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Factorising quadratics

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Explainer Video

Tutor: Bilal

Summary

Factorising quadratics

In a nutshell

A quadratic in the form $ax^2+bx+c$ can be factorised. You have already learnt how to factorise a quadratic when $a=1$ by finding two numbers to multiply to give $c$ and add to give $b$ . This lesson will also discuss how to factorise a quadratic expression where $a>1$ .

Factorise when $a=1$

A quadratic is in the form

$ax^2 + bx +c$

When $a=1$ we can factorise a quadratic as follows:

PROCEDURE

1.	Ensure the quadratic is in the form $ax^2 + bx + c$ and identify $a,b$ and $c$
2.	Set up the answer with two brackets and an $x$ at the start of each bracket, as follows $(x \qquad)(x \qquad)$
3.	Find two numbers which multiply to give $c$ and add to give $b$
4.	Fill these numbers in the double brackets

Example 1

Factorise

$x^2 + 10x + 24$

The quadratic is in the correct form where $b=10$ and $c=24$ .

$6 \times 4 = 24 \space and \space 6 + 4 = 10$ , so the numbers are $6 \space and \space 4$ .

$x^2 + 10x+24 = \underline{(x+6)(x+4)}$

Factorise when $a>1$

When $a>1$ you can factorise a quadratic as follows:

PROCEDURE

1.	Ensure the quadratic is in the form $ax^2+bx+c$ , and identify $a, b$ and $c$ .
2.	Multiply $a$ and $c$ to give a new number $d$ .
3.	Think of two numbers, $m$ and $n$ which multiply to give $d$ and add to give $b$ .
4.	Write the quadratic using $m$ and $n$ in the form $ax^2+mx+nx+c$ .
5.	Factorise $ax^2+mx$ into single brackets.
6.	Factorise $nx+c$ into single brackets.
7.	Use both factorised single brackets expressions to find the double brackets expression. The first bracket consists of the terms outside the single brackets, and the second bracket consists of the expression inside the single brackets.

Example 2

Factorise

$2x^2+13x+15$

The quadratic has $a=2, b=13$ and $c=15$ . Multiply $a$ and $c$ to give $2 \times 15 = 30$ .

Think of two numbers which will multiply to give $30$ and add to give $13$ . The two numbers are $10$ and $3$ , as $10 \times 3 = 30$ and $10+3 = 13$ .

Write the quadratic expression using $10$ and $3$ , and factorise.

$\begin {aligned}&2x^2+13x+15 \\&\underbrace{2x^2 + 10x}_{\text{factorise}} + \underbrace{3x + 15}_{\text{factorise}} \\&2x(x+5) + 3(x+5)\end {aligned}$

The two expressions inside both single brackets should match, in this case $x+5$ .

The final answer consists of two brackets, the first bracket contains the terms on the outside of each bracket, $2x$ and $+3$ . And the second bracket is the repeated expression inside the single brackets factorised.

$\underline{(2x+3)(x+5)}$

Example 3

Factorise and solve

$3x^2+11x-4=0$

The quadratic has $a=3, b=11$ and $c=-4$ . Multiply $a$ and $c$ to give $-12$ .

Think of two numbers which will multiply to give $-12$ and add to give $+11$ . The two numbers are $+12$ and $-1$ , as $12\times -1 = -12$ and $12-1=11$ .

Write the quadratic expression using $12$ and $-1$ , and factorise.

$\begin {aligned}3x^2+11x-4&=0 \\\underbrace{3x^2+12x}_{\text{factorise}}\space \underbrace{-1x-4}_{\text{factorise}}&=0 \\3x(x+4) -1(x+4)&=0 \\(3x-1)(x+4)&=0\end {aligned}$ 

Now the quadratic is factorised, it can be solved. Find the solutions by making either bracket equal to $0$ .

$\underline{x=\frac 1 3 \space or \space x = -4}$ 

Learn with Basics

Length:

Unit 1

X and Y coordinates

Unit 2

Quadratic graphs

Jump Ahead

Optional

Unit 3

Factorising quadratics

Final Test

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How do you factorise a quadratic ax^2+bx+c when a>1?

When a quadratic has a>1, the method of factorising involves splitting b up to make two separate terms in x. The modified expression can then be separated into two separate groups and each group can be factorised using single brackets. The two single bracket expressions can then be used to factorise the quadratic into double brackets.

