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MATHS COMPLEX NUMBERS AND QUADRATIC EQUATIONS

Complex Numbers & Quadratic Equations
Some Important Results
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If n>4, then in 

1 1
 , where k is the reminder when n is divided by 4
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We have i0  1
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A number in the form a + ib, where a and b are real numbers, is said to be a complex number
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In complex number z = a + ib, a is the real part, denoted by Re z and b is the imaginary part denoted by
Im z of the complex number z
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1 =i is called iota, which is a complex number
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The modulus of a complex number z = a + ib denoted by |z| is defined to be a non-negative real
number

a2  b2 , i
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| z |  a2  b2
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For any non-zero complex number z = a + ib (a ≠ 0, b ≠ 0), there exists a complex number
 b , denoted by 1 or z1, called the multiplicative inverse of z such that
a
i 2
2
2
z
a b
a  b2


 a  ib   


a
2

2

a b

i

 b


  1  i0  1
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Conjugate of a complex number z = a + ib, denoted as z , is the complex number a − ib
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topperlearning
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The number z = r(cos θ +isin θ) is the polar form of the complex number z = a + ib
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The plane having a complex number assigned to each of its points is called the Complex plane or
Argand plane
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Let a0 , a1, a2 ,
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Then, the real polynomial of a real variable
with real coefficients is given as

f  x   a0  a1x  a2x2 
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Let a0 , a1, a2 ,
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Then, the complex polynomial of
complex variable with complex coefficients is given as

f  x   a0  a1x  a2x2 
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 anxn is a polynomial of degree n
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A polynomial of second degree is called a quadratic polynomial
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Polynomials of degree 3 and 4 are known as cubic and biquadratic polynomials
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If f(x) is a polynomial, then f(x) = 0 is called a polynomial equation
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If f(x) is a quadratic polynomial, then f(x) = 0 is called a quadratic equation
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The general form of a quadratic equation is ax2  bx  c  0, a  0
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The values of the variable satisfying a given equation are called its roots
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A quadratic equation cannot have more than two roots
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Fundamental Theorem of Algebra states that ‘A polynomial equation of degree n has n roots
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Addition of two complex numbers: If z1 = a + ib and z2 = c + id be any two complex numbers, then
the sum
z1 + z2 = (a + c) + i(b + d)
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Sum of two complex numbers is also a complex number
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Addition of complex numbers satisfies the commutative law
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iii
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z+(-z)=(-z)+z=0
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Then, the product z1 z2 is defined as follows:
z1 z2 = (ac – bd) + i(ad + bc)
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Product of complex numbers is associative, i
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for any three complex numbers z1, z2, z3,
(z1 z2) z3 = z1 (z2 z3)
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There exists a complex number 1 + i0 (denoted as 1), called the multiplicative identity
such that z
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The distributive law: For any three complex numbers z1, z2, z3,
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z2 + z1
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(z1 + z2) z3 = z1
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z3
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= 2
i 2

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Identities for complex numbers
i (z1 + z2)² = z1² + z2² + 2z1
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ii (z1 − z2)² = z1² − 2z1z2 + z2²
iii (z1 + z2)³ = z1³ + 3z1²z2 + 3z1z2² + z2³
iv (z1 − z2)³ = z1³ − 3z1²z2 + 3z1z22 − z2³
v z1² − z2² = (z1 + z2) (z1 − z2)
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|z1 z2| = |z1||z2|
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z1z2  z1 z2
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 1  
provided z2 ≠ 0
 z  z2
2




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z  z  2Re  z 
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z  z  z is purely real
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zz  Re  z   Im  z 




2

2

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Hence there is no meaning in z1  z2
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Two complex numbers z1 and z2 are equal iff Re  z1   Re  z2  and Im  z1   Im  z2 
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The sum and product of two complex numbers are real if and only if they are conjugate of each other
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For any integer k, i4k = 1, i4k+1 = i, i4k+2 = −1, i4k+3 = −i
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If the discriminant D = b² − 4ac  0, then the equation has two real roots given by
x

b  b2  4ac
2a

or x 

b

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Roots of the quadratic equation ax² + bx + c = 0, where a, b and c R, a ≠ 0, when discriminant b² −
4ac < 0, are imaginary given by x 

b  4ac  b2i

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Complex roots occur in pairs
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If a, b and c are rational numbers and b2  4ac is positive and a perfect square, then
rational number and hence the roots are rational and unequal
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If b2  4ac  0 , then the roots of the quadratic equation are real and equal
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If b2  4ac  0 but it is not a perfect square, then the roots of the quadratic equation are irrational
and unequal
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Irrational roots occur in pairs
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A polynomial equation of n degree has n roots
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MATHS COMPLEX NUMBERS AND QUADRATIC EQUATIONS
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Different orientations of
z are as follows

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