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Numerical solution:

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The solution

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 2               
z  + 6*z + 13 = 0
(z2+6z)+13=0\left(z^{2} + 6 z\right) + 13 = 0
Simplify
Detail solution
This equation is of the form
a*z^2 + b*z + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
z1=D−b2az_{1} = \frac{\sqrt{D} - b}{2 a}
z2=−D−b2az_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=1a = 1
b=6b = 6
c=13c = 13
, then
D = b^2 - 4 * a * c = 

(6)^2 - 4 * (1) * (13) = -16

Because D<0, then the equation
has no real roots,
but complex roots is exists.
z1 = (-b + sqrt(D)) / (2*a)

z2 = (-b - sqrt(D)) / (2*a)

or
z1=−3+2iz_{1} = -3 + 2 i
z2=−3−2iz_{2} = -3 - 2 i
Vieta's Theorem
it is reduced quadratic equation
pz+q+z2=0p z + q + z^{2} = 0
where
p=bap = \frac{b}{a}
p=6p = 6
q=caq = \frac{c}{a}
q=13q = 13
Vieta Formulas
z1+z2=−pz_{1} + z_{2} = - p
z1z2=qz_{1} z_{2} = q
z1+z2=−6z_{1} + z_{2} = -6
z1z2=13z_{1} z_{2} = 13
The graph
01-10-9-8-7-6-5-4-3-2-1020
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
-3 - 2*I + -3 + 2*I
(−3−2i)+(−3+2i)\left(-3 - 2 i\right) + \left(-3 + 2 i\right) 
=
-6
−6-6 
product
(-3 - 2*I)*(-3 + 2*I)
(−3−2i)(−3+2i)\left(-3 - 2 i\right) \left(-3 + 2 i\right) 
=
13
1313 
13
Rapid solution [src] 
z1 = -3 - 2*I
z1=−3−2iz_{1} = -3 - 2 i 
z2 = -3 + 2*I
z2=−3+2iz_{2} = -3 + 2 i 
z2 = -3 + 2*i
Numerical answer [src] 
z1 = -3.0 - 2.0*i
z2 = -3.0 + 2.0*i
z2 = -3.0 + 2.0*i
The graph
z^2+6*z+13=0 equation
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