MathematicsChapter 1: Complex Numbers
Linear Systems and Factorising
A pair of equations with complex coefficients is still two (or four real) conditions.
Learning objectives
- State the main ideas and notation used in linear systems and factorising.
- Apply the rules of linear systems and factorising in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
A pair of equations with complex coefficients is still two (or four real) conditions. Solve by substitution or by treating i like a number and collecting. Factorising: z^2 + a^2 = (z + a i)(z - a i). Difference of squares still works in C. Quadratics pz^2 + qz + r = 0 with real p,q,r can have complex roots; complete the square or use the formula.
Formula
r = 0 with real p
Where / symbols
- LINEAR SYSTEMS AND FACTORISING
- A pair of equations with complex coefficients is still two (or four real) conditions. Solve by substitution or by treating i like a number and collecting. Factorising: z^2 + a^2 = (z + a i)(z - a i). Difference of squares still works in C. Quadratics pz^2 + qz + r = 0 with real p,q,r can have complex roots; complete the square or use the formula.
- 1. TWO EQUATIONS
- Example pattern: (1+i)z + w = 2, z + (1-i)w = i. Substitute, or write four real equations. Show each step on its own line.
- 2. FACTORS AND QUADRATICS
- z^2 + 4 = (z+2i)(z-2i). Complete square: z^2 + 2z + 5 = (z+1)^2 + 4 = (z+1)^2 - (2i)^2. Roots come in conjugate pairs if coefficients are real.
Relationship
- z2 + a2 = (z+ai)(z-ai)
Overview
Solve a pair, then factor:
Two Equations
Example pattern: (1+i)z + w = 2, z + (1-i)w = i. Substitute, or write four real equations. Show each step on its own line.
Examples Substitute one unknown · Or equate parts · Check in both equations
Factors and Quadratics
z^2 + 4 = (z+2i)(z-2i). Complete square: z^2 + 2z + 5 = (z+1)^2 + 4 = (z+1)^2 - (2i)^2. Roots come in conjugate pairs if coefficients are real.
Examples z^2+1=(z+i)(z-i) · Complete square · Formula still works
WHAT — LINEAR SYSTEMS AND FACTORISING
A pair of equations with complex coefficients is still two (or four real) conditions. Solve by substitution or by treating i like a number and collecting. Factorising: z^2 + a^2 = (z + a i)(z - a i). Difference of squares still works in C. Quadratics pz^2 + qz + r = 0 with real p,q,r can have complex roots; complete the square or use the formula.
RULE — 1. TWO EQUATIONS
Example pattern: (1+i)z + w = 2, z + (1-i)w = i. Substitute, or write four real equations. Show each step on its own line.
METHOD — 2. FACTORS AND QUADRATICS
z^2 + 4 = (z+2i)(z-2i). Complete square: z^2 + 2z + 5 = (z+1)^2 + 4 = (z+1)^2 - (2i)^2. Roots come in conjugate pairs if coefficients are real.
WORKED EXAMPLE
Worked: factor z^2 + 9. z^2 + 9 = z^2 - (3i)^2 = (z - 3i)(z + 3i) or (z + 3i)(z - 3i) same product
How to answer a linear systems and factorising question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- factor z^2 + 9.
- z2 + 9 = z2 - (3i)2
- = (z - 3i)(z + 3i)
- or (z + 3i)(z - 3i) same product
In simple words
Real-coefficient polynomials: non-real roots come in conjugate pairs.
Key points
- complex coeffs: still algebra
- z2 + a2 = (z+ai)(z-ai)
- collect Re and Im
- real coeffs -> conjugate roots
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Linear Systems and Factorising
Sticky Notes · Quick Revision
Core Idea
- A pair of equations with complex coefficients is still two (or four real) conditions.
- Solve by substitution or by treating i like a number and collecting.
- Factorising: z2 + a2 = (z + a i)(z - a i).
1. TWO EQUATIONS
- Example pattern: (1+i)z + w = 2, z + (1-i)w = i.
- Substitute, or write four real equations.
- Show each step on its own line.
2. FACTORS AND QUADRATICS
- z2 + 4 = (z+2i)(z-2i).
- Complete square: z2 + 2z + 5 = (z+1)2 + 4 = (z+1)2 - (2i)2.
- Roots come in conjugate pairs if coefficients are real.
Worked Example
- Worked: factor z^2 + 9.
- z2 + 9 = z2 - (3i)2
- = (z - 3i)(z + 3i)
- or (z + 3i)(z - 3i) same product
Common Misconception
- Real-coefficient polynomials: non-real roots come in conjugate pairs.
FBISE Exam Check
- State the rule for linear systems and factorising before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- A pair of equations with complex coefficients is still two (or four real) conditions.
- Solve by substitution or by treating i like a number and collecting.
- Factorising: z2 + a2 = (z + a i)(z - a i).
- Real-coefficient polynomials: non-real roots come in conjugate pairs.
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