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Complex Numbers And Quadratic Equations

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Complex Numbers and Quadratic Equations PDF Notes, Important Questions and Formulas

DEFINITION

Complex numbers are defined as expressions of the form a + ib where a, b ∊ R & begin mathsize 12px style straight i equals square root of negative 1 end root end style. It is denoted by z i.e. z=a + ib. ‘a’ is called as real part of z (Re z) and ‘b’ is called as imaginary part of z (Im z).

EVERY COMPLEX NUMBER CAN BE REGARDED AS

Purely real

Purely imaginary

Imaginary

If b = 0

If a = 0

If b ≠ 0

 

ALGEBRA OF COMPLEX NUMBER

The algebraic operations on complex numbers are similar to those on real numbers treating ‘I’ as a polynomial. In equalities in complex numbers are not defined. There is no validity if we say that complex number is positive or negative.

e.g.  z > 0, 4 + 2i < 2+4i are meaningless.

However in real numbers if a2 + b2=0 then a = 0 =b

but in complex numbers,

z12+z22=0 does not imply z1=z2=0

Equality In Complex Number

Two complex numbers z1= a1 + ib1 & z2 =a2 +ib2 are equal if and only if their real & imaginary parts coincide.

 

CONJUGATE COMPLEX

If z=a + ib then its conjugate complex is obtained by changing the sign of its imaginary & is denoted by begin mathsize 12px style straight z with bar on top text   i.e. end text stack text   end text straight z with bar on top equals straight a minus ib end style

GENERAL POLYNOMIAL

A function f defined by f(x) = anxn+an-1xn-1+……..+a1x+a0, where a0, a1, a2….. an ∊ R is called n degree polynomial while coefficient (an ≠0, n ∊ W) is real .If a0 , a2 ….an ∊ C, then it is called complex coefficient polynomial.


Quadratic polynomial

A polynomial of degree two in one variable f(x) = y = ax2 + bx + C, where a ≠ 0 & a, b, c ∊ R

a → leading coefficient, c → absolute term/ constant term

If a = 0 then y = bx + c → linear polynomial b ≠ 0

If a = 0, c = 0 then y = bx → odd linear polynomial


QUADRATIC EQUATION

1. The solution of the quadratic equation, ax2+ bx + c = 0 is given by

begin mathsize 12px style straight x equals fraction numerator negative straight b plus-or-minus square root of straight b squared minus 4 ac end root over denominator 2 straight a end fraction end style
The expression b2-4ac = D is called the discriminant of the quadratic equation.


2. If α & β are the roots of the quadratic equation

Ax2 + bx + c =0 then,

  1. α + β= -b/a
  2. αβ =c/a
  3. begin mathsize 12px style straight alpha minus straight beta equals square root of straight D divided by straight a end style


NATURE OF ROOTS

1. Consider the quadratic equation ax2 + bx + c = 0 where a, b, c ∊ R, a ≠ 0 then;

  1. D > 0 ⇔ roots are real & distinct (unequal)
  2. D = 0 ⇔ roots are real & coincident (equal)
  3. D < 0 ⇔ roots are imaginary
  4. If p +iq  is one root of a quadratic equation, then the other must be the conjugate p – iq & vice versa. ( p, q ∊ R & begin mathsize 12px style straight i equals square root of negative 1 end root end style

2. Consider the quadratic equation ax2+ bx + c = 0 where a, b, c ∊ Q & a≠0 then;

  1. If D > 0 & is a perfect square, then roots are rational & unequal.
  2. If begin mathsize 12px style straight alpha equals straight p plus square root of straight q end style is one root in this case, (where p is rational &  begin mathsize 12px style square root of straight q end style is a surd) then the other root must be the conjugate of it i.e. begin mathsize 12px style straight beta equals straight p minus square root of straight q end style & vice versa.

 

GRAPH OF QUADRATIC EXPRESSION

 begin mathsize 12px style straight y equals straight f left parenthesis straight x right parenthesis equals ax squared plus bx plus straight c text  or  end text left parenthesis straight y plus fraction numerator straight D over denominator 4 straight a end fraction right parenthesis equals straight a left parenthesis straight x plus fraction numerator straight b over denominator 2 straight a end fraction right parenthesis squared end style

  1. The graph between x, y is always a parabola.
  2. If a > 0 then the shape of the parabola is concave upwards & if a < 0 then the shape of the parabola is concave downwards.

  3. The co-ordinate of vertex are begin mathsize 12px style left parenthesis negative fraction numerator straight b over denominator 2 straight a end fraction comma fraction numerator straight D over denominator 4 straight a end fraction right parenthesis end style
  4. The parabola intersect the y-axis at point (0,c)
  5. The x-co-ordinate of point of intersection of parabola with x-axis are the real roots of the quadratic equation f(x) = 0. Hence the parabola may or may not intersect the x-axis at real points.

 

RELATION BETWEEN ROOTS & COEFICIENTS

A quadratic equation whose roots are α & β is

 (x – α)(x - β)=0

i.e. x2-(α + β)X +αβ=0

i.e. x2-(sum of roots)x + product of roots=0

 

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