Complex Number and Quadratic Equation Notes

Page # 1 COMPLEX NUMBER AND QUADRATIC EQUATION COMPLEX NUMBERS AND QUADRATIC EQUATION 1. QUADRATIC EQUATION A quadratic Polynomial f(x) when equated to zero is called Quadratic Equation. e.g 3x2 + 7x + 5 = 0, –9x2 + 7x + 5 = 0, 2 2 x + 2x = 0, 2x = 0 General form : ax2 + bx + c = 0 Where, a, b, c  C and a  0 3.1Roots of a Quadratic Equation The values of variable x which satisfy the quadratic equation is called as Roots (also called solutions or zeros) of a Quadratic Equation. 2. SOLUTION OF QUADRATIC EQUATION 4.1Factorization Method : Let ax2 + bx + c = a(x–) (x –  ) = 0 Then x =  and x =  will satisfy the given equation. Hence factorize the equation and equating each to zero gives roots of equation. e.g. 3x2 – 2x –1= 0 (x – 1)( 3x + 1) = 0 1 x = 1 ,  3 Hindu Method (Sri Dharacharya Method) : By completing the perfect square as b c ax2 + bx + c = 0   x2 + x + = 0 a a Adding and substracting LMF b I MNGH x  2a JK 2  b 2  4ac 4a2 OP PQ FG b IJ H 2a K 2 (A) Suppose a, b, c  R and a  0 then (i) If D > 0  Roots are Real and unequal (ii) If D = 0  Roots are Real and equal and each equal t – b/2a (iii) If D < 0  Roots are imaginary and unequal or complex conjugate. (B) Suppose a, b, c  Q, a  0 then (i) If D > 0 and D is perfect square  Roots are unequal and Rational (ii) If D > 0 and D is not perfect square  Roots are irrational and unequal Conjugate Roots : The Irrational and complex roots of a quadratic equation are always occurs in pairs. Therefore (a, b, c,Q) If One Root then Other Root  + i  – i  + Coefficient of x b = – a Coefficient of x 2 (ii) Product of Roots S      cons tan t term c = a coefficient of x 2 e.g. In equation 3x2 + 4x – 5 = 0 Sum of roots P =   = = 0 b  b2  4ac 2a Hence the Quadratic equation + bx + c = 0 (a S  0) has two roots, given by b  b 2  4ac  = 2a b  b 2  4ac and   2a Note : Every quadratic equation has two and only two roots. 3. NATURE OF ROOTS In Quadratic equation ax2 + bx + c = 0, the term b2 – 4ac is called discriminant of the equation, which plays an important role in finding the nature of the roots. It is denoted by D.  4. SUM AND PRODUCT OF ROOTS If  and  are the roots of quadratic equation ax2 + bx + c = 0, then, (i) Sum of Roots Which gives, x = ax2  –   or =  4 , 3 Product of roots P =  5 3 Relation between Roots and Coefficients If roots of quadratic equation ax2 + bx + c = 0 (a  0) (i) b  g = b  g 2 are  and  then  4 =  (ii)  2   2 = (  )2  2 =  D b2  4 ac = a a b 2  2ac (iii)  2   2 = (  ) (  ) 2  4 =– b b2  4ac a 2 =  D a a2 Page # 2 COMPLEX NUMBER AND QUADRATIC EQUATION b (iv)  3   3 =    g b 3 g – 3     =– e b b 2  3 ac j g (Replace x by x1/n) 3 (v)  3   3 =    + 3     = 2 = (vi)  4   4 = = 2 a + c = 0 {b  g 2 4 (v) k +  , k +   a(x– k)2 + b (x – k) 3  2ac 4 e I JK 2 2 2 e b b 2  2 ac j 2 }  2  2 –2  2  2 (Replace x by (x–k))   (vi) ,  k2 ax2 + kbx + c = 0 k k (Replace x by kx) c2 – 2 (vii)        = (Replace x by x/k) 2 a Fb GH (iv) k  ,k   ax2 + kbx + k2 c = 0 2 2 a2 je 2  2 (vii)  1/n,  1/n ; n N  a(xn)2 + b(xn)+ c = 0 j (Replace x by xn) Symmetric Expressions b 2  4ac The symmetric expressions of the roots  ,  a4 b g of an equation are those expressions in  and 2 (viii) 2 +  +  2 =    –   b  g    2  2  = (ix) =    b (xi) FG  IJ H K + FG  IJ H K 2 = 2  , which do not change by interchanging   2 2   2 of    and  . g  4  4 Some examples of symmetric expressions are– = ( 2   2 ) 2  2  2  2  2 2 5. FORMATION OF AN EQUATION WITH GIVEN ROOTS A