Quadratic equation with complex variableSolve the following equation with ω∈C and γ,ω0∈R for its complex roots ω1,ω2ω2+γω+ω02=0.

The quadratic formula applies also to cases where the roots are complex.ω=−γ±γ2−4ω022Depending on the sign of the γ2−4ω02,ω may be imaginary/real.

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Beryneingmk39

2022-03-30

Quadratic equation with complex variable
Solve the following equation with

ω \in C

and

γ, ω_{0} \in R

for its complex roots

ω_{1}, ω_{2}

ω^{2} + γ ω + ω_{0}^{2} = 0

zevillageobau

Beginner2022-03-31Added 13 answers

The quadratic formula applies also to cases where the roots are complex.

ω = \frac{- γ \pm \sqrt{γ^{2} - 4 ω_{0}^{2}}}{2}

Depending on the sign of the

γ^{2} - 4 ω_{0}^{2}, ω

may be imaginary/real.

arrebusnaavbz

Beginner2022-04-01Added 18 answers

{(x + i y)}^{2} + γ (x + i y) + z^{2} = 0 \Rightarrow x^{2} - y^{2} + 2 x y i + γ x + i γ y + z^{2} = 0

\Rightarrow x^{2} - y^{2} + x + z^{2} = 0; i y (2 x + γ) = 0

by separating the real and imaginary parts.
Two cases are seen here: either

y = 0

(so

ω

is not complex) or

x = - \frac{γ}{2}

, from which we solve the quadratic in y.

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