Complex Numbers and Quadratic Equations Formula Sheet — JEE Main Mathematics

Q: What are the most important Complex Numbers and Quadratic Equations formulas for JEE Main?

This Complex Numbers and Quadratic Equations formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib, Argand plane and polar representation, Algebra of complex numbers, Modulus and argument of complex numbers — each shown with the key result and, where useful, a worked example.

Q: Is this Complex Numbers and Quadratic Equations formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

Q: How should I revise Complex Numbers and Quadratic Equations formulas?

Blurt the Complex Numbers and Quadratic Equations formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Every key Complex Numbers and Quadratic Equations formula, definition and theorem for JEE Main Mathematics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 11 sub-topics

Complex numbers as ordered pairs of reals
Representation of complex numbers in the form a+ib
Argand plane and polar representation
Algebra of complex numbers
Modulus and argument of complex numbers
Square root of a complex number
Triangle inequality
Quadratic equations in real and complex number system and their solutions
Relation between roots and coefficients
Nature of roots
Formation of quadratic equations with given roots

Complex Numbers — i, Algebra & Conjugate

Complex number:

z = a + ib

where

a = Re (z)

b = Im (z)

and

i = - 1

(so

i^{2} = - 1

). Two complex numbers are equal

\Leftrightarrow

their real parts are equal

and

their imaginary parts are equal. As an

ordered pair

of reals,

z = (a, b)

with

(a, b) + (c, d) = (a + c, b + d)

and

(a, b) (c, d) = (a c - b d, a d + b c)

, so

i = (0, 1)

Algebra of

z = a + ib

Operation	Result
$Add / subtract$	$(a + ib) \pm (c + i d) = (a \pm c) + i (b \pm d)$
$Multiply$	$(a + ib) (c + i d) = (a c - b d) + i (a d + b c)$
$Equal$	$a + ib = c + i d \Leftrightarrow a = c, b = d$
$- k (k > 0)$	$= i k$ (so $- 9 = 3 i$ )
$z \overset{z}{ˉ}$	$= a^{2} + b^{2} = ∣ z ∣^{2}$ (real)
$z^{- 1}$	$= \frac{z ˉ}{∣ z ∣ ^{2}}$ — divide via the conjugate

★ Remember · Powers of i repeat every 4

i^{4 k} = 1

i^{4 k + 1} = i

i^{4 k + 2} = - 1

i^{4 k + 3} = - i

. So to find

i^{2026}

, take the remainder of

2026

on division by

4

(here

2

)

\Rightarrow i^{2026} = i^{2} = - 1

🎯 Exam · Conjugate is your division tool

\overset{z}{ˉ} = a - ib

z \overset{z}{ˉ} = a^{2} + b^{2} = ∣ z ∣^{2}

, so

z^{- 1} = \frac{z ˉ}{∣ z ∣ ^{2}}

. To divide, multiply top and bottom by the conjugate of the denominator:

\frac{1}{a + ib} = \frac{a - ib}{a ^{2} + b ^{2}}

⚠️ Watch out · √a·√b ≠ √(ab) for negatives

- k = i k

for

k > 0

, but

- 2 \cdot - 3 \neq = 6

. Convert each to i-form first:

- 2 \cdot - 3 = (i 2) (i 3) = i^{2} 6 = - 6

🚫 Examiner Trap · Examiner traps

(1)

a b = ab

FAILS for negatives:

- 2 - 3 = - 6

, NOT

6

— always pull out i first. (2)

i^{n}

has period 4: reduce the exponent mod 4 (

i^{4 k} = 1

). (3) Equate real and imaginary parts SEPARATELY:

a + ib = c + i d \Rightarrow a = c

AND

b = d

. (4) To divide, multiply top and bottom by the conjugate of the DENOMINATOR;

z \overset{z}{ˉ} = ∣ z ∣^{2}

is real.

Modulus, Argument & Polar Form

A point on the Argand plane with its modulus r and argument theta, the polar and Euler forms, and the modulus/argument property list — The Argand point, the polar form, and the key properties.

Polar / Euler form

▸ $∣ z ∣ = r = x^{2} + y^{2}$ (distance from the origin).
▸ $ar g z = θ = tan^{- 1} \frac{y}{x}$ — fix the quadrant from the signs of x,y.
▸ $z = r (cos θ + i sin θ) = r e^{i θ}$ .
▸Principal argument: $θ \in (- π, π]$ .

