Complex Numbers and Quadratic Equations Formula Sheet

Every must-know formula and result from the Complex Numbers and Quadratic Equations chapter, organised for last-minute revision. Scan top to bottom the night before the exam.

The Imaginary Unit and Powers of i

$$\boxed{i = \sqrt{-1} \quad \Longleftrightarrow \quad i^2 = -1}$$

Powers of $i$ cycle every 4:

$$\boxed{i^{4k} = 1, \quad i^{4k+1} = i, \quad i^{4k+2} = -1, \quad i^{4k+3} = -i}$$

Power	Value
$i^0$	$1$
$i^1$	$i$
$i^2$	$-1$
$i^3$	$-i$
$i^4$	$1$

The pattern repeats: $1, i, -1, -i, 1, i, \dots$ To find $i^n$, divide $n$ by 4 and read the remainder $r \in \{0,1,2,3\}$.

Complex Numbers: Form and Algebra

A complex number: $z = a + ib$, with $a = \text{Re}(z)$, $b = \text{Im}(z)$, and $a, b \in \mathbb{R}$.

$$\mathbb{C} = \{a + ib : a, b \in \mathbb{R}\}$$

Equality:

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

Operation	Formula
Addition	$(a+ib) + (c+id) = (a+c) + i(b+d)$
Subtraction	$(a+ib) - (c+id) = (a-c) + i(b-d)$
Multiplication	$(a+ib)(c+id) = (ac-bd) + i(ad+bc)$
Division	$\dfrac{a+ib}{c+id} = \dfrac{ac+bd}{c^2+d^2} + i\,\dfrac{bc-ad}{c^2+d^2}$

Key special product (always real):

$$\boxed{(a + ib)(a - ib) = a^2 + b^2}$$

Standard identities also hold for complex numbers:

Identity	Formula
$(z_1 + z_2)^2$	$z_1^2 + 2z_1z_2 + z_2^2$
$(z_1 - z_2)^2$	$z_1^2 - 2z_1z_2 + z_2^2$
$z_1^2 - z_2^2$	$(z_1 + z_2)(z_1 - z_2)$
$(z_1 + z_2)^3$	$z_1^3 + 3z_1^2z_2 + 3z_1z_2^2 + z_2^3$

Conjugate

$$\boxed{\bar{z} = a - ib}$$

Geometrically, $\bar{z}$ is the reflection of $z$ across the real axis: $(a,b) \to (a,-b)$.

Property	Result
Addition	$\overline{z_1 + z_2} = \bar{z_1} + \bar{z_2}$
Subtraction	$\overline{z_1 - z_2} = \bar{z_1} - \bar{z_2}$
Product	$\overline{z_1 z_2} = \bar{z_1}\,\bar{z_2}$
Quotient	$\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z_1}}{\bar{z_2}}$
Modulus link	$z\cdot\bar{z} = a^2 + b^2 = \|z\|^2$
Real part	$z + \bar{z} = 2a = 2\text{Re}(z)$
Imaginary part	$z - \bar{z} = 2ib = 2i\,\text{Im}(z)$
Double conjugate	$\overline{\bar{z}} = z$
Real test	$z = \bar{z} \iff z \text{ is real}$

Modulus

$$\boxed{|z| = \sqrt{a^2 + b^2}}$$

This is the distance from the origin to $z$ on the Argand plane.

Property	Result
Non-negative	$\|z\| \ge 0$, and $\|z\| = 0 \iff z = 0$
Conjugate	$\|\bar{z}\| = \|z\|$
Square	$z\cdot\bar{z} = \|z\|^2$
Product	$\|z_1 z_2\| = \|z_1\|\,\|z_2\|$
Quotient	$\left\|\frac{z_1}{z_2}\right\| = \frac{\|z_1\|}{\|z_2\|}$
Power	$\|z^n\| = \|z\|^n$

Distance between two complex numbers:

$$\boxed{|z_1 - z_2| = \sqrt{(a_1 - a_2)^2 + (b_1 - b_2)^2}}$$

Triangle Inequalities

$$\boxed{|z_1 + z_2| \le |z_1| + |z_2|}$$$$|z_1 - z_2| \ge \big||z_1| - |z_2|\big|$$$$|z_1 + z_2| \ge \big||z_1| - |z_2|\big|$$

Equality in $|z_1 + z_2| = |z_1| + |z_2|$ holds when $z_1$ and $z_2$ have the same argument (same direction).