quadratic equation whose roots are  and  is given by (x – ) (x – ) = 0  x2 – x – x +  = 0  x2 – ( + )x +  = 0 i.e x2 – (sum of Roots)x + Product of Roots = 0  x2 – Sx + P = 0 Equation in terms of the Roots of another Equation If ,  are roots of the equation ax2 + bx + c = 0 then the equation whose roots are (i) –  , –   ax2 – bx + c = 0 (Replace x by – x) (ii) 1/  , 1/   cx2 + bx + a = 0 (Replace x by 1/x) and  . To find the value of such an expression, we generally express that in terms  (x)  2   2 =      2 (x1/n) + c = 0 a3 b g b  g  4  {b  g   } eb  acj b  4ac b (iii)  n,  n ; n N  a(x1/n)2 + b 2 2 (i)    2 2 (ii)      (iii) 1 1    FG  IJ  FG  IJ H  K H K 2    (iv)   (v)      (vii)  3   3 (viii)  4   4 2 2 (vi) 2 6. THE REAL NUMBER SYSTEM Natural Number (N) : The number which are used for counting are known as Natural Number (also known as set of Positive Integers) i.e. N = {1, 2, 3, ........................} Whole Number (W) : If ' 0 ' is included in the set of natural numbers then we get the set of Whole Numbers i.e.W = {0, 1, 2, .....................} = {N} + {0} Integers (Z or I) :If negative natural number is included in the set of whole number then we get set of Integers i.e. Z or I = {.............–3, –2, –1, 0, 1, 2, 3,.............} Page # 3 COMPLEX NUMBER AND QUADRATIC EQUATION Rational Numbers (Q) : The numbers which are in the form of p/q (Where p, q  I, q  0) are called as Rational Number e.g. 2 1 , 3, 3 3 , 0.76, 1.2322 etc. Irrational Numbers : The numbers which are not rational i.e. which can not be expressed in p/q form or whose decimal part is non terminating non repeating but which may represent magnitude Integral powers of iota As we have seen i = Note : (i) Number zero is neither positive nor negative but is an even number. (ii) Square of a real number is always positive. (iii) Between two real numbers there lie infinite real numbers. (iv) The real number system is totally ordered, for any two numbers a, b R , we must say, either a < b or b < a or b = a. (v) All real number can be represented by points on a straight line. This line is called as number line. (vi) An Integer (Note) is said to be even, if it is divided by 2 other wise it is odd number. (vii) The magnitude of a physical quantity may be expressed as a real number times, a standard unit. (viii) Number ' 0 ' is an additive quantity (ix)Number '1' is multiplicative quantity. (x) Infinity () is the concept of the number greater than greatest you can imagine. It is not a number, it is just a concept, so we do not associate equality with it. (xi)Division by zero is meaning less. (xii) A non zero integer p is called prime if p  ± 1 and its only divisors are ± 1 and ± p. 7. IMAGINARY NUMBER Square root of a negative real number is an imaginary number, while solving equation x2 + 1 = 0 we get x = ± 1 which is imaginary. So the quantity 1 is denoted by 'i' called 'iota' thus i = 1 Further 3 , 2 , be expressed as ± i ................  4 ....................may 2, ± i 3 , ± 2i = – 1 i3 = – i and i4 = 1 Hence n  N , in = i, – 1, – i, 1 attains four values according to the value of n, so i4n + 1 = i, i4n + 2 = – 1 i4n + 3 = – i, i4n or i4n + 4 = 1 In other words in = (–1)n/2 if n is even integer in = (  1) of physical quantities. e.g. 2 , 51/3, , e,.....etc. Real Numbers (R) : The set of Rational and Irrational Number is called as set of Real Numbers i.e. N  W  Z  Q  R 1 so i2 n 1 2 i if n is odd integer. Note :– (i) i2 = i × i = (ii) a.b = 1 × a . b 1  1 possible iff both a, b are non-negative. (incorrect). It is also true for one positive and one negative no. e.g. (2)(3) =  2. 