★ Remember · Modulus & argument multiply/divide

∣ z_{1} z_{2} ∣ = ∣ z_{1} ∣∣ z_{2} ∣

and

ar g (z_{1} z_{2}) = ar g z_{1} + ar g z_{2}

;

\frac{z _{1}}{z _{2}} = \frac{∣ z _{1} ∣}{∣ z _{2} ∣}

and

ar g \frac{z _{1}}{z _{2}} = ar g z_{1} - ar g z_{2}

. Multiplication adds angles — the seed of rotation.

🎯 Exam · Triangle inequality

∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣

and

∣ z_{1} ∣ - ∣ z_{2} ∣ \leq ∣ z_{1} - z_{2} ∣

. Equality in the first holds when

z_{1}, z_{2}

point the same way (

ar g z_{1} = ar g z_{2}

⚠️ Watch out · tan⁻¹ alone gives the wrong quadrant

tan^{- 1} (y / x)

returns a value in

(- \frac{π}{2}, \frac{π}{2})

, so for z in the 2nd/3rd quadrant you must add or subtract

π

. Always sketch the Argand point before quoting

ar g z

🚫 Examiner Trap · Examiner traps

(1)

ar g z = tan^{- 1} \frac{y}{x}

ONLY gives the right quadrant after you check the signs of x,y — a raw calculator value can be off by

π

. (2) Principal argument lies in

(- π, π]

. (3)

∣ z_{1} + z_{2} ∣ \leq ∣ z_{1} ∣ + ∣ z_{2} ∣

(triangle) with equality only when they are like-directed. (4)

ar g (z_{1} z_{2}) = ar g z_{1} + ar g z_{2}

but may need

\pm 2 π

to stay principal;

∣ z^{n} ∣ = ∣ z ∣^{n}

De Moivre's Theorem, nth Roots & Cube Roots of Unity

De Moivre's theorem:

(cos θ + i sin θ)^{n} = cos n θ + i sin n θ

for integer n. In Euler form this is just

(e^{i θ})^{n} = e^{in θ}

— the quickest way to take powers and roots of complex numbers.

Cube roots of unity 1, ω, ω²

▸ $ω = \frac{- 1 + i 3}{2}$ , $ω^{2} = \frac{- 1 - i 3}{2}$ , $\overset{ω}{ˉ} = ω^{2}$ .
▸ $ω^{3} = 1$ and $1 + ω + ω^{2} = 0$ .
▸ $ω^{3 k} = 1, ω^{3 k + 1} = ω, ω^{3 k + 2} = ω^{2}$ .
▸They are the vertices of an equilateral triangle on $∣ z ∣ = 1$ .

🎯 Exam · n-th roots of unity

The n solutions of

z^{n} = 1

are

e^{2 πik / n}

k = 0, 1, \dots, n - 1

— equally spaced on the unit circle. Their

sum is 0

and they form a regular n-gon.

★ Remember · ω turns sums into zero

Because

1 + ω + ω^{2} = 0

, any symmetric combination collapses:

a + bω + c ω^{2}

simplifies fast, and

a^{3} - b^{3} = (a - b) (a - bω) (a - b ω^{2})

a^{2} + ab + b^{2} = (a - bω) (a - b ω^{2})

⚠️ Watch out · Reduce the exponent mod 3 (or mod 4 for i)

For high powers of

ω

use

ω^{3} = 1

(work mod 3); for high powers of i use

i^{4} = 1

(work mod 4). E.g.

ω^{100} = ω^{99} \cdot ω = ω

since

99 = 3 \times 33

🚫 Examiner Trap · Examiner traps

(1) The n n-th roots are EQUALLY spaced on

∣ z ∣ = 1

\frac{2 π}{n}

apart — find one, then rotate. (2) For

ω

(cube root of unity):

1 + ω + ω^{2} = 0

and

ω^{3} = 1

— reduce any power mod 3. (3) Sum of all n-th roots

= 0

; product

= (- 1)^{n + 1}

. (4)

ω

and

ω^{2}

are conjugates:

\overset{ω}{ˉ} = ω^{2}

; use

a^{2} + ab + b^{2} = (a - bω) (a - b ω^{2})

Argand Plane — Geometry, Rotation & Locus

Rotation of a point by multiplying by e^{i theta}, the standard loci (circle, perpendicular bisector, ray, Apollonius circle), and geometry shortcuts — Rotation about the origin and the standard complex loci.

★ Remember · Multiplying by

e^{i θ}

rotates

z e^{i θ}

rotates z by angle

θ

about the origin (modulus unchanged). To rotate

z_{2}

about another point

z_{1}

z^{'} = z_{1} + (z_{2} - z_{1}) e^{i θ}

. Multiplying by i is a

9 0^{\circ}

rotation.