Argument and Quadrants

$$\tan\theta = \frac{b}{a}, \qquad \boxed{-\pi < \arg(z) \le \pi}\ \text{(principal value)}$$

General argument: $\arg(z) = \theta + 2n\pi,\ n \in \mathbb{Z}$.

The naive $\tan^{-1}(b/a)$ is wrong outside Quadrant I — use the quadrant-aware table:

Position	Condition	Argument $\theta$
Quadrant I	$a>0, b>0$	$\tan^{-1}\frac{b}{a}$
Quadrant II	$a<0, b>0$	$\pi - \tan^{-1}\frac{\|b\|}{\|a\|}$
Quadrant III	$a<0, b<0$	$-\pi + \tan^{-1}\frac{\|b\|}{\|a\|}$
Quadrant IV	$a>0, b<0$	$-\tan^{-1}\frac{\|b\|}{\|a\|}$
+ve real axis	$b=0, a>0$	$0$
-ve real axis	$b=0, a<0$	$\pi$ (or $-\pi$)
+ve imag axis	$a=0, b>0$	$\frac{\pi}{2}$
-ve imag axis	$a=0, b<0$	$-\frac{\pi}{2}$

Properties of Argument

$$\boxed{\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)}$$$$\boxed{\arg\!\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)}$$$$\arg(\bar{z}) = -\arg(z), \qquad \arg(-z) = \arg(z) \pm \pi, \qquad \arg(z^n) = n\,\arg(z)$$

All argument identities hold up to multiples of $2\pi$; adjust the result back into $(-\pi, \pi]$.

Polar and Exponential Form

$$\boxed{z = r(\cos\theta + i\sin\theta) = re^{i\theta}}$$

where $r = |z|$ and $\theta = \arg(z)$. Conversions:

$$a = r\cos\theta, \qquad b = r\sin\theta$$

Euler’s formula and identity:

$$\boxed{e^{i\theta} = \cos\theta + i\sin\theta} \qquad \boxed{e^{i\pi} + 1 = 0}$$

Multiplication and Division in Polar Form

For $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$:

$$\boxed{z_1 z_2 = r_1 r_2\big[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)\big]}$$$$\boxed{\frac{z_1}{z_2} = \frac{r_1}{r_2}\big[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)\big]}$$

Rule: multiply/divide moduli, add/subtract arguments.

De Moivre’s Theorem

$$\boxed{(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)}$$

General form and exponential form:

$$\boxed{z^n = r^n(\cos n\theta + i\sin n\theta)} \qquad (re^{i\theta})^n = r^n e^{in\theta}$$

So $|z^n| = |z|^n$ and $\arg(z^n) = n\theta$.

nth Roots of a Complex Number

If $w = r(\cos\alpha + i\sin\alpha)$, the $n$ distinct $n$th roots of $w$ are:

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

• Modulus of each root is $r^{1/n}$ (the $n$th root of $r$, not $r/n$).
• The $n$ roots are equally spaced by $\frac{2\pi}{n}$ on a circle of radius $r^{1/n}$, forming a regular $n$-gon.

Roots of Unity

The $n$th roots of unity ($z^n = 1$):

$$\boxed{\omega_k = \cos\frac{2k\pi}{n} + i\sin\frac{2k\pi}{n} = e^{2\pi i k/n}}, \quad k = 0, 1, \dots, n-1$$

Property	Result
Sum	$1 + \omega + \omega^2 + \dots + \omega^{n-1} = 0$
Product	$1\cdot\omega\cdot\omega^2\cdots\omega^{n-1} = (-1)^{n-1}$
Structure	Form a GP (first term $1$, ratio $\omega$); vertices of a regular $n$-gon on the unit circle

For a non-real $n$th root $\alpha$ (since $z^n - 1 = (z-1)(z^{n-1} + \dots + z + 1)$):

$$\alpha^{n-1} + \alpha^{n-2} + \dots + \alpha + 1 = 0$$

For a non-real 5th root: $\alpha^4 + \alpha^3 + \alpha^2 + \alpha + 1 = 0$, hence $\alpha + \alpha^2 + \alpha^3 + \alpha^4 = -1$.

Cube Roots of Unity ($\omega$)

$$\boxed{1, \quad \omega = \frac{-1 + i\sqrt{3}}{2}, \quad \omega^2 = \frac{-1 - i\sqrt{3}}{2}}$$

In exponential form $\omega = e^{2\pi i/3}$; the three roots sit at $0$, $\frac{2\pi}{3}$, $\frac{4\pi}{3}$ on the unit circle (equilateral triangle).