3 only invalid when both are negative means a.b  a . b iff a & b both are negative. (iii) ' i ' is neither positive, zero nor negative, Due to this reason order relations are not defined for imaginary numbers. 8. COMPLEX NUMBER A number of the form z = x+iy where x, y R and i = 1 is called a complex number where x is called as real part and y is called imaginary part of complex number and they are expressed as Re ( z ) = x, Im ( z ) = y Here if x = 0 the complex number is purely Imaginary and if y = 0 the complex number is purely Real. A complex number may also be defined as an ordered pair of real numbers any may be denoted by the symbol (a, b). If we write z = (a, b) then a is called the real part and b the imaginary part of the complex number z. Note : (i) Inequalities in complex number are not defined because 'i' is neither positive, zero nor negative so 4 + 3i < 1 + 2i or i< 0 or i > 0 is meaning less. (ii) If two complex numbers are equal, then their real and imaginary parts are separately equal. Thus if a + ib = c + id  a = c and b = d so if z = 0  x + iy =0  x =0 and y = 0 The student must note that x, y  R and x, y  0. Then if Page # 4 COMPLEX NUMBER AND QUADRATIC EQUATION x + y = 0  x = y is correct but x + i y = 0  x =– iy is incorrect Hence a real number cannot be equal to the imaginary number, unless both are zero. (iii) The complex number 0 is purely real and purely imaginary both. Representation of a Complex Number : (a) Cartesian Representation : The complex number z = x + iy = (x, y) is represented by a point P whose coordinates are refered to rectangular axis xox´ and yoy´, which are called real and imaginary axes respectively. Thus a complex number z is represented by a point in a plane, and corresponding to every point in this plane there exists a complex number such a plane is called Argand plane or Argand diagram or complex plane or gussian plane. Note : (i) Distance of any complex number from the origin is called the modulus of complex number and is denoted by |z|. Thus, = (ac  bd) 2 c d 2 + i and multiplicative inverse of z is Cancellation Law : (ii) Angle of any complex number with positive direction of x-axis is called amplitude or argument of z. y Thus, amp (z) = arg (z) =  = tan–1 . x (b)Polar Representation : If z = x + iy is a complex number then z = r (cos  + i sin ) is a polar form of complex number z where x = r cos , y = r sin  and z1  z 2  z1  z3 (c) Exponential Form : If z = x + iy is a complex number then its exponential form is z = r ei where r is modulas and  is amplitude of complex number. (d)Vector Representation : If z = x + iy is a complex number such that it represent point  P(x, y) then its vector representation is z= OP Algebraic operations with Complex Number: Addition (a + ib) + (c + id) = (a + c) + i (b + d) Subtraction (a + ib)–(c + id) = (a – c) + i (b – d) Multiplication (a + ib) (c + id) = ac +iad + ibc +i2 bd =(ac – bd) + i (ad + bc) (a  ib)(c  id) a  ib Division = c  id (c  id)(c  id) (when at least one of c and d is non zero) c 2  d2 Properties of Algebraic operations with Complex Number Let z, z1, z2 and z3 are any complex number then their algebraic operation satisfy following propertiesCommutativity : z1 + z2 = z2 + z1 & z1 z2 = z 2z 1 Associativity : (z1+ z2) + z3 = z1 + (z2 + z3) and (z1 z2) z3 = z1(z2 z3) Identity element : If O = (0, 0) and 1 = (1, 0) then z + 0 = 0 + z = z and z. 1 = 1. z = z. Thus 0 and 1 are the identity elements for addition and multiplication respectively. Inverse element : Additive inverse of z is – z |z| = x2  y 2 . 