Important Complex Numbers & Quadratic Equations Formulas for JEE and NEET

Definition

Complex numbers are defined as expressions of the form a + ib where a, b ∈ R & i = − 1. 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 if b = 0
Purely imaginary if a = 0
Imaginary if b ≠ 0
Note :
(a) The set R of real numbers is a proper subset of the Complex Numbers. Hence the  Complete Number system is  N ⊂ W ⊂ I ⊂ Q ⊂ R ⊂ C.
(b) Zero is both purely real as well as purely imaginary but not imaginary.
(c) i = −1 is called the imaginary unit. Also i² = −1  ;  i³ = −i  ;   i⁴ = 1 etc.
(d) √a √b = √ab only if atleast one of either a or b is non-negative.

Conjugate Complex

If z = a + ib then its conjugate complex is obtained by changing the sign of its imaginary part & is denoted by .
(i)  =  2 Re(z)
(ii)  =  2i Im(z)
(iii)  = a² + b² which is real
(iv) If z lies in the 1st quadrant then  in the 4th quadrant and  lies in the 2nd  quadrant.

Algebraic Operations

The algebraic operations on complex numbers are similiar to those on real numbers treating i as a polynomial. Inequalities 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 Numbers

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

Representation of Complex Numbers in Various Forms

(a) Cartesian Form (Geometric Representation): Every complex number  z = x + iy can be represented by a point on the cartesian plane known as complex plane (Argand diagram) by the  ordered pair (x, y). length OP is called modulus of the complex number denoted  by |z| & θ is called the argument or amplitude
eg.
θ = tan−1(y/x) (angle made by OP with positive x−axis)NOTE : (i) |z| is always non negative. Unlike real numbers |z| =  is not correct.
(ii) Argument of a complex number is a many valued function. If θ is the argument of a complex number then 2nπ + θ ; n ∈ I will also be the argument of that complex number. Any two arguments of a complex number differ by 2nπ.
(iii) The unique value of θ such that  – π < θ ≤ π is called the principal value of the argument.
(iv) Unless otherwise stated, amp z implies principal value of  the argument.
(v) By specifying the modulus & argument a complex number is defined completely. For the complex number 0 + 0 i the argument is not defined and this is the only complex number which is given by its modulus.
(vi) There exists a one-one correspondence between the points of the plane and the members of the set of complex numbers.(b) Trignometric / Polar  Representation :z = r (cos θ + i sin θ)  where |z| = r;   arg z = θ ;  (cosθ − i sinθ)
Note: cosθ + i sinθ is also written as CiS θ.
Also  are known as Euler's identities.(c) Exponential  Representation :z = re^iθ ; |z| = r ; arg z = θ ;

Properties of Conjugate/Moduli/Amplitude:
If z, z1, z₂ ∈ C  then;
(a)
 
(b)
 
|z₁ + z₂|² + |z₁ – z₂|² = 2 [|z₁|² – |z₂|²]
  [TRIANGLE INEQUALITY]
(c) 
(i) amp (z₁ . z₂) = amp z₁ + amp z₂ + 2 kπ. k ∈ I
(ii) amp  = amp z₁ − amp z₂ + 2 kπ ; k ∈ I
(iii) amp(zⁿ) = n amp(z) + 2kπ.
where proper value of k must be chosen so that RHS lies in (−π, π].

Vectorial Representation of Complex Numbers

Every complex number can be considered as if it is the position vector of that point. If the point P represents the complex number z then, NOTE : (i) If  are of unequal magnitude then
(ii) If  A, B, C & D  are four points representing the complex numbers  z1, z2, z3 & z4  then(iii) If z1, z2, z3 are the vertices of an equilateral triangle where z0 is its circumcentre then                                 (a)

Demoivre's Theorem
Statement : cos n θ + i sin n θ is the value or one of the values of (cos θ + i sin θ)ⁿ ¥ n ∈ Q. The theorem is very useful in determining the roots of any complex quantity
Note : Continued product of the roots of a complex quantity should be determined using theory of equations.

Cube Root of Unity 

(i) The cube roots of unity are 1,

(ii) If w is one of the imaginary cube roots of unity then 1 + w + w2 = 0. In general 1 + wr + w2r = 0 ; where r ∈ I but is not the multiple of 3.

(iii) In polar form the cube roots of unity are :

cos 0 + i sin 0 ;  cos

(iv) The three cube roots of unity when plotted on the argand plane constitute the verties of an equilateral triangle.

(v) The following factorisation should be remembered :

(a, b, c ∈ R & ω is the cube root of unity)

a3 − b3 = (a − b) (a − ωb) (a − ω2b) ; x2 + x + 1 = (x − ω) (x − ω2) ;

a3 + b3 = (a + b) (a + ωb) (a + ω2b) ; a3 + b3 + c3 − 3abc = (a + b + c) (a + ωb + ω2c) (a + ω2b + ωc)

n^th Roots of Unity

If 1, α1, α2, α3 ..... αn − 1 are then, n^th root of unity then :

(i) They are in G.P. with common ratio e^i(2π/n) &

(ii)  if p is not an integral multiple of n = n if p is an integral multiple of n

(iii) (1 − α1) (1 − α2) ...... (1 − αn − 1) = n & (1 + α1) (1 + α2) ....... (1 + αn − 1) = 0 if n is even and 1 if n is odd.

