Question

how to find principal argument in chapter complex numbers and quadratic equations ?

Answer 1

let a complex number be = z = x + i y
This is also defined in the complex plane using polar coordinates.

r = |z| = √(x²+y²)

tan Ф = y/x = Imaginary part of complex number / real part of complex number.
here, Ф is called argument of the complex number.      r is the modulus of complex number.

argument = tan⁻¹ y/x
==========================================
In case of quadratic equations, if the determinant is negative, then we have a complex number as the solution.

a x² + b x + c = 0

Determinant = b² - 2 ac  < 0, then let  -Δ = (4ac - b²) = positive

x1 =  [ - b + j √(4ac-b²) ] / 2 a             and    x2  =  [ -b - j √(4ac - b²) ] / 2 a

argument will be (for x1) :  tan Ф = - √(4ac-b²) / b

         Ф =  tan⁻¹ [ - √(4ac-b²) / b ]