How do you factorise into double brackets?

Put the quadratic into the form ax^2 + bx + c. If a=1, then factorise the quadratic by finding two numbers that multiply to give c, and add to give b. These two numbers will go into the double brackets. e.g. x^2 + 7x + 10 = (x+2)(x+5).

What do double brackets mean?

Double brackets consist of two binomial expressions multiplied together, e.g. (x+1)(y-3). Each term in the first bracket should be multiplied by each term in the second bracket.

Home

Maths

Algebra

Factorising quadratics

Videos

Summary

Exercises

Select Lesson

Exam Board

Select an option

Statistics

Sets and Venn diagrams

Sampling and bias

Collecting data: types and classes of data

Mean, median, mode and range

Simple charts and graphs

Pie charts

Scatter graphs

Frequency tables: finding averages

Grouped frequency tables

Box plots - Higher

Cumulative frequency - Higher

Histograms and frequency density - Higher

Interpreting data

Comparing data sets

Probability

Basics of probability

Calculating theoretical probabilities

Probability: Expected and relative frequency

The AND / OR rules

Probability tree diagrams

Conditional probability - Higher

Experimental probability: frequency trees

Trigonometry

Pythagoras' theorem

Sin, cos, tan

Trigonometry: Finding angles and sides

Exact trigonometric values

Sine and cosine rules - Higher

3D Pythagoras - Higher

3D Trigonometry - Higher

Vectors

Vectors - Higher

Angles and geometry

Angles: types, notation and measuring

Basic angle rules

Angles in parallel lines

Circle theorems - Higher

Constructing triangles: SSS, SAS, ASA

Construction: angle and perpendicular bisectors

Construction: Loci

Bearings

Maps and scale drawings

Shapes and area

Properties of 2D shapes

Congruence: conditions for congruent triangles

Similar shapes: Scaling

The four transformations

Area and perimeter: Formulae

Area and circumference of circles: Formulae

3D shapes: faces, edges, vertices

Surface area of 3D shapes: Nets, formulae

Volume of 3D shapes: Formulae

Volume of 3D shapes: Comparing, rates of flow

Area and volume scale factors

Projections and elevations of 3D shapes

Ratio proportion and rates of change

Ratio

Direct and inverse proportion

Finding percentages and percentage change

Compound growth and decay

Converting units: metric and imperial

Converting units: area and volume

Time intervals: converting units of time

Speed, density and pressure: Formulae and units

Graphs

Coordinates and midpoints

Straight line graphs

Drawing straight line graphs

Finding the gradient of a straight line

Equation of a straight line: y = mx + c

Coordinates and ratio

Parallel and perpendicular lines

Quadratic graphs

Reciprocal and cubic graphs

Exponential graphs and circles - Higher

Trigonometric graphs - Higher

Solving equations using graphs

Graph transformations - Higher

Real-life graphs

Distance-time graphs

Velocity-time graphs - Higher

Gradients of real-life graphs - Higher

Algebra

Number

Explainer Video

Tutor: Bilal

Summary

Factorising quadratics

In a nutshell

Factorise when $a=1$

A quadratic is in the form

$ax^2 + bx +c$

When $a=1$ we can factorise a quadratic as follows:

PROCEDURE

1.	Ensure the quadratic is in the form $ax^2 + bx + c$ and identify $a,b$ and $c$
2.	Set up the answer with two brackets and an $x$ at the start of each bracket, as follows $(x \qquad)(x \qquad)$
3.	Find two numbers which multiply to give $c$ and add to give $b$
4.	Fill these numbers in the double brackets

Example 1

Factorise

$x^2 + 10x + 24$

The quadratic is in the correct form where $b=10$ and $c=24$ .

$6 \times 4 = 24 \space and \space 6 + 4 = 10$ , so the numbers are $6 \space and \space 4$ .

$x^2 + 10x+24 = \underline{(x+6)(x+4)}$

Factorise when $a>1$

When $a>1$ you can factorise a quadratic as follows:

PROCEDURE

1.	Ensure the quadratic is in the form $ax^2+bx+c$ , and identify $a, b$ and $c$ .
2.	Multiply $a$ and $c$ to give a new number $d$ .
3.	Think of two numbers, $m$ and $n$ which multiply to give $d$ and add to give $b$ .
4.	Write the quadratic using $m$ and $n$ in the form $ax^2+mx+nx+c$ .
5.	Factorise $ax^2+mx$ into single brackets.
6.	Factorise $nx+c$ into single brackets.
7.	Use both factorised single brackets expressions to find the double brackets expression. The first bracket consists of the terms outside the single brackets, and the second bracket consists of the expression inside the single brackets.