(iv) 1. α1. α2. α3 ......... αn − 1 = 1 or −1 according as n is odd or even.

Sum of Series

(i) cosθ + cos2θ + cos3θ + ..... + cosnθ =

(ii) sinθ + sin2θ + sin3θ + ..... + sin n θ = Note: If θ = (2π/n) then the sum of the above series vanishes.

Straight Lines and Circles in terms of Complex Numbers

(A) If z1 & z2 are two complex numbers then the complex number z =  divides the joins of z1 & z2 in the ratio m : n.

Note: (i) If a, b, c are three real numbers such that az1 + bz2 + cz3 = 0 ; where a + b + c = 0 and a, b, c are not all simultaneously zero, then the complex numbers z1, z2 & z, are collinear.

(ii) If the vertices A, B, C of a ∆ represent the complex nos. z1, z2, z3 respectively, then :

(a) Centroid of the ∆ ABC =

(b) Orthocentre of the ∆ ABC =

(c) Incentre of the ∆ ABC = (az1 + bz2 + cz3) ÷ (a + b + c).

(d) Circumcentre of the ∆ABC = : (Z1 sin 2A + Z2 sin 2B + Z3 sin 2C) ÷ (sin 2A + sin 2B + sin 2C) .

(B) amp(z) = θ  is a ray emanating from the origin inclined at an angle θ to the x− axis.

(C) |z − a| = |z − b| is the perpendicular bisector of the line joining a to b.

(D) The equation of a line joining z1 & z2 is given by ;

z = z1 + t (z1 − z2)  where t is a perameter.

(E) z = z1(1 + it) where t is a real parameter is a line through the point z1 & perpendicular to oz1.

(F) The equation of a line passing through z1 & z2 can be expressed in the determinant form as  = 0. This is also the condition for three complex numbers to be collinear.

(G) Complex equation of a straight line through two given points z1 & z2 can be written as  which on manipulating takes the form as   where r is real and α is a non zero complex constant.

(H) The equation of circle having centre z0 & radius ρ is : |z − z0| = ρ or  which is of the form   r is real centre − α & radius  Circle will be real if

(I) The equation of the circle described on the line segment joining z1 & z2 as diameter is :

(J) Condition for four given points z1, z2, z3 & z4 to be concyclic is, the number  is real. Hence the equation of a circle through 3 non collinear points z1, z2 & z3 can be taken as  is real ⇒

(a) Reflection points for a straight line: Two given points P & Q are the reflection points for a given straight line if the given line is the right bisector of the segment PQ. Note that the two points denoted by the complex numbers z₁ & z₂ will be the reflection points for the straight line  if and only if;  where r is real and α is non zero complex constant.

(b) Inverse points w.r.t. a circle :
Two points P & Q are said to be inverse w.r.t. a circle with centre 'O' and radius ρ, if :
(i) the point O, P, Q are collinear and on the same side of O.
(ii) OP.OQ = ρ².

Note that the two points z₁ & z₂ will be the inverse points w.r.t. the circle  if and only if

Ptolemy's Theorem

It states that the product of the lengths of the diagonals of a convex quadrilateral inscribed in a circle is equal to the sum of the lengths of the two pairs of its opposite sides.

i.e. |z1 − z3| |z2 − z4| = |z1 − z2| |z3 − z4| + |z1 − z4| |z2 − z3|.

Logarithm of a Complex Quantity

(i)(ii) ii represents a set of positive real numbers given by

Theory Of Equations(Quadratic Equations)

The general form of a quadratic equation in x is, ax² + bx + c = 0, where a, b, c ∈ R & a ≠ 0.

RESULTS : 

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

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;
(i) α + β = – b/a
(ii) αβ = c/a
(iii) α – β = √D/a.

NATURE  OF  ROOTS:
(A) Consider the quadratic equation ax² + bx + c = 0 where a, b, c ∈ R & a≠ 0 then
(i) D > 0 ⇔ roots are real & distinct (unequal).
(ii) D = 0 ⇔ roots are real & coincident (equal).
(iii) D < 0 ⇔ roots are imaginary.
(iv) If p + i q is one root of a quadratic equation, then the other must be the conjugate p − i q & vice versa. (p, q ∈ R & i =
(B) Consider the quadratic equation ax² + bx + c = 0 where a, b, c ∈ Q & a ≠ 0 then;
(i) If D > 0 & is a perfect square, then roots are rational & unequal.
(ii) If α = p + q is one root in this case, (where p is rational & √q is a surd)  then the other root must be the conjugate of it i.e. β = p − √q & vice versa.