Standard loci

▸ $∣ z - z_{0} ∣ = r$ : circle, centre $z_{0}$ , radius r.
▸ $∣ z - a ∣ = ∣ z - b ∣$ : perpendicular bisector of segment $ab$ .
▸ $ar g (z - z_{0}) = α$ : a ray starting at $z_{0}$ .
▸ $\frac{z - a}{z - b} = k$ : circle if $k \neq = 1$ , straight line if $k = 1$ .

🎯 Exam · Read the geometry from the algebra

∣ z - z_{1} ∣ + ∣ z - z_{2} ∣ = 2 a

is an ellipse (foci

z_{1}, z_{2}

);

∣ z - z_{1} ∣ - ∣ z - z_{2} ∣ = 2 a

a hyperbola. Three points are collinear

\Leftrightarrow \frac{z _{3} - z _{1}}{z _{2} - z _{1}}

is real.

★ Remember · Distance & section work like coordinates

∣ z_{1} - z_{2} ∣

is the distance between the points, and the section/midpoint formulas carry over directly — so coordinate-geometry intuition transfers straight onto the Argand plane.

🚫 Examiner Trap · Examiner traps

(1) Multiplying by

e^{i θ}

ROTATES by

θ

about O (no scaling, since

∣ e^{i θ} ∣ = 1

). (2)

∣ z - z_{0} ∣ = r

is a CIRCLE;

∣ z - a ∣ = ∣ z - b ∣

is the perpendicular BISECTOR of ab (a line, not a circle). (3)

\frac{z - a}{z - b} = k

is a circle for

k \neq = 1

but a LINE when

k = 1

. (4) Rotate

z_{2}

about

z_{1}

θ

z^{'} = z_{1} + (z_{2} - z_{1}) e^{i θ}

— subtract the centre first.

Quadratic Equations — Discriminant & Nature of Roots

Quadratic equation:

a x^{2} + b x + c = 0

(

a \neq = 0

) has roots

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

. The quantity

D = b^{2} - 4 a c

(the

discriminant

) decides everything about the nature of the roots.

Nature of roots (real

a, b, c

)

Discriminant $D = b^{2} - 4 a c$	Roots
$D > 0$	real and distinct
$D = 0$	real and equal $(= - \frac{b}{2 a})$
$D < 0$	conjugate complex pair $p \pm i q$
$D > 0$ , perfect square	rational (if $a, b, c$ rational)
$D > 0$ , not perfect square	irrational conjugate surds $p \pm q$

🎯 Exam · Conjugate-pair rule

With

real

coefficients, complex roots occur in conjugate pairs

p \pm i q

. With

rational

coefficients, irrational (surd) roots occur in conjugate pairs

p \pm q

. So a quadratic with one root

2 + 3

must have

2 - 3

as the other.

★ Remember · Degree n ⇒ n roots

By the fundamental theorem of algebra, a degree-n polynomial has exactly n roots (counting multiplicity and complex roots). A quadratic always has

two

roots — real or complex.

⚠️ Watch out · ‘Real coefficients’ is the catch

The conjugate-pair rule needs real coefficients. If the coefficients are themselves complex, the roots need

not

be conjugates — solve directly with the formula instead.

🚫 Examiner Trap · Examiner traps

(1)

D = b^{2} - 4 a c

D > 0

real distinct,

D = 0

real equal,

D < 0

complex conjugate — and 'rational roots' needs D a PERFECT SQUARE with rational a,b,c. (2) Complex/surd roots occur in CONJUGATE PAIRS ONLY when coefficients are real/rational. (3)

D = 0

does NOT mean 'no roots' — it means one repeated root

- \frac{b}{2 a}

. (4) A degree-n polynomial has exactly n roots counting multiplicity.

Roots & Coefficients, Formation & Common Roots

★ Remember · Sum and product of roots

α, β

are the roots of

a x^{2} + b x + c = 0

α + β = - \frac{b}{a}

and

α β = \frac{c}{a}

. To build an equation from roots:

x^{2} - (α + β) x + α β = 0

, i.e.

x^{2} - S x + P = 0

Symmetric functions (S = α+β, P = αβ)

▸ $α^{2} + β^{2} = S^{2} - 2 P$ .
▸ $α - β = \pm S^{2} - 4 P$ (note $S^{2} - 4 P = D / a^{2}$ ).
▸ $α^{3} + β^{3} = S^{3} - 3 PS$ .
▸ $\frac{1}{α} + \frac{1}{β} = \frac{S}{P}$ .

