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Revision notes
GCSE
Maths
Edexcel
Higher
Completing the Square

Completing the Square

Maths: Edexcel GCSE Higher, topic 2.3.10

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Previous topic2.3.9 Factorising the Difference of Two Squares Next topic2.4.1 Simplifying Algebraic Fractions by Cancelling

Maths: Edexcel GCSE Higher contents

1 Number
- 1.1 Place Value & Ordering
  1.1.1 Ordering Integers & Decimals
  1.1.2 Equal Symbols
  1.1.3 Place Value
- 1.2 Numerical Operations
  1.2.1 Adding Integers
  1.2.2 Subtracting Integers
  1.2.3 Multiplying Integers
  1.2.4 Dividing Integers
  1.2.5 Adding & Subtracting Decimals
  1.2.6 Multiplying Decimals
  1.2.7 Dividing Decimals
  1.2.8 Adding & Subtracting Negative Numbers
  1.2.9 Multiplying & Dividing Negative Numbers
  1.2.10 Multiplying by Powers of Ten
  1.2.11 Dividing by Powers of Ten
- 1.3 Fractions
  1.3.1 Writing Fractions
  1.3.2 Constructing Fractions
  1.3.3 Simplifying Fractions
  1.3.4 Ordering Fractions
  1.3.5 Fractions & Recurring Decimals
  1.3.6 Mixed Numbers & Improper Fractions
- 1.4 Numerical Operations on Fractions
  1.4.1 Adding & Subtracting Fractions
  1.4.2 Adding & Subtracting Mixed Numbers
  1.4.3 Multiplying Fractions
  1.4.4 Multiplying Mixed Numbers
  1.4.5 Reciprocals
  1.4.6 Dividing Fractions
  1.4.7 Dividing Mixed Numbers
- 1.5 Roots & Powers
  1.5.1 Integer Powers
  1.5.2 Roots
  1.5.3 Estimating Powers & Roots
  1.5.4 Fractional Indices (1/n)
  1.5.5 Fractional Indices (m/n)
  1.5.6 Laws of Indices (Multiplication)
  1.5.7 Laws of Indices (Division)
- 1.6 Structure of Calculations
  1.6.1 Order of Operations (BIDMAS)
- 1.7 Numerical Skills
  1.7.1 Using a Calculator - Powers, Roots & Fractions
  1.7.2 Using a Calculator - Trigonometric Functions
- 1.8 Factors & Multiples
  1.8.1 Finding Factors
  1.8.2 Finding Multiples
- 1.9 Systematic Listing
  1.9.1 Systematic Listing
- 1.10 Surds
  1.10.1 Multiplying & Dividing Surds
  1.10.2 Simplifying Surds
  1.10.3 Expanding Expressions With Surds
  1.10.4 Rationalising the Denominator
- 1.11 Standard Form
  1.11.1 Converting to Standard Form (Positive Powers)
  1.11.2 Converting to Standard Form (Negative Powers)
  1.11.3 Converting From Standard Form (Positive Powers)
  1.11.4 Converting From Standard Form (Negative Powers)
  1.11.5 Adding & Subtracting Standard Form
  1.11.6 Multiplying & Dividing Standard Form
  1.11.7 Standard Form on a Calculator
- 1.12 Fractions, Decimals & Percentages
  1.12.1 Converting Fractions, Decimals & Percentages
  1.12.2 Converting Recurring Decimals
  1.12.3 Numbers as Percentages
  1.12.4 Fractions of Amounts
- 1.13 Percentage Change
  1.13.1 Percentages of Amounts (Calculator)
  1.13.2 Percentages of Amounts (No Calculator)
  1.13.3 Percentage Change
  1.13.4 Original Values After Percentage Change
  1.13.5 Growth & Decay Using Multipliers
- 1.14 Rounding & Estimation
  1.14.1 Rounding Integers
  1.14.2 Estimating Calculations
  1.14.3 Rounding Decimals
  1.14.4 Rounding With Significant Figures
