Factorising Quadratics

Q: Factorising Quadratics

A quadratic expression is of the form ax 2 + bx + c where a, b and c are numbers. For example 2x 2 + 3x - 1.

Q: Completing the Square

Completing the square for y = 2x 2 + 12x + 5:

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Factorising Quadratics

A quadratic expression is of the form ax² + bx + c where a, b and c are numbers. For example 2x² + 3x - 1.

Factorising

Factorising a quadratic expression means writing it as the product of two brackets.

Difference of two squares

When a quadratic expression is the difference of two squares it can be easily factorised:
General case: a² - b² = (a + b)(a - b).

Factorising Quadratics

We can factorise quadratics in the form ax² + bx + c when a = 1.
To factorise x² + 2x - 8:

Find pairs of numbers which multiply to give (-8)

Select the pair which also adds to give 2

These numbers multiply to give (-8) and add to give 2

Write these two numbers in the brackets

Check your answer by expanding the brackets

Completing the Square

Completing the square for y = 2x² + 12x + 5:

Write the equation

Factor out the x² term

Half the x coefficient and put it in the bracket like this

Expand

We need to minus 13 from the red equation to get the purple

Numbers

1.1

Integers

1.1.1Integers 1.1.2Ordering Numbers 1.1.3The Four Rules of Operations 1.1.4Hierarchy of Operations 1.1.5Definitions & Terms 1.1.6HCF & LCM 1.1.7Standard Form 1.1.8Standard Form 2 1.1.9Prime Numbers 1.1.10End of Topic Test - Standard Form

1.2

Fractions

1.2.1Fractions 1.2.2Using Fractions 1.2.3Conversion 1.2.4Reciprocals

1.3

Decimals

1.3.1Decimals 1.3.2Decimals 2 1.3.3Exam-Style Questions - Recurring Decimals 1.3.4Comparing & Converting Decimals 1.3.5End of Topic Test - Fractions & Decimals

1.4

Powers & Roots

1.4.1Square Numbers 1.4.2Index Notation 1.4.3Exam-Style Questions - Index Notation 1.4.4Advanced Indices 1.4.5Surds 1.4.6End of Topic Test - Powers & Roots

1.5

Set Language & Notation

1.5.1Sets

1.6

Percentages

1.6.1Percentages 1.6.2Percentage Change 1.6.3Percentages in Finance 1.6.4Compound Interest

1.7

Ratio & Proportion

1.7.1Manipulating Ratios 1.7.2Proportion

1.8

Degree of Accuracy

1.8.1Rounding & Significant Figures 1.8.2Estimation 1.8.3Upper & Lower Bounds 1.8.4Limits of Accuracy 1.8.5Limits of Accuracy 2 1.8.6End of Topic Test - Rounding & Estimation

Equations, Formulae & Identities

2.1

Algebraic Manipulation

2.1.1Algebra Basics 2.1.2Power Rules & Collecting Terms 2.1.3Expanding Brackets 2.1.4Expanding Brackets 2 2.1.5Factorising Quadratics 2.1.6Algebraic Fractions 2.1.7Proofs 2.1.8End of Topic Test - Algebra & Proofs 2.1.9Exam-Style Questions - Algebraic Proof

2.2

Expressions & Formulae

2.2.1Rearranging Formulae 2.2.2Simple Formulae 2.2.3End of Topic Test - Algebra Manipulation & Formula

2.3

Linear Equations

2.3.1Solving Linear Equations 2.3.2Simultaneous Linear Equations

2.4

Quadratic Equations

2.4.1Solving Quadratics 2.4.2Solving Quadratics 2 2.4.3End of Topic Test - Solving Equations 2.4.4Simulateous Quadratic Equations 2.4.5Exam-Style Questions - Simultaneous Equations 2.4.6Graphs of Simultaneous Equations 2.4.7End of Topic Test - Simultaneous Equations

2.5

Proportion

2.5.1Graphs & Proportion

2.6

Inequalities

2.6.1Solving Inequalities 2.6.2Graphs of Inequalities 2.6.3Exam-Style Questions - Inequalities 2.6.4End of Topic Test - Inequalities

Sequences, Functions & Graphs

3.1

Sequences

3.1.1Sequence Rules 3.1.2Finding Terms 3.1.3Exam-Style Questions - Sequences 3.1.4End of Topic Test - Sequences

