Algebra

Factorising

Q: How do you factorise a quadratic?

For x^2 + bx + c, find two numbers that multiply to c and add to b, then put them in two brackets with x. Expand your answer to check it.

Q: What is the difference of two squares?

It is one square minus another, like x^2 - 49. It always factorises as (a + b)(a - b), so x^2 - 49 = (x + 7)(x - 7).

10 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & Higher4 of 10 from Foundation papers

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Revision guide by The Full Marks

Factorising: what you need to know

Factorising is the reverse of expanding: you put an expression back into brackets. It is used to solve quadratics, simplify algebraic fractions and write proofs, so it is worth getting quick at.

Foundation covers common factors, $x^2 + bx + c$ and the difference of two squares; quadratics with $a \neq 1$ , like $3x^2 - 10x - 8$ , are Higher only.

Key facts

Always take out the highest common factor first: $12y^2 + 18y = 6y(2y + 3)$ .
$x^2 + bx + c = (x + p)(x + q)$ where $p + q = b$ and $pq = c$ .
Difference of two squares: $a^2 - b^2 = (a + b)(a - b)$ . It works with numbers and coefficients too: $9x^2 - 16 = (3x + 4)(3x - 4)$ .
You can always check a factorisation by expanding it again.

Factorising $x^2 + bx + c$

Find two numbers that multiply to $c$ and add to $b$ .
Write them in the brackets: $(x + p)(x + q)$ .
Expand to check.

Factorising $ax^2 + bx + c$ (Higher)

Find two numbers that multiply to $ac$ and add to $b$ .
Split the middle term using those numbers.
Factorise each pair, then take out the common bracket. This is called grouping, and it also works on four-term expressions like $xy + 3x + 2y + 6 = x(y + 3) + 2(y + 3) = (x + 2)(y + 3)$ .

Worked examples

Example 1 Foundation · grade 4

Factorise fully (a) $12y^2 + 18y$ (b) $x^2 + 2x - 15$ (c) $x^2 - 64$

Show the solution

(a) HCF of $12y^2$ and $18y$ is $6y$ .
$12y^2 + 18y = 6y(2y + 3)$
(b) Two numbers that multiply to $-15$ and add to $2$ : $5$ and $-3$ .
$x^2 + 2x - 15 = (x + 5)(x - 3)$
(c) $x^2 - 64 = x^2 - 8^2$ , a difference of two squares.
$x^2 - 64 = (x + 8)(x - 8)$

Example 2 Higher · grade 6

Factorise $3x^2 - 10x - 8$

Show the solution

$ac = 3 \times -8 = -24$ . Numbers that multiply to $-24$ and add to $-10$ : $-12$ and $2$ .
Split: $3x^2 - 12x + 2x - 8$
Factorise pairs: $3x(x - 4) + 2(x - 4)$
$= (3x + 2)(x - 4)$

Common mistakes

Not factorising fully: $3(3x^2 + 2x)$ still has a common factor $x$ , so the answer is $3x(3x + 2)$ .
Stopping at $2(x^2 - 25)$ . Spot the difference of two squares: $2(x + 5)(x - 5)$ .
Mixing up the sum and product, or getting the signs the wrong way round: $x^2 - 3x - 4$ is $(x - 4)(x + 1)$ , not $(x + 4)(x - 1)$ . Expand your answer to check.

Exam tips

"Factorise fully" is a hint that there is more than one factor to take out. A partial factorisation usually earns only some of the marks.
If a question says "hence", use the factorised form from the previous part.
Higher papers use the difference of two squares in other questions too, e.g. to simplify $(x + 5)^2 - (x - 1)^2$ or in a proof. Look out for one square minus another.

Questions people ask

How do you factorise a quadratic?

For $x^2 + bx + c$ , find two numbers that multiply to $c$ and add to $b$ , then put them in two brackets with $x$ . Expand your answer to check it.