- 1.15 Bounds & Accuracy
  1.15.1 Finding Error Intervals
  1.15.2 Upper & Lower Bounds
  1.15.3 Truncating Decimals
  1.15.4 Error Intervals From Truncation
- 1.16 Using Numbers
  1.16.1 Highest Common Factor
  1.16.2 Lowest Common Multiple
  1.16.3 Prime Numbers
2 Algebra
- 2.1 Algebraic Notation
  2.1.1 Using Algebraic Notation
- 2.2 Substitution & Simplification
  2.2.1 Substituting into Expressions
  2.2.2 Substituting into Real-Life Formulae
  2.2.3 Simplifying Expressions by Collecting Like Terms
- 2.3 Brackets
  2.3.1 Expanding Single Brackets
  2.3.2 Expanding & Simplifying with Single Brackets
  2.3.3 Expanding Double Brackets
  2.3.4 Expanding Multiple Brackets
  2.3.5 Factorising into One Bracket
  2.3.6 Factorising Quadratics of the Form x^2 + bx + c
  2.3.7 Factorising Quadratics - AC Method
  2.3.8 Factorising Quadratics - Middle Term Method
  2.3.9 Factorising the Difference of Two Squares
  2.3.10 Completing the Square
- 2.4 Algebraic Fractions
  2.4.1 Simplifying Algebraic Fractions by Cancelling
  2.4.2 Simplifying Algebraic Fractions by Factorising
  2.4.3 Operations with Algebraic Fractions
- 2.5 Rearranging
  2.5.1 Changing the Subject
  2.5.2 Changing the Subject (Repeated Subject)
- 2.6 Functions
  2.6.1 Expressions as Functions
  2.6.2 Finding the Input of a Function
  2.6.3 Finding Composite Functions
  2.6.4 Inverse Functions
- 2.7 Solving Linear Equations
  2.7.1 Solving Linear Equations (One Step)
  2.7.2 Solving Linear Equations (Multi-Step)
  2.7.3 Solving Linear Equations (teacher sample)
  2.7.4 Linear Equations (Unknown Both Sides)
  2.7.5 Solving Equations: (Unknown Denominator)
  2.7.6 Forming & Solving Linear Equations
- 2.8 Inequalities
  2.8.1 Representing Inequalities on a Number Line
  2.8.2 Solving Linear Inequalities
  2.8.3 Linear Inequalities with Unknowns on Both Sides
  2.8.4 Graphs of Linear Inequalities
  2.8.5 Solving Quadratic Inequalities
- 2.9 Solving Quadratic Equations
  2.9.1 Solving Quadratics by Factorising
  2.9.2 Solving Quadratics Using the Formula
- 2.10 Simultaneous Equations
  2.10.1 Solving Simultaneous Equations (Elimination)
  2.10.2 Solving Simultaneous Linear & Quadratic Equations
  2.10.3 Solving Simultaneous Equations Graphically
- 2.11 Iteration
  2.11.1 Iterative Formulae
  2.11.2 Solutions Using Iteration
- 2.12 Sequences
  2.12.1 Term-to-Term Rules
  2.12.2 Position-to-Term Rules
  2.12.3 Finding the Nth Term (Linear Sequences)
  2.12.4 Finding the Nth Term (Quadratic Sequences)
  2.12.5 Quadratic Sequence Formulae
  2.12.6 Simple Geometric Progressions
- 2.13 Coordinates
  2.13.1 Coordinate Reading & Plotting
  2.13.2 Midpoints of Line Segments
- 2.14 Linear Graphs
  2.14.1 Plotting Straight Line Graphs
  2.14.2 Equations of Straight Lines
  2.14.3 Finding Line Equation from Point & Gradient
  2.14.4 Finding Line Equation from Two Points
  2.14.5 Parallel Line Equations
  2.14.6 Perpendicular Line Equations
- 2.15 Quadratic Graphs
  2.15.1 Plotting Quadratics
  2.15.2 Interpreting & Sketching Quadratics
  2.15.3 Finding Turning Points by Completing the Square