Property	Formula	Remember
Definition	$\omega^3 = 1$	Cube gives 1
Sum	$1 + \omega + \omega^2 = 0$	Trinity makes zero
Shortcut	$\omega + \omega^2 = -1$	From the sum
Conjugate	$\omega^2 = \bar{\omega}$	Conjugate pair
Product	$\omega\cdot\omega^2 = 1$	Reciprocals
Power cycle	$\omega^{3k} = 1,\ \omega^{3k+1} = \omega,\ \omega^{3k+2} = \omega^2$	Period 3
Product 1	$(1 - \omega)(1 - \omega^2) = 3$	High-yield
Product 2	$(1 + \omega)(1 + \omega^2) = 1$	High-yield

Sum of $n$th powers:

$$1^n + \omega^n + (\omega^2)^n = \begin{cases} 3 & n \equiv 0 \pmod 3 \\ 0 & \text{otherwise} \end{cases}$$

Useful factorisations via $\omega$:

$$x^3 + 1 = (x+1)(x+\omega)(x+\omega^2)$$$$a^3 + b^3 + c^3 - 3abc = (a+b+c)(a + b\omega + c\omega^2)(a + b\omega^2 + c\omega)$$

Geometric Loci on the Argand Plane

Equation	Geometric meaning
$z = a + ib$	Point $(a, b)$
$\bar{z}$	Reflection of $z$ across the real axis
$-z$	Reflection of $z$ through the origin ($180^\circ$ rotation)
$\|z - z_0\| = r$	Circle, centre $z_0$, radius $r$
$\|z - z_0\| < r$	Interior of that circle
$\|z - z_0\| > r$	Exterior of that circle
$\|z - z_1\| = \|z - z_2\|$	Perpendicular bisector of segment $z_1 z_2$
$\|z - z_1\| + \|z - z_2\| = $ const	Ellipse with foci $z_1, z_2$

Addition of complex numbers obeys the parallelogram law (vector addition); $z_1 - z_2$ is the vector from $z_2$ to $z_1$.

Quadratic Equations

For $ax^2 + bx + c = 0$ ($a \neq 0$):

$$\boxed{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-b \pm \sqrt{D}}{2a}}$$$$\boxed{D = b^2 - 4ac \quad (\text{discriminant})}$$

When $D < 0$: $\;x = \dfrac{-b \pm i\sqrt{|D|}}{2a}$ (complex conjugate roots).

Nature of Roots

Discriminant	Nature of roots
$D > 0$	Two distinct real roots
$D = 0$	Two equal real roots $\left(x = -\frac{b}{2a}\right)$
$D < 0$	Two complex conjugate roots

For rational coefficients ($a, b, c \in \mathbb{Q}$):

Discriminant	Nature of roots
$D > 0$, perfect square	Distinct rational roots
$D > 0$, not a perfect square	Irrational conjugate-surd roots
$D = 0$	Equal rational roots
$D < 0$	Complex conjugate roots

Sum, Product and Formation (Vieta)

$$\boxed{\alpha + \beta = -\frac{b}{a}} \qquad \boxed{\alpha\beta = \frac{c}{a}}$$

For monic $x^2 + px + q = 0$: $\;\alpha + \beta = -p,\ \alpha\beta = q$.

Equation from given roots:

$$\boxed{x^2 - (\alpha + \beta)x + \alpha\beta = 0} \quad\Longleftrightarrow\quad (x - \alpha)(x - \beta) = 0$$

Symmetric-function shortcuts:

$$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$$$$\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)$$

Complex Conjugate Roots (real coefficients)

If $\alpha = p + iq$ is a root, then $\bar{\alpha} = p - iq$ is also a root, and:

• $\alpha + \bar{\alpha} = 2p$ (real)
• $\alpha\bar{\alpha} = p^2 + q^2 = |\alpha|^2$ (real, positive)
• $|\alpha| = |\bar{\alpha}|$ and $\arg(\alpha) = -\arg(\bar{\alpha})$

Special Result Worth Remembering

$x^2 + x + 1 = 0$ has roots $\omega$ and $\omega^2$ (the non-real cube roots of unity), and $D = 1 - 4 = -3 < 0$.

Final Speed Drill

Related Topics

• Complex Numbers Basics
• Argand Diagram
• Modulus, Argument and Polar Form
• De Moivre’s Theorem
• Cube Roots of Unity
• Quadratic Equations with Complex Roots