2 2 r = x  y = |z|. (bc  ad) z2  z1  z 3  z1 UV  z W 2  z3 z1 z2  z1 z 3 z1  0 z z  z z 2 1 3 1 and 1 . z UV  z W 2  z3 Distributivity : z 1 (z2 + z3) = z1 z 2 + z1 z 3 and (z2 + z3) z1 = z2 z 1 + z3 z 1 Conjugate Complex Number : The complex numbers z = (a, b) = a + ib and z = (a, – b) = a – ib where b  0 are said to be complex conjugate of each other (Here the complex conjugate is obtained by just changing the sign of i) e.g.conjugate of z = – 3 + 4i is z = – 3 – 4i. Note : Image of any complex number in x-axis is called its conjugate. Properties of Conjugate Complex Number Let z = a + ib and z = a – ib then (i) b zg = z (ii) z+ z =2a =2 Re(z)=purely real (iii) z – z = 2ib = 2i Im (z) = purely imaginary (iv) z z = a2 + b2 = |z|2 (v) z1  z 2 = z1 + z 2 (vi) z1  z 2 = z1 – z 2 Page # 5 COMPLEX NUMBER AND QUADRATIC EQUATION i = re–i 1 z1 = z 2 (vii) re (ix) zn (xi) z + z = 0 or z = – z  z = 0 or z is purely imaginary (xii) z = z = b zg  (viii) FG z IJ Hz K n 2 (x) z1 z2 = z1 z2 z is purely real 9. MODULUS OF A COMPLEX NUMBER If z = x + iy then modulus of z is equal to x2  y 2 and it is denoted by |z|. Thus z = x + iy  |z| = x2  y 2 Note : Modulus of every complex number is a non negative real number. Properties of modulus of a Complex Number (i) |z|  0 (ii) – |z|  Re (z)  |z| (iii) – |z|  Im (z)  |z| (iv) |z| = | z |= |– z| = |– z | (v) z z = |z|2 (vi) |z1 z2| = |z1| |z2| (vii) z1 z2 z1 = z2 (z2  0) (viii) |z|n = |zn|, n  N (ix) |z| = 1  z = (x) z –1 = 1 z z z 2 (xi) |z1 ± z2|2 = |z1|2 + |z2|2 ± 2Re (z1 z 2) (xii) |z1+z2|2 + |z1–z2|2 = 2 [|z1|2 + |z2|2] (xiii) |rei|= r 10.AMPLITUDE OR ARGUMENT OF A COMPLEX NUMBER The amplitude or argument of a complex number z is the inclination of the directed line segment representing z, with real axis For finding the argument of any complex number first check that the complex number is in which quadrant and then find the angle  and amplitude using the adjacent figure. Note : (i) Principle value of any complex number lies between –  <    (ii) Amplitude of a complex number is a many valued function. If  is the argument of a complex number then (2n + ) is also argument of complex number. (iii) Argument of zero is not defined. (iv) If a complex number is multiplied by iota (i) its amplitude will be increased by /2 and will be decreased by /2, if is multiplied by –i. (v) Amplitude of complex number in I and II quadrant is always positive and in III and IV is always negative. Properties of argument of a Complex Number (i) amp (any real positive number) = 0 (ii) amp (any real negative number) =  (iii) (iv) amp (z1 . z2) = amp (z1) + amp (z2) amp (z – z ) = ± /2 (v) amp Fz I GH z JK 1 = amp (z1) 2 – amp (z2) (vi) amp ( z ) = – amp (z) = amp (1/z) (vii) amp (– z) = amp (z) ±  (viii) amp (zn) = n amp (z) (ix) amp (iy) = /2 if y > 0 = – /2, if y < 0 (x) amp (z) + amp ( z ) = 0 11.SQUARE ROOT OF A COMPLEX NUMBER The square root of z = a + ib is a  ib = ± and ± LM MN LM MN z a 2 z a 2 i i z a 2 z a 2 OP PQ OP PQ for b > 0 for b < 0 Note : (i) The square root of i is ± FG 1  i IJ H 2K (Here b = 1) (ii) The square root of – i is ± FG 1  iIJ H 2K (Here b = –1) (iii) The square root of  is ± 2 (iv) The square root of 2 is ±  12.TRIANGLE INEQUALITIES (i) |z1 ± z2|  |z1| + |z2| (ii) |z1 ± z2|  |z1| – |z2|