Mathematics · JEE

Complex Numbers and Quadratic Equations Formula Sheet for JEE

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

This page lists 65+ JEE-relevant Complex Numbers and Quadratic Equations formulas organised by subtopic. Start with essential formulas, then important identities before supplementary shortcuts.

Q: Is this Complex Numbers and Quadratic Equations formula sheet aligned with JEE Main?

Yes. Every formula is mapped to the JEE Main Mathematics syllabus for Complex Numbers and Quadratic Equations, covering Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, Algebra of complex numbers, and more.

Q: How should I revise the Complex Numbers and Quadratic Equations formula sheet before JEE?

Revise essential formulas daily, important ones every 2–3 days, and supplementary results weekly. After each pass, solve 10–15 MCQs to test recall under exam conditions.

Q: Where can I practise Complex Numbers and Quadratic Equations MCQs after revising formulas?

Use the Online Practice or MCQs pages for the same unit on Goodmarks to convert formula recall into problem-solving speed.

Q: Does this replace NCERT for Complex Numbers and Quadratic Equations?

No — use this formula sheet for quick revision alongside NCERT and your coaching notes. Formulas here are a condensed reference, not a substitute for concept building.

65+ JEE formulas in this unit

Quick answer

The Complex Numbers and Quadratic Equations JEE formula sheet lists 65+ important formulas for JEE Main and Advanced, including essential identities from Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, Algebra of complex numbers, and more. Revise essential formulas first, then practise MCQs on Goodmarks.

Download-free JEE mathematics formula revision for Complex Numbers and Quadratic Equations. This unit-wise formula list covers 65+ exam-relevant results across Complex numbers as ordered pairs of reals, Representation of complex numbers in the form a+ib and their representation in a plane, Argand diagram, Algebra of complex numbers, and more, organised by subtopic for quick last-minute revision.

JEE Formula Sheet

65 formulas across 8 subtopics — organised for JEE Main & Advanced revision

Practise MCQs for this unit

Essential: 26Important: 26Supplementary: 13

Complex numbers as ordered pairs of reals

Ordered pair representation of a complex numberEssential

z=(a,b)\equiv a+ib

Variables: $a,b\in\mathbb R$ , $i=\sqrt{-1}$
Where used in JEE: Basic representation and conversion between ordered pair and algebraic form.

Equality of complex numbersEssential

a+ib=c+id\iff a=c\text{ and }b=d

Variables: $a,b,c,d\in\mathbb R$
Where used in JEE: Equating real and imaginary parts, solving unknowns in complex equations.

Addition in ordered pair formImportant

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

Variables: $a,b,c,d\in\mathbb R$
Where used in JEE: Basic operations in ordered pair form.

Multiplication in ordered pair formEssential

(a,b)(c,d)=(ac-bd,ad+bc)

Variables: $a,b,c,d\in\mathbb R$
Where used in JEE: Product of complex numbers using ordered pairs.

Additive identity and inverseSupplementary

(a,b)+(0,0)=(a,b),\quad -(a,b)=(-a,-b)

Variables: $a,b\in\mathbb R$
Where used in JEE: Basic algebraic properties of complex numbers.

Multiplicative identitySupplementary

(a,b)(1,0)=(a,b)

Variables: $a,b\in\mathbb R$
Where used in JEE: Basic algebraic properties of complex numbers.

Representation of complex numbers in the form a+ib and their representation in a plane

Power of imaginary unitEssential

i^2=-1,\ i^3=-i,\ i^4=1,\ i^{4n}=1,\ i^{4n+1}=i,\ i^{4n+2}=-1,\ i^{4n+3}=-i

Variables: $n\in\mathbb Z$
Where used in JEE: Simplification of powers of $i$.

Real and imaginary partsEssential

\Re(z)=a,\ \Im(z)=b\text{ for }z=a+ib

Variables: $a,b\in\mathbb R$
Where used in JEE: Separating real and imaginary parts.