- 2.16 More Non-Linear Graphs
  2.16.1 Cubic Graphs
  2.16.2 Trigonometric & Exponential Functions
  2.16.3 Equations of Circles
  2.16.4 Equations of Tangents
- 2.17 Graph Transformations
  2.17.1 Translating Graphs
  2.17.2 Reflecting Graphs
  2.17.3 Transforming Graphs
- 2.18 Real-Life Graphs
  2.18.1 Distance-Time Graphs
  2.18.2 Acceleration from Velocity-Time Graphs
  2.18.3 Distance from Velocity-Time Graphs
3 Ratio
- 3.1 Unit Conversions
  3.1.1 Converting Units of Time
  3.1.2 Metric Conversions
  3.1.3 Converting Units of Area & Volume
- 3.2 Compound Units
  3.2.1 Calculating Speed
  3.2.2 Best Buy & Unit Pricing
  3.2.3 Density, Mass & Volume
  3.2.4 Pressure, Force & Area
- 3.3 Ratio Fundamentals
  3.3.1 Ratio Notation & Simplification
  3.3.2 Ratios as Fractions & Linear Equations
  3.3.3 Sharing in a Ratio
  3.3.4 Proportion Problems
- 3.4 Advanced Ratio & Scale
  3.4.1 Map Scales & Real Distances
- 3.5 Proportion
  3.5.1 Currency Conversions
  3.5.2 Direct & Inverse Proportion
  3.5.3 Graphs of Direct & Inverse Proportion
- 3.6 Algebraic Proportion
  3.6.1 Direct Proportion Equations
  3.6.2 Inverse Proportion Equations
  3.6.3 Direct Proportion with Powers & Roots
  3.6.4 Inverse Proportion with Powers & Roots
- 3.7 Percentages & Growth
  3.7.1 Compound Interest & Depreciation
- 3.8 Manipulating Ratios
  3.8.1 Differences in Ratios
  3.8.2 Fractions in Ratios
- 3.9 Percentage & Interest
  3.9.1 Simple Interest
4 Geometry
- 4.1 Area & Perimeter
  4.1.1 Perimeter of Simple Shapes
  4.1.2 Perimeter of Compound Shapes
  4.1.3 Area of Triangles
  4.1.4 Area of Rectangles & Parallelograms
  4.1.5 Area of Trapezia
  4.1.6 Area of Compound Shapes
  4.1.7 Area of Shapes Involving Triangles
- 4.2 Properties of Shapes
  4.2.1 Properties of Lines
  4.2.2 Symmetry
  4.2.3 Rotational Symmetry
  4.2.4 Shortest Distance from a Point to a Line
  4.2.5 Properties of 3D Shapes
  4.2.6 Cross Section
- 4.3 Angles
  4.3.1 Classification of Angles
  4.3.2 Angles on a Line & Around a Point
  4.3.3 Combining Angle Facts in Triangles
  4.3.4 Angles on Parallel Lines
  4.3.5 Interior & Exterior Angles in Polygons
- 4.4 Trigonometry
  4.4.1 Sides in Right-Angled Triangles
  4.4.2 Angles in Right-Angled Triangles
  4.4.3 Trigonometric Graphs
  4.4.4 Exact Values of Trigonometric Ratios
  4.4.5 Trigonometry in 3D Shapes
  4.4.6 Using the Sine Rule
  4.4.7 Using the Cosine Rule
  4.4.8 Area of a Triangle Using Sine
- 4.5 Similarity & Congruence
  4.5.1 Congruent Shapes
  4.5.2 Length Scale Factors in Similar Shapes
  4.5.3 Area & Volume Scale Factors
- 4.6 Circle Geometry
  4.6.1 Identifying Parts of a Circle
  4.6.2 Calculating Circumference
  4.6.3 Calculating Area of a Circle
  4.6.4 Calculating Arc Lengths
  4.6.5 Calculating Area of Sectors
- 4.7 Circle Theorems
  4.7.1 Standard Circle Theorems
  4.7.2 Cyclic Quadrilaterals & Segment Theorems
  4.7.3 Chord & Tangent Properties
- 4.8 Pythagoras
  4.8.1 Pythagoras' Theorem in 2D
  4.8.2 Pythagoras’ Theorem in 3D
- 4.9 3D Representations
  4.9.1 Nets of 3D Shapes
- 4.10 Surface Area
