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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.
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Linear Systems and Factorising
What is it?
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.
In simple words
- 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.
Important terms
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.
What you should learn
- 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.
Examples
- Substitute one unknown
- Or equate parts
- Check in both equations
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.
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.
Recommended for Class 11
Full school-level explanation
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.
Solve a pair, then factor:
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
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.
Remember
- collect Re and Im
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.
Remember
- real coeffs -> conjugate roots
Examples
- z2+1=(z+i)(z-i)
- Complete square
- Formula still works
How to answer a linear systems and factorising question
How it works
- 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.
Remember
- complex coeffs: still algebra
- z2 + a2 = (z+ai)(z-ai)
- (z+3i)(z-3i)
- conjugate pairs
- collect Re and Im
- real coeffs -> conjugate roots
- complex coeffs: still algebra
- z2 + a2 = (z+ai)(z-ai)
How the ideas connect
Worked: factor z^2 + 9. z^2 + 9 = z^2 - (3i)^2 = (z - 3i)(z + 3i) or (z + 3i)(z - 3i) same product
Deeper understanding
Go beyond memorising Linear Systems and Factorising. Connect the definition to reasons, relationships, exceptions, and exam wording.
How the ideas connect
Worked: factor z^2 + 9. z^2 + 9 = z^2 - (3i)^2 = (z - 3i)(z + 3i) or (z + 3i)(z - 3i) same product
Common mistakes
Exam focus
Core Idea
1. TWO EQUATIONS
2. FACTORS AND QUADRATICS
Worked Example
Exam focus
Quick revision
- complex coeffs: still algebra
- z2 + a2 = (z+ai)(z-ai)
- (z+3i)(z-3i)
- conjugate pairs
- LINEAR SYSTEMS AND FACTORISING: A pair of equations with complex coefficients is still two (or four real) conditions. Solve by subst
- 1. TWO EQUATIONS: Example pattern: (1+i)z + w = 2, z + (1-i)w = i. Substitute, or write four real equations. Show each
Real-coefficient polynomials: non-real roots come in conjugate pairs.
For long answers on Linear Systems and Factorising: define clearly → explain the main parts → give one example → state one exam caution.
Exam Focus
Exam Focus
Important points
- complex coeffs: still algebra
- z2 + a2 = (z+ai)(z-ai)
- (z+3i)(z-3i)
- conjugate pairs
Important formulas
- complex coeffs: still algebra
- z2 + a2 = (z+ai)(z-ai)
- (z+3i)(z-3i)
Common exam traps
- Real-coefficient polynomials: non-real roots come in conjugate pairs.
- Do not use a remembered formula before checking that its conditions fit the question.
Long question structure
Definition → key parts → example → exam caution
Common Mistakes
Common Mistakes
- Mistake
Real-coefficient polynomials: non-real roots come in conjugate pairs.
- Mistake
Do not use a remembered formula before checking that its conditions fit the question.
- Mistake
Do not hide sign changes or restrictions; write them on the line where they are used.
- Mistake
Real-coefficient polynomials: non-real roots come in conjugate pairs.
Revision Zone
60-Second Revision
Quick cues from this topic — use them for last-minute revision.
complex coeffs: still algebra
z2 + a2 = (z+ai)(z-ai)
(z+3i)(z-3i)
conjugate pairs
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. Facto
z2 + a2 = (z+ai)(z-ai)
- 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. Facto
- Formula
- z2 + a2 = (z+ai)(z-ai)
- Example
- Substitute one unknown
- Critical warning
- Real-coefficient polynomials: non-real roots come in conjugate pairs.
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Exam Practice
Define
Q1. Define Linear Systems and Factorising.
Explain
Q2. Explain: TWO EQUATIONS.
Example
Q3. Give one clear example of Linear Systems and Factorising and say why it fits.
Mistake
Q4. State one common mistake students make about Linear Systems and Factorising.
Exam note
Q5. Write a short exam note on Linear Systems and Factorising: definition, two points, one example.
Sources & References
School-standard facts for Matric and Intermediate. Original explanations — not copied from textbooks.