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Algebra Tricks Every O Level & IGCSE Maths Student Must Know

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Core Algebra Foundations for Fast, Error-Free Working

  • Build automatic fluency with basic algebraic manipulation

    • Simplify expressions before substituting

    • Remove brackets cleanly using distributive law

    • Keep like terms together to avoid scattered working

  • Maintain strict sign awareness

    • Track negative signs clearly

    • Use brackets when substituting negative values

  • Apply BIDMAS consistently

    • Identify operations inside brackets first

    • Prioritise multiplication/division before addition/subtraction

  • Maintain neat, stepwise transformations

    • Prevent accidental term loss

    • Ensure examiner sees logical structure

  • Convert between forms when helpful

    • Rewrite complicated expressions into easier equivalents

High-Value Tricks for Expanding and Factorising

  • Expand brackets confidently

    • Multiply each term inside bracket

      • First × First

      • Outer × Outer

      • Inner × Inner

      • Last × Last

  • Expand double brackets using FOIL systematically

  • Identify perfect square patterns

    • (a + b)² = a² + 2ab + b²

    • (a − b)² = a² − 2ab + b²

  • Recognise difference of two squares

    • a² − b² = (a + b)(a − b)

  • Factorise quadratics by inspection

    • Identify pair of numbers that multiply to constant term and add to middle term

  • Use grouping method for four-term factorisation

    • Group into two brackets

    • Factorise each mini-group

    • Identify common binomial factor

  • Factorise algebraic fractions before simplification

    • Cancel only after fully breaking expressions into factors

  • Combine expansion and factorisation to reverse complex transformations

Solving Linear Equations with Maximum Speed

  • Isolate variable strategically

    • Move terms in smallest number of steps

    • Collect x terms on one side and constants on the other

  • Handel negatives with care

    • Use bracket expansion to avoid misinterpreting minus signs

  • Remove fractions early

    • Multiply entire equation by denominator to simplify

  • Deal with unknowns in denominators

    • Multiply both sides appropriately to clear fractions

  • Solve equations involving brackets

    • Expand first

    • Simplify

    • Rearrange

  • Reverse operations cleanly

    • Use inverse steps one at a time

  • Check solution validity by substitution

Simultaneous Equation Tricks for Perfect Accuracy

  • Choose elimination when possible

    • Align coefficients using multiplication

    • Add or subtract equations to remove variable

  • Use substitution when elimination becomes messy

    • Rearrange one equation into x = … or y = …

    • Substitute into the second

  • Manage signs carefully during subtraction

  • Solve word-based simultaneous problems effectively

    • Translate text into equations correctly

    • Identify consistent variable meaning

  • Use checking method to avoid errors

    • Substitute solution pair back into both equations

    • Confirm equality holds exactly

  • Avoid decimal working where fractions provide cleaner results

Mastering Algebraic Fractions for Simplification and Solving

  • Identify LCM of denominators first

  • Multiply entire equation by LCM for clean simplification

  • Simplify numerator and denominator separately

  • Factorise before cancellation

  • Avoid cancelling terms unless they are genuine factors

  • Recognise compound fractions

    • Simplify top and bottom before dividing

  • Rewrite complex fractions into single line expressions

  • Use reciprocal when dividing fractions

  • Maintain structure to avoid mixing division and multiplication incorrectly

Index Laws and Powers: Rules Every Student Must Memorise

  • Apply index laws consistently

    • aᵐ × aⁿ = aᵐ⁺ⁿ

    • aᵐ ÷ aⁿ = aᵐ⁻ⁿ

    • (aᵐ)ⁿ = aᵐⁿ

    • a⁻ⁿ = 1⁄aⁿ

    • a⁰ = 1

  • Work confidently with fractional indices

    • a¹ᐟ² = √a

    • a¹ᐟ³ = ³√a

    • aᵐᐟⁿ = ⁿ√(aᵐ)