  4.10.1 Surface Area of Cubes & Cuboids
  4.10.2 Surface Area of Prisms
  4.10.3 Surface Area of Cylinders
  4.10.4 Surface Area of Spheres & Cones
  4.10.5 Surface Area of Composite Solids
- 4.11 Volume
  4.11.1 Volume of Prisms
  4.11.2 Volume of Cylinders
  4.11.3 Volume of Frustums & Composite Solids
- 4.12 Vectors
  4.12.1 Vectors & Column Notation
  4.12.2 Operations with Column Vectors
- 4.13 Transformations
  4.13.1 Translation Using Vectors
  4.13.2 Rotations
  4.13.3 Reflections
  4.13.4 Enlargements & Positive Scale Factors
  4.13.5 Enlargements & Negative Scale Factors
  4.13.6 Combining Transformations
- 4.14 Scale Drawings
  4.14.1 Scale Diagrams
- 4.15 Bearings
  4.15.1 Bearings
- 4.16 Geometry Proof
  4.16.1 Geometric Proofs Using Vectors
- 4.17 Introduction to Geometry
  4.17.1 Geometry Basics
  4.17.2 Geometry Basics 2
  4.17.3 Bisectors & Perpendiculars
  4.17.4 Locus
- 4.18 Triangles & Quadrilaterals
  4.18.1 Quadrilaterals
- 4.19 Transformations
  4.19.1 Invariant Points
- 4.20 Measurements & Units
  4.20.1 Time
- 4.21 3D Shapes
  4.21.1 Types of 3D Shape
  4.21.2 Plans & Elevations
  4.21.3 Volume of a Cuboid
  4.21.4 Volume of a Sphere
  4.21.5 Volume of a Hemisphere
  4.21.6 Volume of a Cone
  4.21.7 Volume of a Pyramid
5 Probability
- 5.1 Single Events
  5.1.1 Writing Probabilities as Fractions
  5.1.2 Mutually Exclusive Events
  5.1.3 Expected Results
- 5.2 Combined Events
  5.2.1 Venn Diagrams
  5.2.2 Using Set Notation
  5.2.3 Using Set Notation 2
  5.2.4 Independent & Mutually Exclusive Events
  5.2.5 Frequency Trees
  5.2.6 Tree Diagrams (Independent)
  5.2.7 Tree Diagrams (Dependent)
- 5.3 Experimental Probability
  5.3.1 Fairness and Bias
  5.3.2 Relative Frequency & Convergence
- 5.4 Counting Outcomes
  5.4.1 Product Rule for Counting
- 5.5 Conditional Probability
  5.5.1 Conditional Probability: Venn Diagrams
  5.5.2 Conditional Probability: Tree Diagrams
  5.5.3 Conditional Probability from Tables
- 5.6 Representing Probabilities
  5.6.1 Probability Scale
  5.6.2 Sample Space Diagrams
- 5.7 Probability Calculations
  5.7.1 Complex Tree Diagrams
6 Statistics
- 6.1 Averages & Range
  6.1.1 Range
  6.1.2 Median
  6.1.3 Mode
  6.1.4 Mean
  6.1.5 Averages from Frequency Tables
  6.1.6 Interpreting Two-Way Tables & Frequency Tables
- 6.2 Bar Charts
  6.2.1 Bar Charts
- 6.3 Scatter Graphs
  6.3.1 Scatter Graphs
  6.3.2 Lines of Best Fit
- 6.4 Histograms
  6.4.1 Interpreting Histograms
  6.4.2 Drawing Histograms
- 6.5 Frequency Polygons
  6.5.1 Frequency Polygons
- 6.6 Cumulative Frequency
  6.6.1 Drawing Cumulative Frequency
  6.6.2 Interpreting Cumulative Frequency
- 6.7 Box Plots
  6.7.1 Interpreting Box Plots & Comparing Distributions
- 6.8 Collecting & Presenting Data
  6.8.1 Types of Data
  6.8.2 Designing Questionnaires
  6.8.3 Capture-Recapture
- 6.9 Charts & Diagrams
  6.9.1 Pie Charts
  6.9.2 Stem & Leaf Diagrams
  6.9.3 Back to Back Stem & Leaf Diagrams
  6.9.4 Pictograms
  6.9.5 Time Series Graphs
  6.9.6 Correlation & Causation

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