Sec 2 · G2 (N(A))

Expansion and factorisation of quadratic expressions

Practise expansion and factorisation of quadratic expressions with step-by-step help. Free account.

1. Expanding two brackets

Each term × each term

\displaystyle (x + 3)(x + 2)

A rectangle x + 3 wide and x + 2 tall split into x², 3x, 2x and 6.

e.g.

\displaystyle (x + 3)(x + 2)

\displaystyle = x^{2} + 2x + 3x + 6

\displaystyle = x^{2} + 5x + 6

Watch the signs

\displaystyle (2x - 5)(x + 4)

e.g.

\displaystyle (2x - 5)(x + 4)

\displaystyle = 2x^{2} + 8x - 5x - 20

\displaystyle = 2x^{2} + 3x - 20

📖 Worked example

Expand and simplify $(x - 7)(x - 1)$ .

x

-1

x

-7

Step 1 of 5

Set up the box

Write one bracket down the side (

x

and

-7

) and the other across the top (

x

and

-1

Practise this →

2. Special products

Identity	Example
$(a + b)^{2} = a^{2} + 2ab + b^{2}$	$(x + 4)^{2} = x^{2} + 8x + 16$
$(a - b)^{2} = a^{2} - 2ab + b^{2}$	$(3x - 1)^{2} = 9x^{2} - 6x + 1$
$(a + b)(a - b) = a^{2} - b^{2}$	$(x + 7)(x - 7) = x^{2} - 49$

📖 Worked example

Expand $(4x - 3)^2$ .

4x

-3

4x

-3

Step 1 of 5

Set up the box

Write one bracket down the side (

4x

and

-3

) and the other across the top (

4x

and

-3

Practise this →

3. Factorising quadratics

Two numbers that multiply to 6 and add to 5

\displaystyle x^{2} + 5x + 6

e.g.

\displaystyle x^{2} + 5x + 6

\displaystyle = (x + 2)(x + 3)

Numbers that multiply to −12 and add to −1

\displaystyle x^{2} - x - 12

e.g.

\displaystyle x^{2} - x - 12

\displaystyle = (x - 4)(x + 3)

Coefficient of x² not 1: try the factor pairs

\displaystyle 2x^{2} + x - 3

e.g.

\displaystyle 2x^{2} + x - 3

\displaystyle = (2x + 3)(x - 1)

Difference of two squares

\displaystyle a^{2} - b^{2} = (a + b)(a - b)

e.g.

\displaystyle 9x^{2} - 25

\displaystyle = (3x + 5)(3x - 5)

📖 Worked example

Factorise $x^{2} - 7x - 18$ .

x^{2}

-18

Step 1 of 6

Set up the box

Draw a 2 × 2 box. Put

x^{2}

in the top-left and

-18

in the bottom-right.

Practise this →

4. Grouping, a² − b² and the common factor first

Form	Method	Example
$ax + ay + bx + by$	pairs, then the common bracket	$3x + 3y + ax + ay = (x + y)(3 + a)$
$ax - 2a - bx + 2b$	take out $-b$ : the sign inside changes	$a(x - 2) - b(x - 2) = (x - 2)(a - b)$
$a^{2} - b^{2}$	$(a + b)(a - b)$	$9x^{2} - 25y^{2} = (3x + 5y)(3x - 5y)$
common factor	take it out FIRST	$2x^{2} - 18 = 2(x + 3)(x - 3)$
perfect square − square	$(u - v)^{2} - w^{2}$	$x^{2} - 2xy + y^{2} - 9 = (x - y + 3)(x - y - 3)$

📖 Worked example

Factorise $4mx - 7x - 12my + 21y$ completely.

Step 1 of 2

Group the terms in pairs and take out the HCF of each pair (a minus HCF changes the signs in its bracket)

4mx - 7x - 12my + 21y = x(4m - 7) - 3y(4m - 7)

📖 Worked example

Factorise $100x^{3} - 16x$ completely.

Step 1 of 2

Take out the HCF of the terms

100x^{3} - 16x = 4x(25x^{2} - 4)

Practise this →

5. Using the identities

Given	Use	Example
$a + b$ , $ab$	$a^{2} + b^{2} = (a + b)^{2} - 2ab$	$a + b = 7$ , $ab = 10$ : $49 - 20 = 29$
$a - b$ , $ab$	$a^{2} + b^{2} = (a - b)^{2} + 2ab$	$a - b = 3$ , $ab = 10$ : $9 + 20 = 29$
$A^{2} - B^{2}$ (numbers)	$(A + B)(A - B)$	$73^{2} - 27^{2} = 100 \times 46$
multiple of $k$	simplify, take out $k$	$(n + 3)^{2} - (n - 1)^{2} = 8(n + 1)$

📖 Worked example

Given that $a + 4b = 26$ and $a - 4b = -38$ , find the value of $a^{2} - 16b^{2}$ .

Step 1 of 3

Write a² - 16b² using what you are given

a^{2} - 16b^{2} = (a + 4b)(a - 4b)

📖 Worked example

Show that $(n + 1)^{2} - (n - 7)^{2}$ is a multiple of $16$ for every positive integer $n$ .