3.2

Functions

3.2.1Functions 3.2.2Composite & Inverse 3.2.3End of Topic Test - Functions

3.3

Graphs

3.3.1Plotting Graphs 3.3.2Straight Line Graphs 3.3.3Properties of Graphs 3.3.4Quadratic Graphs 3.3.5Practical Graphs 3.3.6Graphs of Polynomials 3.3.7End of Topic Test Graphs

3.4

Common Graphs

3.4.1Graphs of Common Functions 3.4.2Transformations of Graphs 3.4.3End of Topic Test - Transformations and Graph

3.5

Calculus

3.5.1Derivatives of Functions 3.5.2Maxima & Minima

Geometry

4.1

Angles, Lines & Triangles

4.1.1Angles & Lines 4.1.2Angle Rules 4.1.3Angles in Triangles 4.1.4Exam-Style Questions - Angles

4.2

Polygons

4.2.1Polygons 4.2.2Quadrilaterals 4.2.3Congruence 4.2.4Scale Factor 4.2.5Exam-Style Questions - Enlargements 4.2.6End of Topic Test - Geometry

4.3

Symmetry

4.3.1Symmetry 4.3.2Exam-Style Questions - Reflection

4.4

Measures

4.4.1Measurements 4.4.2Time 4.4.3Bearings 4.4.4Speed

4.5

Circle Properties

4.5.1Properties of a Circle 4.5.2Circle Theorems 4.5.3Circle Theorems 2 4.5.4Circle Theorems 3 4.5.5Exam-Style Questions - Circle Theorems

4.6

Trigonometry & Pythagoras’ theorem

4.6.1Pythagoras' Theorem 4.6.2Trigonometry 4.6.3Sine & Cosine for Obtuse Angles 4.6.4Triangle Formulae 4.6.5Exam-Style Questions - Trigonometry & Area

4.7

Mensuration of 2D Shapes

4.7.1Standard Units 4.7.2Perimeter of 2D Shapes 4.7.3Area of 2D Shapes 4.7.4Area & Circumference of Circle 4.7.5End of Topic Test - Trigonometry & Shapes

4.8

3D Shapes & Volume

4.8.13D Shapes 4.8.2Surface Area 4.8.3Volume 4.8.4Exam-Style Questions - Volume

Vectors & Transformation Geometry

5.1

Vectors

5.1.1Vectors 5.1.2Vectors 2 5.1.3Exam-Style Questions - Vectors

5.2

Transformation Geometry

5.2.1Transformations 5.2.2Rotations 5.2.3Translations & Reflections 5.2.4Enlargements 5.2.5Effects of Transformations

Statistics & Probability

6.1

Statistical Measures

6.1.1Basic Statistics 6.1.2Frequency Table

6.2

Graphical Representation of Data

6.2.1Pictograms, Bar Charts & Pie Charts 6.2.2Histograms 6.2.3Cumulative Frequency 6.2.4Cumulative Frequency 2 6.2.5Exam-Style Questions - Cumulative Frequency 6.2.6End of Topic - Datasets

6.3

Probability

6.3.1Probability 6.3.2Theoretical Probability 6.3.3Experimental Probability 6.3.4Venn Diagrams 6.3.5Sample Space Diagrams 6.3.6Tree Diagrams 6.3.7Independent Events 6.3.8Conditional Probability 6.3.9End of Topic Test - Probability 6.3.10Exam-Style Questions - Probability

Jump to other topics

Numbers

1.1

Integers

1.2

Fractions

1.2.1Fractions 1.2.2Using Fractions 1.2.3Conversion 1.2.4Reciprocals

1.3

Decimals

1.3.1Decimals 1.3.2Decimals 2 1.3.3Exam-Style Questions - Recurring Decimals 1.3.4Comparing & Converting Decimals 1.3.5End of Topic Test - Fractions & Decimals

1.4

Powers & Roots

1.4.1Square Numbers 1.4.2Index Notation 1.4.3Exam-Style Questions - Index Notation 1.4.4Advanced Indices 1.4.5Surds 1.4.6End of Topic Test - Powers & Roots

1.5

Set Language & Notation

1.5.1Sets

1.6

Percentages

1.6.1Percentages 1.6.2Percentage Change 1.6.3Percentages in Finance 1.6.4Compound Interest

1.7

Ratio & Proportion

1.7.1Manipulating Ratios 1.7.2Proportion

1.8

Degree of Accuracy

1.8.1Rounding & Significant Figures 1.8.2Estimation 1.8.3Upper & Lower Bounds 1.8.4Limits of Accuracy 1.8.5Limits of Accuracy 2 1.8.6End of Topic Test - Rounding & Estimation