4. 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.

5. Remember that a quadratic equation cannot have three different roots & if it has, it becomes an  identity.

6. Consider the quadratic expression, y = ax² + bx + c, a ≠ 0 &  a, b, c ∈ R then
(i) The graph between  x, y is always a parabola. If a > 0 then the shape of the parabola is concave upwards & if a < 0  then the shape of the parabola is concave downwards.
(ii) ∀ x ∈ R,  y > 0 only if a > 0 & b² − 4ac < 0 (figure 3).
(iii) ∀ x ∈ R,  y < 0 only if a < 0 & b² − 4ac < 0 (figure 6).
Carefully go through the 6 different shapes of the parabola given below.
  
  

SOLUTION OF QUADRATIC INEQUALITIES: 
ax² + bx + c > 0 (a ≠ 0).
(i) If D > 0, then the equation ax² + bx + c = 0 has two different roots x₁ < x₂.
Then a > 0  ⇒ x ∈ (−∞, x₁) ∪ (x₂, ∞)
a < 0 ⇒ x ∈ (x₁, x₂)
(ii) If D = 0, then roots are equal, i.e.  x₁ = x₂.
In that case a > 0 ⇒ x ∈ (−∞, x₁) ∪ (x₁, ∞)
a < 0 ⇒ x ∈ φ
(iii) Inequalities of the form  can be quickly solved using the method of intervals.

MAXIMUM & MINIMUM VALUE 
Maximum and Minimum value of y = ax² + bx + c occurs at  x = − (b/2a) according as ;  a < 0 or a > 0.

COMMON ROOTS OF 2 QUADRATIC EQUATIONS [ONLY ONE COMMON ROOT]: 
Let α be the common root of ax² + bx + c = 0 & a′x² + b′x + c′ = 0 Therefore a α² + bα + c = 0; a′α² + b′α + c′ = 0. By Cramer’s Rule  Therefore, α = 

 So the condition for a common root is (ca′ − c′a)² = (ab′ − a′b)(bc′ − b′c).

The condition that a quadratic function f (x, y) = ax² + 2 hxy + by² + 2gx + 2 fy + c may be resolved into two linear factors is that ;
abc + 2 fgh − af² − bg² − ch² = 0 OR

Theory of Equations

If  α1, α2, α3, ......αn are the roots of the equation; f(x) = a0xn + a1xn-1 + a2xn-2 + .... + an-1x + an = 0 where a0, a1, .... an are all real & a0 ≠ 0 then, ∑α1 =  Note :

(i) If α is a root of the equation f(x) = 0, then the polynomial f(x) is exactly divisible by (x − α) or (x − α) is a factor of f(x) and conversely .

(ii) Every equation of nth degree (n ≥ 1) has exactly n roots & if the equation has more than n roots, it is an identity.

(iii) If the coefficients of the equation f(x) = 0 are all real and α + iβ is its root, then α − iβ is also a root. i.e. imaginary roots occur in conjugate pairs.

(iv) If the coefficients in the equation are all rational & α + √β is one of its roots, then α − √β is also a root where  α, β ∈ Q & β  is not a perfect square.

(v) If there be any two real numbers 'a' & 'b' such that f(a) & f(b) are of opposite signs,  then f(x) = 0 must have atleast one real root between 'a' and 'b'.

(vi) Every equation f(x) = 0 of degree odd has atleast one real root of a sign opposite to that of its last term.

LOCATION OF ROOTS :
Let f (x) = ax² + bx + c, where  a > 0 & a, b, c ∈ R.
(i) Conditions for both the roots of f(x) = 0 to be greater than a specified number ‘d’ are b² − 4ac ≥ 0;  f (d) > 0  &  (− b/2a) > d.
(ii) Conditions for both roots of f (x) = 0 to lie on either side of the number ‘d’ (in other words the number ‘d’ lies between the roots of f (x) = 0) is f (d) < 0.
(iii) Conditions for exactly one root of f (x) = 0 to lie in the interval (d, e) i.e. d < x < e are  b² − 4ac > 0 & f (d). f (e) < 0.
(iv) Conditions that both roots of f (x) = 0 to be confined between the numbers p & q are (p < q). b² − 4ac ≥ 0; f (p) > 0; f (q) > 0 & p < (− b/2a) < q.

Logarithmic Inequalities

(i) For a > 1 the inequality 0 < x < y & log_a x < log_a y are equivalent.
(ii) For 0 < a < 1 the inequality 0 < x < y & log_a x > log_a y are equivalent.
(iii) If a > 1 then log_a x < p ⇒ 0 < x < a^p 
(iv) If a > 1 then log_a x > p ⇒ x > a^p 
(v) If 0 < a < 1 then log_a x < p ⇒ x > a^p 
(vi) If 0 < a < 1 then log_a x > p ⇒ 0 < x < a^p