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Complex Numbers and Quadratic Equations formulas for JEE

Every Complex Numbers and Quadratic Equations formula you need for JEE, grouped by concept.

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Addition of complex numbersz1+z2=(a+c)+i(b+d)z_1 + z_2 = (a+c) + i(b+d)Component-wise addition of two complex numbers.complexaddition
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All 36 Complex Numbers and Quadratic Equations formulas
01

Complex Numbers Basics

26 formulas

Addition of complex numbers

z1+z2=(a+c)+i(b+d)z_1 + z_2 = (a+c) + i(b+d)

Component-wise addition of two complex numbers.

complexaddition

Conjugate of quotient

(z1z2)‾=z1‾z2‾\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\overline{z_1}}{\overline{z_2}}

The conjugate distributes over division.

applies whenz2≠0z_2 \neq 0
complexconjugate

Conjugate of product

z1z2‾=z1‾⋅z2‾\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}

The conjugate distributes over multiplication.

complexconjugate

Conjugate of sum/difference

z1±z2‾=z1‾±z2‾\overline{z_1 \pm z_2} = \overline{z_1} \pm \overline{z_2}

The conjugate distributes over addition and subtraction.

complexconjugate

Conjugate

zˉ=a−ib\bar{z} = a - ib

The complex conjugate of z, reflecting it across the real axis.

complexconjugate

Division of complex numbers

z1z2=z1z2−1\frac{z_1}{z_2} = z_1 z_2^{-1}

Quotient of two complex numbers.

applies whenz2≠0z_2 \neq 0
complexdivision

Equality of complex numbers

z1=z2  ⟺  a=c and b=dz_1 = z_2 \iff a = c \text{ and } b = d

Two complex numbers are equal if and only if their real and imaginary parts are respectively equal.

applies whenz1=a+ibz_1 = a+ib, z2=c+idz_2 = c+id
complexequality

Cube of difference

(z1−z2)3=z13−3z12z2+3z1z22−z23(z_1-z_2)^3 = z_1^3 - 3z_1^2z_2 + 3z_1z_2^2 - z_2^3

Algebraic expansion for the cube of a difference of two complex numbers.

complexidentityalgebra

Cube of sum

(z1+z2)3=z13+3z12z2+3z1z22+z23(z_1+z_2)^3 = z_1^3 + 3z_1^2z_2 + 3z_1z_2^2 + z_2^3

Algebraic expansion for the cube of a sum of two complex numbers.

complexidentityalgebra

Difference of squares

z12−z22=(z1−z2)(z1+z2)z_1^2 - z_2^2 = (z_1-z_2)(z_1+z_2)

Factorization of the difference of squares in the complex plane.

complexidentityalgebra

Imaginary unit

i=−1i = \sqrt{-1}

Definition of the fundamental imaginary unit.

compleximaginary-unit

Square of difference

(z1−z2)2=z12−2z1z2+z22(z_1-z_2)^2 = z_1^2 - 2z_1z_2 + z_2^2

Algebraic expansion for the square of a difference of two complex numbers.

complexidentityalgebra

Square of sum

(z1+z2)2=z12+2z1z2+z22(z_1+z_2)^2 = z_1^2 + 2z_1z_2 + z_2^2

Algebraic expansion for the square of a sum of two complex numbers.

complexidentityalgebra

Multiplicative inverse

z−1=a−iba2+b2z^{-1} = \frac{a - ib}{a^2+b^2}

The reciprocal of a non-zero complex number in standard form.

applies whenz≠0z \neq 0
complexinverse

Powers of i

i4k=1,  i4k+1=i,  i4k+2=−1,  i4k+3=−ii^{4k}=1, \; i^{4k+1}=i, \; i^{4k+2}=-1, \; i^{4k+3}=-i

Cyclic reduction of integer powers of the imaginary unit.

applies whenk∈Zk \in \mathbb{Z}
complexpowers-of-i

Modulus-Conjugate Relation

zzˉ=∣z∣2z\bar{z} = |z|^2

The product of a complex number and its conjugate equals the square of its modulus.

complexmodulusconjugate

Modulus of quotient

∣z1z2∣=∣z1∣∣z2∣\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}

The modulus of a quotient is the quotient of the moduli.

applies whenz2≠0z_2 \neq 0
complexmodulus

Modulus of product

∣z1z2∣=∣z1∣∣z2∣|z_1 z_2| = |z_1| |z_2|

The modulus of a product is the product of the moduli.