Step 1 of 3

Expand each square

(n + 1)^{2} - (n - 7)^{2} = (n^{2} + 2n + 1) - (n^{2} - 14n + 49)

Practise this →

Practise expansion and factorisation of quadratic expressions with step-by-step help

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Sec 2 · G2 (N(A))

Expansion and factorisation of quadratic expressions

Practise expansion and factorisation of quadratic expressions with step-by-step help. Free account.

Start free

1. Expanding two brackets

Each term × each term

\displaystyle (x + 3)(x + 2)

e.g.

\displaystyle (x + 3)(x + 2)

\displaystyle = x^{2} + 2x + 3x + 6

\displaystyle = x^{2} + 5x + 6

Watch the signs

\displaystyle (2x - 5)(x + 4)

e.g.

\displaystyle (2x - 5)(x + 4)

\displaystyle = 2x^{2} + 8x - 5x - 20

\displaystyle = 2x^{2} + 3x - 20

📖 Worked example

Expand and simplify $(x - 7)(x - 1)$ .

x

-1

x

-7

Step 1 of 5

Set up the box

Write one bracket down the side (

x

and

-7

) and the other across the top (

x

and

-1

Practise this →

2. Special products

Identity	Example
$(a + b)^{2} = a^{2} + 2ab + b^{2}$	$(x + 4)^{2} = x^{2} + 8x + 16$
$(a - b)^{2} = a^{2} - 2ab + b^{2}$	$(3x - 1)^{2} = 9x^{2} - 6x + 1$
$(a + b)(a - b) = a^{2} - b^{2}$	$(x + 7)(x - 7) = x^{2} - 49$

📖 Worked example

Expand $(4x - 3)^2$ .

4x

-3

4x

-3

Step 1 of 5

Set up the box

Write one bracket down the side (

4x

and

-3

) and the other across the top (

4x

and

-3

Practise this →

3. Factorising quadratics

Two numbers that multiply to 6 and add to 5

\displaystyle x^{2} + 5x + 6

e.g.

\displaystyle x^{2} + 5x + 6

\displaystyle = (x + 2)(x + 3)

Numbers that multiply to −12 and add to −1

\displaystyle x^{2} - x - 12

e.g.

\displaystyle x^{2} - x - 12

\displaystyle = (x - 4)(x + 3)

Coefficient of x² not 1: try the factor pairs

\displaystyle 2x^{2} + x - 3

e.g.

\displaystyle 2x^{2} + x - 3

\displaystyle = (2x + 3)(x - 1)

Difference of two squares

\displaystyle a^{2} - b^{2} = (a + b)(a - b)

e.g.

\displaystyle 9x^{2} - 25

\displaystyle = (3x + 5)(3x - 5)

📖 Worked example

Factorise $x^{2} - 7x - 18$ .

x^{2}

-18

Step 1 of 6

Set up the box

Draw a 2 × 2 box. Put

x^{2}

in the top-left and

-18

in the bottom-right.

Practise this →

4. Grouping, a² − b² and the common factor first

Form	Method	Example
$ax + ay + bx + by$	pairs, then the common bracket	$3x + 3y + ax + ay = (x + y)(3 + a)$
$ax - 2a - bx + 2b$	take out $-b$ : the sign inside changes	$a(x - 2) - b(x - 2) = (x - 2)(a - b)$
$a^{2} - b^{2}$	$(a + b)(a - b)$	$9x^{2} - 25y^{2} = (3x + 5y)(3x - 5y)$
common factor	take it out FIRST	$2x^{2} - 18 = 2(x + 3)(x - 3)$
perfect square − square	$(u - v)^{2} - w^{2}$	$x^{2} - 2xy + y^{2} - 9 = (x - y + 3)(x - y - 3)$

📖 Worked example

Factorise $4mx - 7x - 12my + 21y$ completely.

Step 1 of 2

Group the terms in pairs and take out the HCF of each pair (a minus HCF changes the signs in its bracket)

4mx - 7x - 12my + 21y = x(4m - 7) - 3y(4m - 7)

📖 Worked example

Factorise $100x^{3} - 16x$ completely.

Step 1 of 2

Take out the HCF of the terms

100x^{3} - 16x = 4x(25x^{2} - 4)

Practise this →

5. Using the identities

Given	Use	Example
$a + b$ , $ab$	$a^{2} + b^{2} = (a + b)^{2} - 2ab$	$a + b = 7$ , $ab = 10$ : $49 - 20 = 29$
$a - b$ , $ab$	$a^{2} + b^{2} = (a - b)^{2} + 2ab$	$a - b = 3$ , $ab = 10$ : $9 + 20 = 29$
$A^{2} - B^{2}$ (numbers)	$(A + B)(A - B)$	$73^{2} - 27^{2} = 100 \times 46$
multiple of $k$	simplify, take out $k$	$(n + 3)^{2} - (n - 1)^{2} = 8(n + 1)$

📖 Worked example

Given that $a + 4b = 26$ and $a - 4b = -38$ , find the value of $a^{2} - 16b^{2}$ .

Step 1 of 3

Write a² - 16b² using what you are given

a^{2} - 16b^{2} = (a + 4b)(a - 4b)

📖 Worked example

Show that $(n + 1)^{2} - (n - 7)^{2}$ is a multiple of $16$ for every positive integer $n$ .

Step 1 of 3

Expand each square

(n + 1)^{2} - (n - 7)^{2} = (n^{2} + 2n + 1) - (n^{2} - 14n + 49)

Practise this →

Practise expansion and factorisation of quadratic expressions with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free