What is the difference of two squares?

It is one square minus another, like $x^2 - 49$ . It always factorises as $(a + b)(a - b)$ , so $x^2 - 49 = (x + 7)(x - 7)$ .

Exam questions: Factorising (10, easiest first)

Factorising2 marksGrade 4CalculatorFoundation

June 2024 · Paper 3 Foundation · Q21

Factorise a linear expression and a simple quadratic expression.

Show answer

(a)	$3(2x - 5)$	1 mark
(b)	$m(m + 5)$	1 mark

Worked solution Checked against the mark scheme

(a) The HCF of 6 and 15 is 3: $6x - 15 = 3(2x - 5)$

(b) Both terms contain $m$ : $m^2 + 5m = m(m + 5)$

Watch out: Only partly factorising, or writing $m(m + 5m)$ .

Factorising4 marksGrade 4CalculatorFoundation

November 2024 · Paper 2 Foundation · Q19

Factorise a quadratic expression fully and show an inequality on a number line.

Show answer

(a)	$5w(3w - 1)$	2 marks
(b)	Open circle at $-2$ , closed circle at $4$ , joined by a line	2 marks

Worked solution Checked against the mark scheme

(a) The HCF of $15w^2$ and $5w$ is $5w$ .

$15w^2 - 5w = 5w(3w - 1)$

(b) $-2$ is not included: open circle. $4$ is included: filled circle. Join them with a line.

Watch out: In (a), taking out only 5 or only $w$ , which is not fully factorised.

Factorising1 markGrade 5No calculatorFoundation

Specimen · Paper 1 Foundation · Q28

Show answer

Answer	$(x + 4)(x - 4)$	1 mark

Worked solution Checked against the mark scheme

$x^2 - 16 = x^2 - 4^2$ , a difference of two squares.

So $x^2 - 16 = (x + 4)(x - 4)$

Watch out: Writing (x − 4)(x − 4) or (x − 8)(x + 8).

Factorising2 marksGrade 5CalculatorFoundation

Specimen · Paper 3 Foundation · Q26

Factorise a quadratic expression with a coefficient of 1 on x squared.

Show answer

Answer	$(x - 1)(x + 4)$	2 marks

Worked solution Checked against the mark scheme

Find two numbers that multiply to −4 and add to 3: $4$ and $-1$ .

$x^2 + 3x - 4 = (x + 4)(x - 1)$

Watch out: Getting the signs the wrong way round: $(x + 1)(x - 4)$ .

Factorising2 marksGrade 5CalculatorHigher

Specimen · Paper 3 Higher · Q6

Factorise a quadratic expression with a coefficient of 1 on the squared term.

Show answer

Answer	$(x + 4)(x - 1)$	2 marks

Worked solution Checked against the mark scheme

Find two numbers that multiply to $-4$ and add to $3$ : these are $4$ and $-1$ .

$x^2 + 3x - 4 = (x + 4)(x - 1)$

Watch out: Getting the signs the wrong way round: (x − 4)(x + 1).

Factorising5 marksGrade 6No calculatorHigher

November 2024 · Paper 1 Higher · Q11

Correct a mix-up about the sum and product when factorising a quadratic, then factorise a difference of two squares and an expression by grouping.

Show answer

(a)	The sum and product are swapped: $a + b$ should be 5 and $ab$ should be 6	1 mark
(b)	$2(m - 1)(m + 1)$	2 marks
(c)	$(a + b)(x - y)$	2 marks

Worked solution Checked against the mark scheme

(a) Expanding $(x + a)(x + b)$ gives $x^2 + (a + b)x + ab$ , so $a + b = 5$ and $ab = 6$ . Kate has them the wrong way round.

(b) $2m^2 - 2 = 2(m^2 - 1) = 2(m - 1)(m + 1)$

Watch out: In (b), stopping at 2(m² − 1) and not using the difference of two squares.

Factorising5 marksGrade 6CalculatorHigher

June 2022 · Paper 3 Higher · Q14

Fully factorise a difference of two squares with a common factor, and expand a product of three brackets into a cubic.

Show answer

(a)	$4(p-3)(p+3)$	2 marks
(b)	$6m^3 + 11m^2 - 57m - 20$	3 marks

Worked solution Checked against the mark scheme

(a) $4p^2 - 36 = 4(p^2 - 9) = 4(p - 3)(p + 3)$

(b) $(m + 4)(2m - 5) = 2m^2 + 3m - 20$

$(2m^2 + 3m - 20)(3m + 1) = 6m^3 + 2m^2 + 9m^2 + 3m - 60m - 20$

$= 6m^3 + 11m^2 - 57m - 20$

Watch out: In (a), stopping at 4(p² − 9) without factorising fully.

Factorising3 marksGrade 6No calculatorHigher

November 2018 · Paper 1 Higher · Q10

$Simplify an algebraic fraction and factorise an expression fully using the difference of two squares.$

Show answer

(a)	$\dfrac{1}{5(x-1)}$	1 mark
(b)	$2(5+y)(5-y)$	2 marks

Worked solution Checked against the mark scheme

(a) Cancel one factor of $(x-1)$ from top and bottom: $\dfrac{1}{5(x-1)}$

(b) Take out the common factor 2: $2(25 - y^2)$

$25 - y^2$ is a difference of two squares: $2(5+y)(5-y)$

Watch out: Stopping at $2(25 - y^2)$ without factorising fully.

Factorising3 marksGrade 7CalculatorHigher

June 2023 · Paper 3 Higher · Q15

Factorise a difference of two squares, then use it to show that a power of 2 minus 1 is the product of two consecutive odd numbers.

Show answer

(a)	$(a - b)(a + b)$	1 mark
(b)	$2^{40} - 1 = (2^{20} - 1)(2^{20} + 1)$ ; $2^{20}$ is even, so both factors are odd and differ by 2	2 marks

Worked solution Checked against the mark scheme

(a) $a^2 - b^2 = (a - b)(a + b)$

(b) $2^{40} = (2^{20})^2$ , so $2^{40} - 1 = (2^{20} - 1)(2^{20} + 1)$

$2^{20}$ is even, so $2^{20} - 1$ and $2^{20} + 1$ are both odd, and they differ by 2, so they are consecutive odd numbers ( $1\,048\,575$ and $1\,048\,577$ ).

Watch out: In (b), factorising but not explaining why the two factors are odd and consecutive.

Factorising4 marksGrade 7No calculatorHigher

June 2018 · Paper 1 Higher · Q15

Factorise a difference of two squares and use it to simplify the difference of two squared brackets.

Show answer

(a)	$(a - b)(a + b)$	1 mark
(b)	$12(x^2 + 1)$	3 marks

Worked solution Checked against the mark scheme

(a) $a^2 - b^2 = (a - b)(a + b)$

(b) Let $a = x^2 + 4$ and $b = x^2 - 2$ .

$a + b = 2x^2 + 2$ and $a - b = 6$

$(a + b)(a - b) = 6(2x^2 + 2) = 12x^2 + 12 = 12(x^2 + 1)$

Watch out: Not changing the sign of the −2 when subtracting the second bracket.

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

November 2024 · Paper 1 Higher · Q13: Form and solve an equation from the difference in volume of two cuboids with algebraic side lengths.
November 2024 · Paper 2 Foundation · Q11: Simplify a product of two terms and collect like terms in an expression.
June 2024 · Paper 1 Foundation · Q17: Solve a linear equation with brackets.
June 2024 · Paper 1 Higher · Q20: Use index laws with a cube root and a power of root 2 to find an unknown power.
June 2024 · Paper 3 Higher · Q15: Simplify an algebraic fraction, factorise a quadratic with a non-unit coefficient, and divide algebraic fractions.
November 2023 · Paper 3 Higher · Q23: Solve an equation of algebraic fractions in x squared and y squared to find the ratio x to y in its simplest form.
June 2023 · Paper 2 Foundation · Q4: Simplify a product of a number and an algebraic term.
June 2023 · Paper 3 Higher · Q24: Find an algebraic probability of picking two counters of the same colour without replacement, using a ratio to write everything in one variable.
November 2022 · Paper 1 Higher · Q16: Prove an algebraic identity for the difference of two squared expressions, then judge a claim about odd numbers that follows from it.
November 2022 · Paper 1 Foundation · Q21: Simplify a squared product of powers of 2 and give the answer as a single power.
November 2022 · Paper 1 Foundation · Q26: Solve a linear inequality with a fraction, then factorise a quadratic.
June 2022 · Paper 1 Foundation · Q2: Collect four identical letter terms into a single term.
June 2022 · Paper 3 Foundation · Q28: Show that the area of a shaded trapezium in a rectangle is a given quadratic expression in x.
November 2021 · Paper 3 Foundation · Q6: Simplify a product of letters and a number, then collect like terms in an expression.
November 2020 · Paper 1 Foundation · Q30: Rearrange a linear formula to change the subject, and simplify a negative power raised to a negative power.
November 2020 · Paper 2 Higher · Q10: Simplify an expression raised to the power 0, cancel an algebraic fraction, and raise a product of powers to a power.
November 2020 · Paper 2 Higher · Q14: Set up and solve an equation for two squares whose sides differ by 4 cm and areas differ by 70 cm², then find the larger area.
November 2020 · Paper 3 Higher · Q12: Add two algebraic fractions into one simplified fraction, and expand a product of three brackets.
November 2019 · Paper 1 Foundation · Q13: Simplify a product of two terms, then collect like terms in an expression.
November 2019 · Paper 1 Higher · Q13: Add two algebraic fractions, then factorise an expression with a repeated bracket.
November 2019 · Paper 3 Higher · Q10: Solve a linear equation with a fraction, and simplify an algebraic fraction with a repeated bracket.
June 2019 · Paper 1 Higher · Q21: Find an inverse function, then use composite functions to show that a given quadratic equation holds.
June 2019 · Paper 2 Foundation · Q8: Simplify three algebraic products, the last one a fraction of letters that cancels.
November 2018 · Paper 2 Foundation · Q19: Solve a linear equation with a bracket and factorise an expression fully.
June 2018 · Paper 1 Foundation · Q16: Substitute values into a formula, expand a single bracket and solve a linear equation with brackets.
Specimen · Paper 1 Higher · Q14: Solve an equation with two algebraic fractions that have x in the denominators.
Specimen · Paper 1 Higher · Q21: Find an algebraic expression for the probability of picking one pen of each colour without replacement.
Specimen · Paper 2 Foundation · Q4: Simplify three short algebraic expressions by collecting like terms and multiplying.
Specimen · Paper 2 Foundation · Q16: Solve two linear equations and simplify a power raised to a power.
Specimen · Paper 2 Higher · Q18: Substitute x + 2 into a quadratic function and simplify to the form ax² + bx.
Specimen · Paper 2 Higher · Q19: Form and solve a quadratic from the area of a right-angled triangle, then use Pythagoras to find its perimeter.
Specimen · Paper 3 Foundation · Q7: Solve two simple linear equations and simplify a product of powers of d.

Factorising

Factorising: what you need to know

Key facts

Factorising x2+bx+cx^2 + bx + cx2+bx+c

Factorising ax2+bx+cax^2 + bx + cax2+bx+c (Higher)

Worked examples

Common mistakes

Exam tips

Questions people ask

Exam questions: Factorising (10, easiest first)

Also tested in these questions

Factorising $x^2 + bx + c$

Factorising $ax^2 + bx + c$ (Higher)