Complex Numbers, Quadratics & Inequalities

1. Core Concept

Complex Numbers expand the real number system to solve equations like x²+1=0 by introducing i = √-1, bridging algebra and 2D geometry (the Argand plane). Quadratic Equations deal with finding roots of degree-2 polynomials. Linear Inequalities define ranges of valid solutions.

2. Complex Numbers

Algebraic Form: z = a + ib.
Polar Form: z = r(cos θ + i sin θ).
Modulus: |z| = √(a² + b²) (Distance from origin).
Conjugate: z̅ = a - ib (Reflection across the Real x-axis).
Identity: z · z̅ = |z|².

Fundamental Theorem of Algebra (FTOA)

Statement: Every polynomial equation of degree n ≥ 1 with complex coefficients has at least one complex root.

Key Corollaries for IAT:

Exactly n Roots: A polynomial of degree n has exactly n complex roots (counting multiplicities).
Conjugate Root Theorem: If the coefficients are real, complex roots must occur in conjugate pairs (a ± ib).
Irrational Pair Theorem: If the coefficients are rational, irrational roots of the form a ± √b occur in pairs.

3. Quadratic Equations

x = [-b ± √(b² - 4ac)] / 2a

Discriminant D = b² - 4ac.

Roots Relations:

Sum of roots (α + β) = -b/a
Product of roots (αβ) = c/a
Difference of roots (|α - β|) = √D / |a|

Nature of Roots (for real coefficients):

D > 0: Real & distinct.
D = 0: Real & equal.
D < 0: Complex conjugate pairs.

4. Linear Inequalities

Algebraic Solution Steps

To solve a linear inequality (e.g., ax + b < c):

Simplify both sides by removing brackets and collecting like terms.
Add or subtract the same number from both sides (inequality sign remains the same).
Multiply or divide both sides by the same positive number.
Golden Rule: If you multiply or divide by a negative number, you must reverse the inequality sign (< becomes >).

Interval Notation

Solutions are expressed using the following intervals:

(a, b) → a < x < b (Open: endpoints excluded)
[a, b] → a ≤ x ≤ b (Closed: endpoints included)
[a, b) → a ≤ x < b (Semi-open)
(−∞, a] → x ≤ a

Graphical Representation

Inequalities can be visualized on a number line as follows:

$Number Line Representation$

Comparing Open (x > a) vs. Closed (x ≥ b) endpoints.

Open Circle (○): Used for < or > (Endpoint not part of solution).
Closed Circle (●): Used for ≤ or ≥ (Endpoint is part of solution).

Wavy Curve Method (Method of Intervals):

To solve rational inequalities like (x-1)(x-3) ≤ 0:

Locate critical points (roots of numerator and denominator).
Place them on a number line. The sign usually alternates across intervals.
Pro Tip: Use open intervals () for roots that come from the denominator, even for slack inequalities.

5. Conceptual Insights

Geometry of i: Multiplying a complex number by i rotates its vector representation by 90° counter-clockwise.
Triangle Inequality: ||z₁| - |z₂|| ≤ |z₁ ± z₂| ≤ |z₁| + |z₂|.
Cross-Multiplication Trap: Never cross-multiply an inequality variable unless you are 100% sure it is strictly positive. Instead, bring everything to one side and use the Wavy Curve method.

5. Common Mistakes to Avoid

Inequality Sign Reversal: Forgetting to reverse the sign when multiplying/dividing by a negative number.
Conjugate Root Assumption: Assuming complex roots are always conjugates. This only applies if all coefficients are real.
Triangle Inequality: Confusing the direction; remember |z₁ + z₂| ≤ |z₁| + |z₂|.
Wavy Curve Errors: Not accounting for even multiplicities of roots (where sign does NOT change).

6. IAT Exam Focus Points

Cube Roots of Unity (1, ω, ω²):

Memorize identities: 1 + ω + ω² = 0 and ω³ = 1. If you see x²+x+1=0, the roots are ω and ω².

Locus in Argand Plane:

|z - z₁| = |z - z₂| describes the perpendicular bisector of the line segment joining z₁ and z₂.

Complex Conjugates:

Complex roots always occur in conjugate pairs (p ± iq) only if the coefficients of the quadratic are real numbers.

7. Practice Mock Test

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