  • Simplify using prime factorisation

    • Break bases into primes where needed

  • Recognise exponential patterns

    • Understand behaviour of powers during simplification

  • Avoid mixing powers incorrectly

    • Keep multiplication and addition rules distinct

Quadratic Equation Mastery for A* Candidates

  • Identify quadratic form ax² + bx + c

  • Choose correct solving method based on structure

    • Factorisation for simple quadratics

    • Completing the square for specific formats

    • Quadratic formula for general cases

  • Handle discriminants correctly

    • b² − 4ac identifies number of solutions

  • Simplify solutions cleanly

    • Cancel factors inside square roots where possible

  • Identify which solution is acceptable in context questions

    • Lengths cannot be negative

    • Time cannot be negative

  • Use substitution to verify calculated solutions

Understanding Expressions vs Equations

  • Recognise that expressions cannot be solved

  • Use simplification techniques instead of solving

  • Identify equations by presence of equals sign

  • Treat identities separately

    • Understand expression equality for all variable values

  • Avoid manipulating expressions as equations

Algebraic Manipulation in Word Problems

  • Translate description carefully

  • Identify variable meaning clearly

  • Create algebraic equations from proportional relationships

  • Use substitution effectively for multi-step descriptions

  • Avoid copying irrelevant story information

  • Check units before forming equations

  • Ensure final answers relate to context

    • Avoid non-physical values

  • Write concluding interpretation if question requires reasoning

Sequences and nth Term Tricks

  • Identify arithmetic sequences

    • Use constant difference to form nth term

  • Identify geometric sequences

    • Recognise constant multiplier

  • Use general form

    • Linear: an = dn + (first term − d)

    • Quadratic: identify second difference

  • Avoid mixing arithmetic and geometric behaviour

  • Use nth term to test membership of sequence

    • Solve for n

    • Check that n is whole number

Inequalities Tricks for Perfect Region Identification

  • Solve inequality same way as linear equation

    • Reverse inequality sign when multiplying or dividing by negative

  • Represent solutions correctly

    • Open circles for < and >

    • Closed circles for ≤ and ≥

  • Shade correct region on number line

  • Write solution in interval or inequality form consistently

  • Handle compound inequalities step by step

  • Avoid switching sign unnecessarily

Working with Algebraic Graphs Efficiently

  • Identify type of graph from structure

    • Linear

    • Quadratic

    • Cubic

    • Reciprocal

  • Compare gradient from coefficients

  • Use substitution to find coordinate pairs quickly

  • Identify axis intercepts accurately

    • y-intercept from constant term

    • x-intercept by substituting y = 0

  • Use simultaneous interpretation to find intersections

  • Avoid joining points with incorrect shape

    • Draw smooth curves for quadratics

    • Avoid straight-line shortcuts

Avoiding Common Algebra Errors

  • Dropping terms accidentally during rearrangement

  • Cancelling incorrectly inside addition expressions

  • Forgetting bracket expansion

  • Mixing up sign operations

  • Dividing only part of the equation instead of whole

  • Confusing negative and fractional powers

  • Misinterpreting expressions with multiple brackets

Speed Tricks for Mental Algebra in Non-Calculator Papers

  • Factorise simple expressions mentally before writing

  • Use difference of squares pattern to reduce working

  • Break composite expressions into familiar algebraic identities

  • Estimate solutions quickly

  • Identify structure before calculation

  • Store common coefficient pairs in memory

  • Recognise opportunities for cancellation early

Error Checking Techniques for Algebra

  • Substitute final solution back into original equation

  • Check algebraic fractions by reconstructing numerator

  • Validate signs throughout solution

  • Confirm dimensions and units where applicable

  • Use approximate values to confirm reasonableness

  • Compare simplified form to alternative methods

  • Confirm factorisation by expanding back

  • Check if two algebraic expressions truly match

Building A* Level Algebra Approach

  • Maintain method-first mindset

  • Organise working neatly

  • Use consistent transformations

  • Apply algebra within real context constraints

  • Combine substitution, factorisation and rearrangement seamlessly

  • Review past paper algebra questions repeatedly

  • Analyse examiner reports for common algebraic weaknesses

  • Develop pattern recognition for repeated algebra formats

  • Treat algebra as foundation for entire Maths syllabus at O Level & IGCSE

Hunain Zia

Hunain Zia has previously achieved considerable success in the high-school educational stream, gaining a massive 154 Total A Grades and 7 Major Distinctions, a process in which he broke/ set 11 different world records, including the most A grades ever achieved, most subjects ever appeared in, most distinctions, gaining distinctions across two separate boards (Pearson Edexcel and CAIE) within the same year/ ever, and even 19 hours of constant examination. All records stand intact today. Hunain pursued his Honors Accounting Degree from the prestigious Bentley University, and then completed an LLB (in Honors) from University of London. Currently, he manages multiple social and commercial projects, has founded digital streams and works tirelessly in the education sector.

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