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Complex Numbers and Quadratic Equations
Quadratic Equations
An equation of the form ax2 + bx + c = 0 is called a quadratic equation, where a, b and c are
real or complex numbers such that a ≠ 0 and x is a variable
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x = α and x = β are the roots of the quadratic equation f(x) = 0 if f(α) = f(β) = 0
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It is given by:
D = b2 − 4ac
Nature of Roots
The nature of roots of a quadratic equation depends on its discriminant (D), which can be
observed as follows:
Nature of D
If D = 0, then
b2 = 4ac
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If D > 0, then

Roots are distinct and real
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If D < 0, then

Roots are imaginary and given by:

b2 < 4ac

and

Imaginary roots of a quadratic equation are complex conjugates of each other; i
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,

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Let us study about the cube roots and nth roots of unity in detail
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The roots 1, ,

are called the cube roots of unity
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Properties of cube roots of unity:
Sum of the roots, 1+ ω + ω2 = 0
Product of the roots, 1 × ω × ω2 = ω3 =1
Representation of cube roots of unity on the Argand plane:
If we plot the roots of the equation x3 = 1 on the Argand plane, we obtain an equilateral triangle
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nth root of unity
Consider the equation xn = 1
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xn = 1

If r = 0, then x = 1

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If r = n − 1, then
The roots
If

then the nth roots of unity will be represented as 1, α, α 2,…, α n −1
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C
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For the complex numbers z = a + ib, a is called the real part (denoted by Re z) and b is
called the imaginary part (denoted by Im z) of the complex number z
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Addition of complex numbers
Two complex numbers z1 = a + ib and z2 = c + id can be added as,
z1 + z2 = (a + ib) + (c + id) = (a + c) + i(b + d)

Properties of addition of complex numbers:
Closure Law: Sum of two complex numbers is a complex number
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Commutative Law: For two complex numbers z1 and z2,
z1 + z2 = z2 + z1
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Existence of Additive Identity: There exists a complex number 0 + i0
,
called the additive identity or zero complex number, such that for every complex
number z, z + 0 = z
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Subtraction of complex numbers
Given any two complex numbers z1, and z2, the difference z1 - z2 is defined as
z1 - z2 = z1 + (–z2)
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In fact, for two complex numbers z1 and z2, such that z1 = a + ib and z2 = c + id, we
obtain z1 z2 = (ac - bd) + i(ad + bc)
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Associative Law: For any three complex numbers z1, z2 and z3,
(z1z2) z3 = z1 (z2z3)
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1 = z
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Example: The multiplicative inverse of the complex number z = 2 – 3i can be found as,

Distributive Law: For any three complex numbers z1, z2 and z3,
z1 (z2 + z3) = z1z2 + z1z3
(z1 + z2) z3 = z1z3 + z2z3
Division of Complex Numbers
Given any two complex number z1 and z2, where

, the quotient

Example: For z1 = 1 + i and z2 = 2 – 3i, the quotient

is defined as

can be found as,

Property of Complex Numbers
For any integerk,
If a and b are negative real numbers, then


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e
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Example 1: If z = 2 – 3i, then
Conjugate of Complex Number
The conjugate of a complex number z = a + ib, is denoted by
number a – ib, i
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,

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Solution: We have

Thus, the conjugate of

is


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An
Argand plane is shown in the following figure
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Complex Number on Argand plane: The complex number z = a + ib can be represented
on an Argand plane as

In this figure,

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Conjugate of Complex Number on Argand plane:
The conjugate of a complex number z = a + ib is
the points P(a, b) and Q(a, −b) on the Argand plane as

, z and can be represented by

Thus, on the Argand plane, the conjugate of a complex number is the mirror image of the
complex number with respect to the real axis
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The value of θ is such that –π < θ ≤ π, which is called the principle argument of z
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Thus,

Thus, the required polar form is


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The solutions of the quadratic equation ax2 + bx + c = 0, where a, b, c

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