Equations, Formulae & Identities

2.1

Algebraic Manipulation

2.2

Expressions & Formulae

2.2.1Rearranging Formulae 2.2.2Simple Formulae 2.2.3End of Topic Test - Algebra Manipulation & Formula

2.3

Linear Equations

2.3.1Solving Linear Equations 2.3.2Simultaneous Linear Equations

2.4

Quadratic Equations

2.5

Proportion

2.5.1Graphs & Proportion

2.6

Inequalities

2.6.1Solving Inequalities 2.6.2Graphs of Inequalities 2.6.3Exam-Style Questions - Inequalities 2.6.4End of Topic Test - Inequalities

Sequences, Functions & Graphs

3.1

Sequences

3.1.1Sequence Rules 3.1.2Finding Terms 3.1.3Exam-Style Questions - Sequences 3.1.4End of Topic Test - Sequences

3.2

Functions

3.2.1Functions 3.2.2Composite & Inverse 3.2.3End of Topic Test - Functions

3.3

Graphs

3.3.1Plotting Graphs 3.3.2Straight Line Graphs 3.3.3Properties of Graphs 3.3.4Quadratic Graphs 3.3.5Practical Graphs 3.3.6Graphs of Polynomials 3.3.7End of Topic Test Graphs

3.4

Common Graphs

3.4.1Graphs of Common Functions 3.4.2Transformations of Graphs 3.4.3End of Topic Test - Transformations and Graph

3.5

Calculus

3.5.1Derivatives of Functions 3.5.2Maxima & Minima

Geometry

4.1

Angles, Lines & Triangles

4.1.1Angles & Lines 4.1.2Angle Rules 4.1.3Angles in Triangles 4.1.4Exam-Style Questions - Angles

4.2

Polygons

4.2.1Polygons 4.2.2Quadrilaterals 4.2.3Congruence 4.2.4Scale Factor 4.2.5Exam-Style Questions - Enlargements 4.2.6End of Topic Test - Geometry

4.3

Symmetry

4.3.1Symmetry 4.3.2Exam-Style Questions - Reflection

4.4

Measures

4.4.1Measurements 4.4.2Time 4.4.3Bearings 4.4.4Speed

4.5

Circle Properties

4.5.1Properties of a Circle 4.5.2Circle Theorems 4.5.3Circle Theorems 2 4.5.4Circle Theorems 3 4.5.5Exam-Style Questions - Circle Theorems

4.6

Trigonometry & Pythagoras’ theorem

4.6.1Pythagoras' Theorem 4.6.2Trigonometry 4.6.3Sine & Cosine for Obtuse Angles 4.6.4Triangle Formulae 4.6.5Exam-Style Questions - Trigonometry & Area

4.7

Mensuration of 2D Shapes

4.7.1Standard Units 4.7.2Perimeter of 2D Shapes 4.7.3Area of 2D Shapes 4.7.4Area & Circumference of Circle 4.7.5End of Topic Test - Trigonometry & Shapes

4.8

3D Shapes & Volume

4.8.13D Shapes 4.8.2Surface Area 4.8.3Volume 4.8.4Exam-Style Questions - Volume

Vectors & Transformation Geometry

5.1

Vectors

5.1.1Vectors 5.1.2Vectors 2 5.1.3Exam-Style Questions - Vectors

5.2

Transformation Geometry

5.2.1Transformations 5.2.2Rotations 5.2.3Translations & Reflections 5.2.4Enlargements 5.2.5Effects of Transformations

Statistics & Probability

6.1

Statistical Measures

6.1.1Basic Statistics 6.1.2Frequency Table

6.2

Graphical Representation of Data

6.2.1Pictograms, Bar Charts & Pie Charts 6.2.2Histograms 6.2.3Cumulative Frequency 6.2.4Cumulative Frequency 2 6.2.5Exam-Style Questions - Cumulative Frequency 6.2.6End of Topic - Datasets

6.3

Probability

Practice questions on Factorising Quadratics

Can you answer these? Test yourself with free interactive practice on Seneca — used by over 10 million students.

1
Which of these is an example of difference of two squares?Multiple choice
2
Which of these is an example of difference of two squares?Multiple choice
3
Which of these is an example of difference of two squares?Multiple choice
4
Which is the correct factorisation of this expression? <br> x² + 3x + 2Multiple choice
5
x² - 3x - 4</br>Which of the following is NOT a correct factorisation of this expression? Multiple choice

Answer all questions on Factorising Quadratics

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Expanding Brackets 2

Algebraic Fractions