🎯 Exam · Common roots

One

common root of

a_{1} x^{2} + b_{1} x + c_{1}

and

a_{2} x^{2} + b_{2} x + c_{2}

(c_{1} a_{2} - c_{2} a_{1})^{2} = (b_{1} c_{2} - b_{2} c_{1}) (a_{1} b_{2} - a_{2} b_{1})

Both

roots common:

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}

★ Remember · Read conditions off the coefficients

Reciprocal roots

\Rightarrow c = a

; roots equal in magnitude, opposite sign

\Rightarrow b = 0

; one root zero

\Rightarrow c = 0

; roots of opposite sign

\Rightarrow \frac{c}{a} < 0

(product negative).

🚫 Examiner Trap · Examiner traps

(1) Sign trap:

α + β = - \frac{b}{a}

(NEGATIVE

b / a

α β = \frac{c}{a}

. (2) Build the equation as

x^{2} - S x + P = 0

with

S =

sum,

P =

product — don't drop the minus on S. (3)

α^{2} + β^{2} = S^{2} - 2 P

and

∣ α - β ∣ = \frac{D}{∣ a ∣} = S^{2} - 4 P

. (4) Condition for a COMMON root:

(c_{1} a_{2} - c_{2} a_{1})^{2} = (b_{1} c_{2} - b_{2} c_{1}) (a_{1} b_{2} - a_{2} b_{1})

Quadratic Expression — Graph, Sign & Max/Min

Two parabolas with real roots alpha and beta. For a>0 it opens up with the vertex at x=-b/2a below the axis (minimum -D/4a); for a<0 it opens down with the vertex above (maximum -D/4a). f has the sign of a outside the roots and the opposite sign between them. — The parabola, its vertex/extremum, and sign analysis.

★ Remember · Vertex and extreme value

f (x) = a x^{2} + b x + c

is a parabola opening

up

a > 0

down

a < 0

. Vertex at

x = - \frac{b}{2 a}

; the extreme value is

- \frac{D}{4 a}

(a minimum if

a > 0

, a maximum if

a < 0

Sign of f(x)

▸ $D < 0$ : f(x) keeps the sign of a for $all$ x (no real root).
▸ $D \geq 0$ : f has the sign of a outside the roots, the opposite sign between them.
▸Always positive $\Leftrightarrow a > 0$ and $D < 0$ ; always negative $\Leftrightarrow a < 0$ and $D < 0$ .
▸Range of f: $[- D /4 a, \infty)$ if $a > 0$ , $(- \infty, - D /4 a]$ if $a < 0$ .

🎯 Exam · Where do the roots sit?

A number k lies

between

the roots

\Leftrightarrow a f (k) < 0

. Both roots

> k

needs

D \geq 0

a f (k) > 0

and

- \frac{b}{2 a} > k

(the vertex on the right side).

⚠️ Watch out · Don't forget the a > 0 / a < 0 split

Every sign and position statement flips with the sign of a. When you reduce a question to '

f (k) > 0

', remember it really means '

a f (k) > 0

' — check whether the parabola opens up or down first.

🚫 Examiner Trap · Examiner traps

(1) Parabola opens UP if

a > 0

(minimum) and DOWN if

a < 0

(maximum) — the extreme value is

- \frac{D}{4 a}

x = - \frac{b}{2 a}

. (2) f(x) keeps the sign of a for ALL x when

D < 0

. (3) 'k lies between the roots'

\Leftrightarrow a f (k) < 0

(a single condition); both roots

> k

needs

D \geq 0

a f (k) > 0

AND

- \frac{b}{2 a} > k

. (4) Don't forget the

a \neq = 0

requirement for a quadratic.

More JEE Main Mathematics formula sheets

Frequently Asked Questions

What are the most important Complex Numbers and Quadratic Equations formulas for JEE Main?

This Complex Numbers and Quadratic Equations formula sheet covers all the high-yield Mathematics formulas, definitions and theorems you need for JEE Main, across Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib, Argand plane and polar representation, Algebra of complex numbers, Modulus and argument of complex numbers — each shown with the key result and, where useful, a worked example.

Is this Complex Numbers and Quadratic Equations formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

How should I revise Complex Numbers and Quadratic Equations formulas?

Blurt the Complex Numbers and Quadratic Equations formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Also useful: all formula sheetsJEE Main previous-year papersmost important chapters.