Factorising ax2 + bx + c Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Factorising ax2 + bx + c questions. See exactly how to solve problems on factorising quadratics, AC method, splitting the middle term, common factor.

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GCSE Higher70 questionsStep-by-step solutions

Question 1

1 markeasy

Factorise

2x^2 + 3x + 1

Worked solution

Set up the brackets
$2x^2 + 3x + 1 = (2x\ \ )(x\ \ )$
The $x^2$ term is $2x^2 = 2x \times x$ , so each bracket starts with $2x$ and $x$ .
Find the number terms
$1 \times 1 = 1, \quad 2x + 1x = 3x$
The two numbers must multiply to give $1$ and, after cross-multiplying, give the middle term $3x$ . Those numbers are $1$ and $1$ .
Write the factorised form
$(2x + 1)(x + 1)$
So $2x^2 + 3x + 1 = (2x + 1)(x + 1)$ . Expanding $(2x + 1)(x + 1)$ returns $2x^2 + 3x + 1$ , confirming the factorisation.

Answer

(2x + 1)(x + 1)

Question 2

1 markeasy

Which of the following is the correct factorisation of

2x^2 + 5x + 2

Worked solution

Set up the brackets
$2x^2 + 5x + 2 = (2x\ \ )(x\ \ )$
The $x^2$ term is $2x^2 = 2x \times x$ , so each bracket starts with $2x$ and $x$ .
Find the number terms
$1 \times 2 = 2, \quad 4x + 1x = 5x$
The two numbers must multiply to give $2$ and, after cross-multiplying, give the middle term $5x$ . Those numbers are $1$ and $2$ .
Write the factorised form
$(2x + 1)(x + 2)$
So $2x^2 + 5x + 2 = (2x + 1)(x + 2)$ . Expanding $(2x + 1)(x + 2)$ returns $2x^2 + 5x + 2$ , confirming the factorisation.

Answer

(2x + 1)(x + 2)

Question 3

1 markeasy

Factorise

2x^2 + 7x + 3

Worked solution

Set up the brackets
$2x^2 + 7x + 3 = (2x\ \ )(x\ \ )$
The $x^2$ term is $2x^2 = 2x \times x$ , so each bracket starts with $2x$ and $x$ .
Find the number terms
$1 \times 3 = 3, \quad 6x + 1x = 7x$
The two numbers must multiply to give $3$ and, after cross-multiplying, give the middle term $7x$ . Those numbers are $1$ and $3$ .
Write the factorised form
$(2x + 1)(x + 3)$
So $2x^2 + 7x + 3 = (2x + 1)(x + 3)$ . Expanding $(2x + 1)(x + 3)$ returns $2x^2 + 7x + 3$ , confirming the factorisation.

Answer

(2x + 1)(x + 3)

Question 4

1 markeasy

Factorise

3x^2 + 4x + 1

Worked solution

Set up the brackets
$3x^2 + 4x + 1 = (3x\ \ )(x\ \ )$
The $x^2$ term is $3x^2 = 3x \times x$ , so each bracket starts with $3x$ and $x$ .
Find the number terms
$1 \times 1 = 1, \quad 3x + 1x = 4x$
The two numbers must multiply to give $1$ and, after cross-multiplying, give the middle term $4x$ . Those numbers are $1$ and $1$ .
Write the factorised form
$(3x + 1)(x + 1)$
So $3x^2 + 4x + 1 = (3x + 1)(x + 1)$ . Expanding $(3x + 1)(x + 1)$ returns $3x^2 + 4x + 1$ , confirming the factorisation.

Answer

(3x + 1)(x + 1)

Question 5

1 markeasy

Factorise

2x^2 + 7x + 5

Worked solution

Set up the brackets
$2x^2 + 7x + 5 = (2x\ \ )(x\ \ )$
The $x^2$ term is $2x^2 = 2x \times x$ , so each bracket starts with $2x$ and $x$ .
Find the number terms
$5 \times 1 = 5, \quad 2x + 5x = 7x$
The two numbers must multiply to give $5$ and, after cross-multiplying, give the middle term $7x$ . Those numbers are $5$ and $1$ .
Write the factorised form
$(2x + 5)(x + 1)$
So $2x^2 + 7x + 5 = (2x + 5)(x + 1)$ . Expanding $(2x + 5)(x + 1)$ returns $2x^2 + 7x + 5$ , confirming the factorisation.

Answer

(2x + 5)(x + 1)

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