Purely real and purely imaginary conditionsImportant

z\text{ is real }\iff \Im(z)=0;\quad z\text{ is purely imaginary }\iff \Re(z)=0

Variables: $z=a+ib$
Where used in JEE: Classification of complex numbers.

Conjugate of a complex numberEssential

\overline{z}=a-ib\text{ for }z=a+ib

Variables: $a,b\in\mathbb R$
Where used in JEE: Rationalization, modulus identities, equation solving.

Point representation in planeEssential

z=a+ib\leftrightarrow P(a,b)

Variables: $a,b\in\mathbb R$
Where used in JEE: Geometric interpretation in the complex plane.

Argand diagram

Distance between two complex numbersImportant

|z_1-z_2|=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}

Variables: $z_1=x_1+iy_1,\ z_2=x_2+iy_2$
Where used in JEE: Locus problems and geometric interpretation in Argand plane.

Section formula in complex planeSupplementary

z=\dfrac{mz_2+nz_1}{m+n}

Variables: $z_1,z_2$ are endpoints, $z$ divides the segment internally in ratio $m:n$
Conditions: $m+n\neq 0$
Where used in JEE: Geometry of points in Argand plane.

Midpoint formula in complex planeSupplementary

z=\dfrac{z_1+z_2}{2}

Variables: $z_1,z_2\in\mathbb C$
Where used in JEE: Geometric problems on Argand diagram.

Collinearity condition in Argand planeImportant

z_1,z_2,z_3\text{ are collinear }\iff \dfrac{z_1-z_3}{z_2-z_3}\in\mathbb R

Variables: $z_1,z_2,z_3\in\mathbb C$
Conditions: $z_2\neq z_3$
Where used in JEE: Straight line and geometry problems in complex plane.

Perpendicularity condition in Argand planeSupplementary

\dfrac{z_1-z_3}{z_2-z_3}\in i\mathbb R

Variables: $z_1,z_2,z_3\in\mathbb C$
Conditions: $z_2\neq z_3$
Where used in JEE: Checking right angle in triangle formed by complex points.

Algebra of complex numbers

Basic algebraic operationsEssential

(a+ib)\pm(c+id)=(a\pm c)+i(b\pm d),\qquad (a+ib)(c+id)=(ac-bd)+i(ad+bc)

Variables: $a,b,c,d\in\mathbb R$
Where used in JEE: Fundamental operations in complex algebra.

Division of complex numbersEssential

\dfrac{a+ib}{c+id}=\dfrac{(ac+bd)+i(bc-ad)}{c^2+d^2}

Variables: $a,b,c,d\in\mathbb R$
Conditions: $c,d$ not both zero
Where used in JEE: Simplifying quotients and expressing in standard form.

Reciprocal of a complex numberImportant

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

Variables: $a,b\in\mathbb R$
Conditions: $a,b$ not both zero
Where used in JEE: Rationalization and simplification.

Conjugate identitiesEssential

\overline{z_1+z_2}=\overline z_1+\overline z_2,\quad \overline{z_1-z_2}=\overline z_1-\overline z_2,\quad \overline{z_1z_2}=\overline z_1\,\overline z_2,\quad \overline{\left(\frac{z_1}{z_2}\right)}=\frac{\overline z_1}{\overline z_2}

Variables: $z_1,z_2\in\mathbb C$
Conditions: For quotient, $z_2\neq 0$
Where used in JEE: Manipulation of expressions involving conjugates.

Double conjugation and real-imaginary extractionImportant

\overline{\overline z}=z,\quad z+\overline z=2\Re(z),\quad z-\overline z=2i\Im(z)

Variables: $z\in\mathbb C$
Where used in JEE: Separating real and imaginary parts.

Product with conjugateEssential

z\overline z=|z|^2

Variables: $z\in\mathbb C$
Where used in JEE: Finding modulus, simplifying fractions.

Polynomial with real coefficients and conjugate rootsEssential

P(z)\in\mathbb R[x],\ P(\alpha+i\beta)=0\implies P(\alpha-i\beta)=0

Variables: $\alpha,\beta\in\mathbb R$
Where used in JEE: Finding roots of equations with real coefficients.