complexmodulus

Modulus

∣z∣=a2+b2|z| = \sqrt{a^2+b^2}

The magnitude or absolute value of a complex number.

complexmodulus

Multiplication of complex numbers

z1z2=(ac−bd)+i(ad+bc)z_1 z_2 = (ac-bd) + i(ad+bc)

Product of two complex numbers in standard form.

complexmultiplication

Real and Imaginary parts via conjugate

Re(z)=z+zˉ2,  Im(z)=z−zˉ2i\text{Re}(z) = \frac{z+\bar{z}}{2}, \; \text{Im}(z) = \frac{z-\bar{z}}{2i}

Expressing Cartesian components using the complex conjugate.

complexconjugatejee-advanced

Real part of product

Re(z1z2)=Re(z1)Re(z2)−Im(z1)Im(z2)\text{Re}(z_1 z_2) = \text{Re}(z_1)\text{Re}(z_2) - \text{Im}(z_1)\text{Im}(z_2)

Expanding the real part of the product of two complex numbers.

complexreal-part

Standard form of a complex number

z=a+ibz = a + ib

Algebraic representation of a complex number where a and b are real numbers.

applies whena,b∈Ra, b \in \mathbb{R}
complexstandard-form

Difference of complex numbers

z1−z2=(a−c)+i(b−d)z_1 - z_2 = (a-c) + i(b-d)

Component-wise subtraction of two complex numbers.

complexsubtraction

Triangle inequality (Maximum)

∣z1±z2∣≤∣z1∣+∣z2∣|z_1 \pm z_2| \leq |z_1| + |z_2|

Upper bound for the modulus of a sum or difference.

complexinequalitiesjee-advanced

Triangle inequality (Minimum)

∣z1±z2∣≥∣∣z1∣−∣z2∣∣|z_1 \pm z_2| \geq ||z_1| - |z_2||

Lower bound for the modulus of a sum or difference.

complexinequalitiesjee-advanced
02

Complex Number Representation

6 formulas

De Moivre's Theorem

(cos⁡θ+isin⁡θ)n=cos⁡(nθ)+isin⁡(nθ)(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)

Formula for raising a complex number in polar form to an integer power.

applies whenn∈Zn \in \mathbb{Z}
complexde-moivrejee-advanced

Distance between complex numbers

d=∣z1−z2∣d = |z_1 - z_2|

Geometric distance between two points in the complex plane.

complexdistancejee-advanced

Distance from origin

d=x2+y2d = \sqrt{x^2+y^2}

Distance of a point P(x,y) from the origin in the Argand plane.

complexargand-planedistance

Euler form

z=reiθz = r e^{i\theta}

Exponential representation of a complex number.

applies whenr=∣z∣r = |z|
complexeulerjee-advanced

Polar form

z=r(cos⁡θ+isin⁡θ)z = r(\cos\theta + i\sin\theta)

Trigonometric representation of a complex number where r is modulus and theta is argument.

applies whenr=∣z∣r = |z|
complexpolarjee-advanced

Rotation Theorem (Coni Method)

z3−z1z2−z1=∣z3−z1z2−z1∣eiα\frac{z_3-z_1}{z_2-z_1} = \left|\frac{z_3-z_1}{z_2-z_1}\right| e^{i\alpha}

Rotation of the vector from z1 to z2 by angle alpha to align with the vector from z1 to z3.

applies whenRotation is counter-clockwise by angle α\alpha
complexrotationjee-advanced
03

Quadratic Equations and Complex Roots

4 formulas

Complex cube root of unity

ω=ei2π/3=−1+i32\omega = e^{i2\pi/3} = \frac{-1 + i\sqrt{3}}{2}

The primary non-real cube root of unity.

complexcube-rootsjee-advanced

Properties of cube roots of unity

1+ω+ω2=0 and ω3=11 + \omega + \omega^2 = 0 \text{ and } \omega^3 = 1

Sum and product relationships for the cube roots of 1.

complexcube-rootsjee-advanced

Roots of a quadratic equation

x=−b±i∣D∣2ax = \frac{-b \pm i\sqrt{|D|}}{2a}

Complex roots of a quadratic equation when the discriminant is negative.

applies whenD=b2−4ac<0D = b^2 - 4ac < 0
complexquadraticjee-advanced

Square root of negative real

−a=ia\sqrt{-a} = i\sqrt{a}

Expressing the square root of a negative real number using i.

applies whena>0a > 0
complexsquare-root
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