Complex Numbers and Quadratic Equations
Complex Numbers and Quadratic Equations PDF Notes, Important Questions and Synopsis
SYNOPSIS
 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) 
Complex conjugate
Argument
Magnitude
If z = x + iy, then the conjugate of z is
= x  iyamp(z) = arg(z) = q =
General argument: 2nπ + θ, n ϵ ℕ
Principal argument: π < θ ≤ π
Least positive argument: 0 < θ ≤ 2πz = x + iy
z=  Representation of Complex Number
Polar Representation
Exponential Form
Vector Representation
x = r cos θ, y = r sin θ
z = r e^{iθ}
(where = cos e^{iθ} + I sin θ)
z = x + iy is considered a position vector of point p
 Square roots of a complex number
Let z = x + iy, then square root of z is
,for y>0
, for y<0
 Properties of the argument of a Complex Number:
 arg(any real positive number) = 0
 arg(any real negative number) = π
 Inequalities
I. Triangle inequalities
1. z_{1 }± z_{2} £  z_{1} ±  z_{2}
2. z_{1 }± z_{2} ³  z_{1}   z_{2}
II. Parallelogram inequalities
 z_{1 }+ z_{2}^{2}+  z_{1 } z_{2}^{2} = 2 [z_{1}^{2}+ z_{2}^{2}]  If ABC is an equilateral triangle having vertices z_{1}, z_{2}, z_{3}, then or
 If z_{1}, z_{2}, z_{3}, z_{4} are vertices of a parallelogram, then z_{1} + z_{3} = z_{2} + z_{4}.
 If z_{1}, z_{2}, z_{3} are affixes of the points A, B and C in the Argand plane, then
i. ÐBAC =ii. , where α = ÐBAC  The equation of a circle whose centre is at a point having affix z_{0} and radius R = z  z_{0}.
 If a, b are positive real numbers, then.
 Integral powers of iota
Hence,
Quadratic Equations
 An equation of the form is called a quadratic equation, where a, b, c are real numbers and a ≠ 0.
 Values of the variable which satisfies the quadratic equation are called its roots.

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.

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 .

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