Example 2

Factorise

$2x^2+13x+15$

The quadratic has $a=2, b=13$ and $c=15$ . Multiply $a$ and $c$ to give $2 \times 15 = 30$ .

Think of two numbers which will multiply to give $30$ and add to give $13$ . The two numbers are $10$ and $3$ , as $10 \times 3 = 30$ and $10+3 = 13$ .

Write the quadratic expression using $10$ and $3$ , and factorise.

$\begin {aligned}&2x^2+13x+15 \\&\underbrace{2x^2 + 10x}_{\text{factorise}} + \underbrace{3x + 15}_{\text{factorise}} \\&2x(x+5) + 3(x+5)\end {aligned}$

The two expressions inside both single brackets should match, in this case $x+5$ .

$\underline{(2x+3)(x+5)}$

Example 3

Factorise and solve

$3x^2+11x-4=0$

The quadratic has $a=3, b=11$ and $c=-4$ . Multiply $a$ and $c$ to give $-12$ .

Think of two numbers which will multiply to give $-12$ and add to give $+11$ . The two numbers are $+12$ and $-1$ , as $12\times -1 = -12$ and $12-1=11$ .

Write the quadratic expression using $12$ and $-1$ , and factorise.

$\begin {aligned}3x^2+11x-4&=0 \\\underbrace{3x^2+12x}_{\text{factorise}}\space \underbrace{-1x-4}_{\text{factorise}}&=0 \\3x(x+4) -1(x+4)&=0 \\(3x-1)(x+4)&=0\end {aligned}$ 

Now the quadratic is factorised, it can be solved. Find the solutions by making either bracket equal to $0$ .

$\underline{x=\frac 1 3 \space or \space x = -4}$ 

Learn with Basics

Length:

Unit 1

X and Y coordinates

Unit 2

Quadratic graphs

Jump Ahead

Optional

Unit 3

Factorising quadratics

Final Test

Create an account to complete the exercises

FAQs - Frequently Asked Questions

How do you factorise a quadratic ax^2+bx+c when a>1?

How do you factorise into double brackets?

What do double brackets mean?

Double brackets consist of two binomial expressions multiplied together, e.g. (x+1)(y-3). Each term in the first bracket should be multiplied by each term in the second bracket.

Factorising quadratics

Videos

Summary

Exercises

Select Lesson

Exam Board

Statistics

Probability

Trigonometry

Angles and geometry

Shapes and area

Ratio proportion and rates of change

Graphs

Algebra

Number

Explainer Video

Summary

Factorising quadratics

In a nutshell

Factorise when a=1a=1a=1​​

PROCEDURE

Example 1

Factorise when a>1a>1a>1

PROCEDURE

Example 2

Example 3

Learn with Basics

Unit 1

X and Y coordinates

Unit 2

Quadratic graphs

Jump Ahead

Unit 3

Factorising quadratics

Final Test

FAQs - Frequently Asked Questions

How do you factorise a quadratic ax^2+bx+c when a>1?

How do you factorise into double brackets?

What do double brackets mean?

Factorising quadratics

Videos

Summary

Exercises

Select Lesson

Exam Board

Statistics

Probability

Trigonometry

Angles and geometry

Shapes and area

Ratio proportion and rates of change

Graphs

Algebra

Number

Explainer Video

Summary

Factorising quadratics

In a nutshell

Factorise when a=1a=1a=1​​

PROCEDURE

Example 1

Factorise when a>1a>1a>1

PROCEDURE

Example 2

Example 3

Learn with Basics

Unit 1

X and Y coordinates

Unit 2

Quadratic graphs

Jump Ahead

Unit 3

Factorising quadratics

Final Test

FAQs - Frequently Asked Questions

How do you factorise a quadratic ax^2+bx+c when a>1?

How do you factorise into double brackets?

What do double brackets mean?

Factorise when $a=1$

Factorise when $a>1$

Factorise when $a=1$

Factorise when $a>1$