Complex Numbers and Quadratic Equations formulas for JEE Main & Advanced

24 formula entries for Complex Numbers and Quadratic Equations, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Complex Number Definition

z = a + i b

When to use. When representing numbers that involve square roots of negatives or solving equations with no real solutions.

Trap. Confusing $i$ with a real number; $i^{2} = - 1$ is fundamental.

Powers of i

i^{4 k} = 1

i^{4 k + 1} = i

i^{4 k + 2} = - 1

i^{4 k + 3} = - i

When to use. Simplifying expressions with high powers of $i$ .

Trap. Forgetting that $i^{0} = 1$ .

Conjugate of a Complex Number

\overset{ˉ}{z} = a - i b

When to use. Rationalizing denominators, finding modulus, proving properties.

Trap. Forgetting that $\overline{z_{1} + z_{2}} = \overset{ˉ}{z_{1}} + \overset{ˉ}{z_{2}}$ but $\overline{z_{1} z_{2}} = \overset{ˉ}{z_{1}} \overset{ˉ}{z_{2}}$ .

Modulus of a Complex Number

∣ z ∣ = \sqrt{a^{2} + b^{2}}

When to use. Finding magnitude, distance between points, or in inequalities.

Trap. Confusing modulus with absolute value; $∣ z ∣$ is always non-negative real.

Argument of a Complex Number

\arg (z) = \tan^{- 1} (\frac{b}{a})

(with quadrant adjustment)

When to use. When converting to polar form or solving equations involving angles.

Trap. Using $\tan^{- 1} (b / a)$ directly without considering quadrant of $(a, b)$ .

Polar and Euler Forms

z = r (\cos θ + i \sin θ) = r e^{i θ}

When to use. Multiplying/dividing complex numbers, finding powers/roots via De Moivre.

Trap. Forgetting $\cos θ + i \sin θ$ is not $\cos θ - i \sin θ$ .

De Moivre's Theorem

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

When to use. Finding powers, roots, or simplifying trigonometric expressions.

Trap. Applying to $(\cos θ - i \sin θ)$ without converting to $\cos (- θ) + i \sin (- θ)$ .

nth Roots of Unity

z_{k} = e^{2 π i k / n} = \cos \frac{2 π k}{n} + i \sin \frac{2 π k}{n}

When to use. Solving $z^{n} = 1$ , factorizing $z^{n} - 1$ , or in geometry problems.

Trap. Forgetting that sum of all $n$ th roots of unity is 0.

Cube Roots of Unity

ω = \frac{- 1 + i \sqrt{3}}{2}

ω^{2} = \frac{- 1 - i \sqrt{3}}{2}

When to use. Simplifying expressions like $a^{3} + b^{3}$ factorization, or problems with $x^{2} + x + 1 = 0$ .

Trap. Forgetting $ω^{2} = \overset{ˉ}{ω}$ .

Quadratic Formula

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

D = b^{2} - 4 a c

When to use. Solving any quadratic equation with real coefficients.

Trap. Forgetting the $\pm$ sign; both roots must be considered.

Discriminant and Nature of Roots

D = b^{2} - 4 a c

When to use. Quickly determining root type without solving.

Trap. Thinking $D = 0$ gives one root; it gives two equal roots.

Sum and Product of Roots

α + β = - \frac{b}{a}

α β = \frac{c}{a}

When to use. Forming quadratic equations from roots, or finding symmetric functions of roots.

Trap. Forgetting the negative sign in sum.

Forming Quadratic from Roots

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

, where

S = α + β

P = α β

When to use. Given roots, find the quadratic equation.

Trap. Forgetting the minus sign before $S$ .

Complex Roots Occur in Conjugate Pairs

P (a + i b) = 0

, then

P (a - i b) = 0

for real coefficients.

When to use. When given one complex root of a real polynomial, find others.

Trap. Applying this to polynomials with complex coefficients; it fails.

Solving Quadratic with Complex Coefficients

z = \frac{- b \pm \sqrt{D}}{2 a}

where

D

may be complex.

When to use. When coefficients are complex numbers.

Trap. Assuming $\sqrt{D}$ is real; it's complex and has two values.

Square Root of a Complex Number

\sqrt{a + i b} = \pm (\sqrt{\frac{∣ z ∣ + a}{2}} + i sgn (b) \sqrt{\frac{∣ z ∣ - a}{2}})

When to use. Solving quadratic with complex discriminant, or finding roots of complex numbers.

Trap. Forgetting the $\pm$ sign; square root has two values.

Condition for Common Roots

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

When to use. When given two quadratics and asked if they share a root.

Trap. Forgetting that this condition holds for at least one common root; if both roots are common, $\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}$ .

Location of Roots (Quadratic Inequalities)

For both roots

> k

D \geq 0

a f (k) > 0

- b / (2 a) > k

When to use. Problems asking 'find $k$ such that both roots are greater than 2' etc.

Trap. Forgetting to check $D \geq 0$ for real roots.

Argand Plane and Geometry

∣ z_{1} - z_{2} ∣ = \sqrt{(a_{1} - a_{2})^{2} + (b_{1} - b_{2})^{2}}

When to use. Geometry problems involving circles, lines, or triangles in complex plane.

Trap. Thinking $∣ z_{1} + z_{2} ∣$ is sum of distances; it's not.

Locus Problems

Circle:

∣ z - z_{0} ∣ = r

; Line:

∣ z - z_{1} ∣ = ∣ z - z_{2} ∣

; Ray:

\arg (z - z_{0}) = θ

When to use. When asked to find the geometric figure described by a complex equation.

Trap. Forgetting that $∣ z - z_{1} ∣ = ∣ z - z_{2} ∣$ gives a line, not a circle.

Triangle Inequality

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

When to use. Proving bounds or inequalities involving moduli.

Trap. Forgetting the reverse triangle inequality $∣ z_{1} ∣ - ∣ z_{2} ∣ \leq ∣ z_{1} - z_{2} ∣$ .

De Moivre for Trigonometric Identities

\cos n θ + i \sin n θ = \sum_{k = 0}^{n} (\binom{n}{k}) \cos^{n - k} θ (i \sin θ)^{k}

When to use. Deriving formulas like $\cos 3 θ = 4 \cos^{3} θ - 3 \cos θ$ .

Trap. Forgetting to separate real and imaginary parts correctly.

Key Properties Summary

z \overset{ˉ}{z} = ∣ z ∣^{2}

When to use. In almost every complex number problem for simplification.

Trap. Forgetting that $\arg (0)$ is undefined.

Quadratic with Real Coefficients: Complex Roots

α \pm i β = \frac{- b \pm i \sqrt{∣ D ∣}}{2 a}

When to use. When solving quadratics with negative discriminant.

Trap. Forgetting that roots are conjugates only if coefficients are real.

Values to remember

$i = \sqrt{- 1}$ Complex Number Definition
$i^{2} = - 1$ Complex Number Definition
$i^{4} = 1$ Powers of i
$e^{i π} = - 1$ Polar and Euler Forms
$e^{i π / 2} = i$ Polar and Euler Forms
$ω = e^{2 π i / 3}$ nth Roots of Unity
$1 + ω + ω^{2} = 0$ nth Roots of Unity
$ω^{3} = 1$ Cube Roots of Unity

Read the full chapter: Complex Numbers and Quadratic Equations for JEE, with what JEE asks and the common traps.