Modulus and argument (or amplitude) of a complex number

Modulus of a complex numberEssential

|z|=\sqrt{a^2+b^2}\text{ for }z=a+ib

Variables: $a,b\in\mathbb R$
Where used in JEE: Magnitude, locus, trigonometric form.

Argument of a complex numberEssential

\arg z=\theta

where

z=|z|(\cos\theta+i\sin\theta)

Variables: $\theta\in\mathbb R$
Conditions: Defined for $z\neq 0$
Where used in JEE: Polar/trigonometric form and geometry in complex plane.

Principal argument rangeImportant

\Arg z\in(-\pi,\pi]

Variables: $z\in\mathbb C\setminus\{0\}$
Where used in JEE: Determining unique principal value of argument.

Trigonometric form of a complex numberEssential

z=r(\cos\theta+i\sin\theta)

Variables: $r=|z|,\ \theta=\arg z$
Conditions: $z\neq 0$
Where used in JEE: Multiplication, division, powers, roots.

Real and imaginary parts via modulus and argumentImportant

a=r\cos\theta,\quad b=r\sin\theta

Variables: $z=a+ib=r(\cos\theta+i\sin\theta)$
Conditions: $r\ge 0$
Where used in JEE: Conversion between Cartesian and polar forms.

Tangent of argumentImportant

\tan\theta=\dfrac{b}{a}

Variables: $z=a+ib=r(\cos\theta+i\sin\theta)$
Conditions: Use quadrant of $(a,b)$ ; valid when $a\neq 0$
Where used in JEE: Finding argument from Cartesian form.

Quadrant-wise argumentsImportant

\arg(a+ib)=\begin{cases}\tan^{-1}(b/a),&a>0\\ \pi+\tan^{-1}(b/a),&a<0,\ b\ge 0\\ -\pi+\tan^{-1}(b/a),&a<0,\ b<0\\ \pi/2,&a=0,\ b>0\\ -\pi/2,&a=0,\ b<0\end{cases}

Variables: $a,b\in\mathbb R$
Conditions: $a,b$ not both zero
Where used in JEE: Correct determination of argument in JEE problems.

Modulus of conjugate and negativeSupplementary

|\overline z|=|z|,\quad |-z|=|z|

Variables: $z\in\mathbb C$
Where used in JEE: Simplification in modulus problems.

Argument of conjugate and negativeImportant

\arg(\overline z)=-\arg z,\quad \arg(-z)=\arg z\pm\pi\pmod{2\pi}

Variables: $z\neq 0$
Where used in JEE: Transformations of arguments.

Multiplicative property of modulusEssential

|z_1z_2|=|z_1|\,|z_2|

Variables: $z_1,z_2\in\mathbb C$
Where used in JEE: Products, powers, trigonometric form.

Quotient property of modulusEssential

\left|\dfrac{z_1}{z_2}\right|=\dfrac{|z_1|}{|z_2|}

Variables: $z_1,z_2\in\mathbb C$
Conditions: $z_2\neq 0$
Where used in JEE: Division and simplification in modulus form.

Triangle inequalityImportant

|z_1+z_2|\le |z_1|+|z_2|

Variables: $z_1,z_2\in\mathbb C$
Where used in JEE: Bounding values and locus problems.

Reverse triangle inequalityImportant

\big||z_1|-|z_2|\big|\le |z_1-z_2|

Variables: $z_1,z_2\in\mathbb C$
Where used in JEE: Inequalities and geometric distance bounds.

Argument of product and quotientEssential

\arg(z_1z_2)=\arg z_1+\arg z_2\pmod{2\pi},\quad \arg\left(\dfrac{z_1}{z_2}\right)=\arg z_1-\arg z_2\pmod{2\pi}

Variables: $z_1,z_2\in\mathbb C\setminus\{0\}$
Where used in JEE: Polar form operations.

De Moivre's theoremEssential

[r(\cos\theta+i\sin\theta)]^n=r^n[\cos(n\theta)+i\sin(n\theta)]

Variables: $r\ge 0,\ \theta\in\mathbb R,\ n\in\mathbb Z$
Conditions: For negative $n$ , require $r\neq 0$
Where used in JEE: Powers of complex numbers and trigonometric identities.

nth roots of a complex numberImportant

z_k=r^{1/n}\left[\cos\left(\frac{\theta+2k\pi}{n}\right)+i\sin\left(\frac{\theta+2k\pi}{n}\right)\right],\quad k=0,1,\dots,n-1

Variables: $z=r(\cos\theta+i\sin\theta)$ , $n\in\mathbb N$
Conditions: $r>0$
Where used in JEE: Finding all roots of complex numbers.

Square roots of a complex numberSupplementary

\sqrt{a+ib}=\pm\left(\sqrt{\dfrac{r+a}{2}}+i\,\operatorname{sgn}(b)\sqrt{\dfrac{r-a}{2}}\right)

,\ \text{where }r=\sqrt{a^2+b^2}\)

Variables: $a,b\in\mathbb R$
Conditions: For $b=0$ , choose signs consistently
Where used in JEE: Explicit evaluation of complex square roots.

Quadratic equations in real and complex number systems and their solutions

Quadratic equation standard formEssential

ax^2+bx+c=0

Variables: $a,b,c\in\mathbb R$ or $\mathbb C$ , $a\neq 0$
Where used in JEE: General form of quadratic equations.

Quadratic formulaEssential

x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}

Variables: $a,b,c$ are coefficients
Conditions: $a\neq 0$
Where used in JEE: Finding roots in real or complex domain.

DiscriminantEssential

D=b^2-4ac

Variables: $a,b,c$ are coefficients
Conditions: $a\neq 0$
Where used in JEE: Determining nature of roots and applying quadratic formula.

Roots in terms of discriminantImportant

x=\dfrac{-b\pm\sqrt D}{2a}

Variables: $D=b^2-4ac$
Conditions: $a\neq 0$
Where used in JEE: Compact root representation.

Nature of roots for real coefficientsEssential

\begin{cases}D>0&\Rightarrow \text{two distinct real roots}\\ D=0&\Rightarrow \text{two equal real roots}\\ D<0&\Rightarrow \text{two non-real complex conjugate roots}\end{cases}

Variables: $D=b^2-4ac$
Conditions: $a,b,c\in\mathbb R,\ a\neq 0$
Where used in JEE: Classifying roots in quadratic equations.

Repeated root valueImportant

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

Variables: $a,b\in\mathbb R$ or $\mathbb C$
Conditions: Applicable when $D=0$
Where used in JEE: Equal roots and tangency-type problems.

Complex roots for negative discriminantImportant

x=\dfrac{-b}{2a}\pm i\dfrac{\sqrt{4ac-b^2}}{2a}

Variables: $a,b,c\in\mathbb R$
Conditions: $a\neq 0,\ D<0$
Where used in JEE: Expressing non-real roots explicitly.

Relations between roots and coefficients, nature of roots

Sum and product of rootsEssential

\alpha+\beta=-\dfrac{b}{a},\quad \alpha\beta=\dfrac{c}{a}

Variables: $\alpha,\beta$ are roots of $ax^2+bx+c=0$
Conditions: $a\neq 0$
Where used in JEE: Relation-based root problems.

Quadratic in factored formImportant

a(x-\alpha)(x-\beta)=0

Variables: $\alpha,\beta$ are roots, $a\neq 0$
Where used in JEE: Connecting roots with coefficients and forming equations.

Difference of rootsImportant

(\alpha-\beta)^2=\dfrac{b^2-4ac}{a^2}=\dfrac{D}{a^2}

Variables: $\alpha,\beta$ are roots, $D=b^2-4ac$
Conditions: $a\neq 0$
Where used in JEE: Comparing roots and checking equality/distinctness.

Condition for equal rootsEssential

\alpha=\beta\iff b^2-4ac=0

Variables: $\alpha,\beta$ are roots
Conditions: $a\neq 0$
Where used in JEE: Repeated root problems.

