Complex Numbers and Quadratic Equations

SYNOPSIS

1. A number of the form x + iy, where x, y Î ℝ and  (i is iota), is called a complex number.
   It is denoted by z, and a set of complex numbers is denoted by ℂ.
   x = real part or Re(z), y = imaginary part or Im(z)
   
   

2. 	Complex conjugate	Argument	Magnitude
   	If z = x + iy, then the conjugate of z is = x - iy	amp(z) = arg(z) = q = General argument: 2nπ + θ, n ϵ ℕ Principal argument: -π < θ ≤ π Least positive argument: 0 < θ ≤ 2π	z = x + iy |z|=

3. Representation of Complex Number
   
   

   Polar Representation	Exponential Form	Vector Representation
   x = r cos  θ, y = r sin  θ	z = r eiθ (where = cos eiθ + I sin θ)	z = x + iy is considered a position vector of point p

   
   
   
4. Square roots of a complex number
   Let z = x + iy, then square root of z is
   

   ,for y>0

   , for y<0

5.  Properties of the argument of a Complex Number: 
   

   arg(any real positive number) = 0	arg(any real negative number) = π

   
   
   
   
6. Inequalities
   

   I. Triangle inequalities       1. |z1 ± z2| £ | z1| ± | z2|       2. |z1 ± z2| ³ | z1| - | z2| II. Parallelogram inequalities       | z1 + z2|2+ | z1 - z2|2 = 2 [|z1|2+| z2|2]

7. If ABC is an equilateral triangle having vertices z₁, z₂, z₃, then  or 
   
   
8. If z₁, z₂, z₃, z₄ are vertices of a parallelogram, then z₁ + z₃ = z₂ + z₄.
   
   
9. If z₁, z₂, z₃ are affixes of the points A, B and C in the Argand plane, then 
   
   

   i. ÐBAC =

   ii.   , where α = ÐBAC

10. The equation of a circle whose centre is at a point having affix z₀ and radius R = |z - z₀|.
    
11. If a, b are positive real numbers, then. 
12. Integral powers of iota
    
    Hence,

Quadratic Equations

1. An equation of the form  is called a quadratic equation, where a, b, c are real numbers and a ≠ 0.
2. Values of the variable which satisfies the quadratic equation are called its roots.

3. Nature of Roots
   Let f(x) =  be the quadratic equation, the discriminant D = .

   If a > 0	If a < 0
   1.	1.
   2.	2.
   3.	3.

4. Let α, β be the roots of the quadratic equation then

   i. Roots are given by the quadratic formula:       formula:      a, b =

   ii.  Relation between roots and coefficients: 1. Sum of the roots =a+b = - 2.  Product of the roots = a×b =

   Note: Quadratic equation can be rewritten as .

5. Quadratic inequalities
   Let y =  be the quadratic polynomial. There are two inequalities: