MathsICSE Class 8 · Factorisation

Factorising x² + px + q into (x + a)(x + b) (ICSE Class 8 Maths)

Q: What if the leading coefficient is not 1?

Class 8 stays with x^2 + px + q (after taking out a common factor).

Q: Can the answer have fractions?

In Class 8 questions the numbers are always integers.

Updated 17 October 2026 · Fact-checked

To factorise $x^2 + px + q$, find two numbers $a$ and $b$ whose product is $q$ and whose sum is $p$; then $x^2 + px + q = (x + a)(x + b)$. Take out any common factor first, and check by expanding.

Find two numbers with product 12 and sum 7 (3 and 4); they are the side pieces of the rectangle, so the factors are (x + 3)(x + 4).

Illustration drawn for QuestionBuddy.

Understand Factorising x² + px + q

The curriculum includes factorisation of the type $(x + a) (x + b)$ .

Why it works. $(x + a) (x + b) = x^{2} + (a + b) x + ab$ . So in $x^{2} + p x + q$ we need $a + b = p$ and $ab = q$ .

Finding the numbers. List the factor pairs of $q$ and choose the pair whose sum is $p$ .

If $q > 0$ , $a$ and $b$ have the same sign as $p$ : $x^{2} + 7 x + 12 = (x + 3) (x + 4)$ and $x^{2} - 11 x + 28 = (x - 4) (x - 7)$ .
If $q < 0$ , $a$ and $b$ have opposite signs, and the larger (in size) has the sign of $p$ : $x^{2} - 5 x - 24 = (x - 8) (x + 3)$ .

Two letters. $a^{2} + 2 ab - 8 b^{2}$ : treat $b$ as part of the numbers: need numbers with sum $2$ and product $- 8$ , namely $4$ and $- 2$ , so $(a + 4 b) (a - 2 b)$ .

With a common factor. $3 x^{2} - 12 x - 63 = 3 (x^{2} - 4 x - 21) = 3 (x - 7) (x + 3)$ .

**Which values of $k$ ?** For $x^{2} + k x + 12$ to factorise over the integers, $k$ is the sum of a factor pair of $12$ : $1 + 12$ , $2 + 6$ , $3 + 4$ and their negatives, giving $k = \pm 13, \pm 8, \pm 7$ .

Check. Expand $(x + a) (x + b)$ and compare with the original.

Factorising x² + px + q: formulas to remember

Product of two binomials: $(x + a) (x + b) = x^{2} + (a + b) x + ab$; Reverse it to factorise.
Condition for factors: $a + b = p, ab = q$; Find the pair.
Sign rule: $q > 0 : same signs; q < 0 : opposite signs$; Decides the signs of $a$ and $b$ .

How to solve Factorising x² + px + q questions

List factor pairs of q, pick the one that adds to p, then write the two brackets.

1Take out any common factor.
2List factor pairs of the constant term $q$ .
3Choose the pair with sum equal to the coefficient of $x$ (watch signs).
4Write $(x + a) (x + b)$ .
5Expand to verify.

Factorising x² + px + q in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Dimensions of a plot

Suppose a rectangular plot has area $x^{2} + 9 x + 20$ square metres, where the sides are of the form $x + a$ and $x + b$ .

Find the sides, and the area and perimeter for $x = 6$ .

Try it first, then show the solutionSolution

Numbers with sum $9$ and product $20$ are $4$ and $5$ : sides $(x + 4)$ and $(x + 5)$ .
For $x = 6$ : sides $10$ and $11$ ; area $110$ (check $36 + 54 + 20 = 110$ ); perimeter $42$ .

Answer: Sides $(x + 4)$ and $(x + 5)$ ; area 110 m $^{2}$ , perimeter 42 m at $x = 6$ .

Memory hook: Sum for the middle, product for the end.

Tricks to remember Factorising x² + px + q

Short memory aids. Say each one aloud once, then use it on a question the same day.

Sum and product
Middle coefficient is the sum, last term is the product.
Use it when: Every x² + px + q.
Same or opposite
Positive constant: same signs. Negative constant: opposite signs.
Use it when: Choosing signs quickly.

Factorising x² + px + q: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Positive sum and product · 1 mark · MCQBoard level

$x^{2} + 7 x + 12$ factorises as:
(a) $(x + 2) (x + 6)$ (b) $(x + 1) (x + 12)$ (c) $(x + 3) (x + 4)$ (d) $(x - 3) (x - 4)$

Try it first, then show the solutionSolution

Numbers with product $12$ and sum $7$ : $3$ and $4$ .

Answer: (c)

Example 2: Negative constant · 3 marks · Short answerBoard level

Factorise $x^{2} - 5 x - 24$ .

Try it first, then show the solutionSolution

Need product $- 24$ , sum $- 5$ : pairs of opposite sign: $- 8$ and $3$ ( $- 8 + 3 = - 5$ ).
$(x - 8) (x + 3)$ .
Check: $x^{2} + 3 x - 8 x - 24 = x^{2} - 5 x - 24$ .

Answer: $(x - 8) (x + 3)$

Example 3: Two letters · 3 marks · Short answerHarder board level

Factorise $a^{2} + 2 ab - 8 b^{2}$ .

Try it first, then show the solutionSolution

Need numbers with sum $2$ and product $- 8$ : $4$ and $- 2$ .
$(a + 4 b) (a - 2 b)$ .
Check: $a^{2} - 2 ab + 4 ab - 8 b^{2} = a^{2} + 2 ab - 8 b^{2}$ .

Answer: $(a + 4 b) (a - 2 b)$

Example 4: With a common factor · 4 marks · Short answerHarder board level

Factorise completely $3 x^{2} - 12 x - 63$ .

Try it first, then show the solutionSolution

Common factor $3$ : $3 (x^{2} - 4 x - 21)$ .
Numbers with sum $- 4$ , product $- 21$ : $- 7$ and $3$ .
$3 (x - 7) (x + 3)$ .
Check at $x = 1$ : LHS $3 - 12 - 63 = - 72$ ; RHS $3 (- 6) (4) = - 72$ .

Answer: $3 (x - 7) (x + 3)$

Example 5: Which values of k? · 4 marks · Long answerHOTS / competency

Find all integer values of $k$ for which $x^{2} + k x + 12$ can be factorised into two linear factors with integer coefficients.

Try it first, then show the solutionSolution

Factor pairs of $12$ : $(1, 12), (2, 6), (3, 4)$ and $(- 1, - 12), (- 2, - 6), (- 3, - 4)$ .
$k$ is the sum of a pair: $13, 8, 7, - 13, - 8, - 7$ .
Pairs with opposite signs give a product of $- 12$ , not $12$ , so they are excluded.

Answer: $k = \pm 7, \pm 8, \pm 13$

Where students lose marks in Factorising x² + px + q

The slips examiners see most often, and what to write instead.

Mistake: Choosing a pair with the right product but wrong sum
Both conditions must hold.
Wrong working
$x^{2} - 5 x - 24 = (x - 6) (x + 4)$ .
Right working
$(x - 8) (x + 3)$ ; $(x - 6) (x + 4)$ expands to $x^{2} - 2 x - 24$ .
Do this: Expand to check the middle term.
Usually costs: Answer mark.
Mistake: Wrong signs for a negative constant
Opposite signs are needed, and the larger one takes the sign of the middle term.
Wrong working
$x^{2} - x - 12 = (x + 4) (x - 3)$ .
Right working
$(x - 4) (x + 3)$ .
Do this: Place the larger number with the sign of p.
Usually costs: Answer mark.

Exam tips

List factor pairs systematically; do not guess.
Check by expanding to confirm the middle term.
When a leading common factor exists, keep it in the final answer.

Practice questions on Factorising x² + px + q

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
$x^{2} - x - 12$ factorises as:
(a) $(x - 3) (x + 4)$ (b) $(x - 4) (x + 3)$ (c) $(x - 6) (x + 2)$ (d) $(x + 4) (x + 3)$
Hint
Product -12, sum -1.
Answer
(b)
Q2 · 2 marks · Short answerBoard level
Factorise $y^{2} + 9 y + 14$ .
Hint
Product 14, sum 9.
Answer
$(y + 2) (y + 7)$ .
Q3 · 3 marks · Short answerHarder board level
Factorise $m^{2} - 14 m + 45$ .
Hint
Both numbers are negative.
Answer
$(m - 5) (m - 9)$ (sum $- 14$ , product $45$ ).
Q4 · 3 marks · Short answerHarder board level
Factorise completely $2 x^{3} - 2 x^{2} - 24 x$ .
Hint
Take out 2x first.
Answer
$2 x (x^{2} - x - 12) = 2 x (x - 4) (x + 3)$ .
Q5 · 4 marks · Case-basedHOTS / competency
A rectangle has area $x^{2} + 2 x - 35$ and one side $x + 7$ . Find the other side, and the area and perimeter for $x = 9$ .
Hint
Factorise the area.
Answer
$x^{2} + 2 x - 35 = (x + 7) (x - 5)$ , so the other side is $x - 5$ . For $x = 9$ : sides $16$ and $4$ ; area $64$ (check $81 + 18 - 35 = 64$ ); perimeter $40$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Factorise x^2 + 5x + 6.Show answer
(x + 2)(x + 3).
2.Factors of x^2 - 7x + 12?Show answer
(x - 3)(x - 4).
3.If the constant term is negative, the signs of a and b are?Show answer
Opposite.
4.Sum and product conditions for (x + a)(x + b)?Show answer
a + b = p, ab = q.

Factorising x² + px + q into (x + a)(x + b) (ICSE Class 8 Maths)

Understand Factorising x² + px + q

Factorising x² + px + q: formulas to remember

How to solve Factorising x² + px + q questions

Factorising x² + px + q in real life

Tricks to remember Factorising x² + px + q

Factorising x² + px + q: solved examples

Where students lose marks in Factorising x² + px + q

Exam tips

Practice questions on Factorising x² + px + q

Test your recall

Factorising x² + px + q: frequently asked questions

What if the leading coefficient is not 1?

Can the answer have fractions?