Condition for roots of opposite signImportant

\alpha\beta<0\iff \dfrac{c}{a}<0

Variables: $\alpha,\beta\in\mathbb R$
Conditions: Roots must be real
Where used in JEE: Sign analysis of roots.

Condition for both roots positiveImportant

\alpha,\beta>0\iff D\ge 0,\ -\dfrac{b}{a}>0,\ \dfrac{c}{a}>0

Variables: $\alpha,\beta$ are roots
Conditions: $a,b,c\in\mathbb R,\ a\neq 0$
Where used in JEE: Determining sign of roots.

Condition for both roots negativeImportant

\alpha,\beta<0\iff D\ge 0,\ -\dfrac{b}{a}<0,\ \dfrac{c}{a}>0

Variables: $\alpha,\beta$ are roots
Conditions: $a,b,c\in\mathbb R,\ a\neq 0$
Where used in JEE: Determining sign of roots.

Condition for reciprocal rootsSupplementary

\alpha\beta=1\iff c=a

Variables: $\alpha,\beta$ are roots of $ax^2+bx+c=0$
Conditions: $a\neq 0$
Where used in JEE: Special quadratic equations.

Condition for one root being negative reciprocal of the otherSupplementary

\alpha\beta=-1\iff c=-a

Variables: $\alpha,\beta$ are roots
Conditions: $a\neq 0$
Where used in JEE: Special root relation problems.

Condition for one root greater than k and the other less than kSupplementary

(k-\alpha)(k-\beta)<0\iff ak^2+bk+c\text{ has sign opposite to }a

Variables: $\alpha,\beta$ are real roots, $k\in\mathbb R$
Conditions: $a\neq 0$
Where used in JEE: Location of roots relative to a number.

The formation of quadratic equations with given roots

Equation formed from given rootsEssential

x^2-(\alpha+\beta)x+\alpha\beta=0

Variables: $\alpha,\beta$ are the given roots
Conditions: Monic equation
Where used in JEE: Forming quadratic equation from roots.

General quadratic with given rootsImportant

k\big[x^2-(\alpha+\beta)x+\alpha\beta\big]=0

Variables: $\alpha,\beta$ are roots, $k\neq 0$
Where used in JEE: Forming any equivalent quadratic equation with specified roots.

Equation with roots m times the original rootsImportant

x^2-m(\alpha+\beta)x+m^2\alpha\beta=0

Variables: $\alpha,\beta$ are original roots, new roots are $m\alpha,m\beta$
Where used in JEE: Transforming roots by scaling.

Equation with roots reciprocals of original rootsImportant

cx^2+bx+a=0

Variables: $\alpha,\beta$ are roots of $ax^2+bx+c=0$ ; new roots are $1/\alpha,1/\beta$
Conditions: $c\neq 0,\ \alpha\beta\neq 0$
Where used in JEE: Forming equation with reciprocal roots.

Equation with roots negatives of original rootsSupplementary

ax^2-bx+c=0

Variables: $\alpha,\beta$ are roots of $ax^2+bx+c=0$ ; new roots are $-\alpha,-\beta$
Conditions: $a\neq 0$
Where used in JEE: Root transformation problems.

Equation with roots shifted by hImportant

a(x-h)^2+b(x-h)+c=0

Variables: $\alpha,\beta$ are roots of $ax^2+bx+c=0$ ; new roots are $\alpha+h,\beta+h$
Conditions: $a\neq 0$
Where used in JEE: Forming equations under translation of roots.

Equation with roots transformed by x to x-h substitutionSupplementary

f(x-h)=0

gives roots

\alpha+h,\beta+h

\ \text{if }f(x)=0\text{ has roots }\alpha,\beta\)

Variables: $f(x)=ax^2+bx+c$
Where used in JEE: Standard root transformation technique.

Equation with roots 1/alpha and 1/beta from monic formSupplementary

\alpha\beta x^2-(\alpha+\beta)x+1=0

Variables: $\alpha,\beta\neq 0$
Where used in JEE: Direct formation from known roots.

Frequently asked questions

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