Year 10 - Quadratic factorising - Free practice

Year 10 Quadratic Factorising Worksheets

Expand, factorise, difference of squares - the building blocks.

Quick answer

How do you factorise quadratic expressions?

To factorise a monic quadratic like x² + bx + c, find two numbers that multiply to c and add to b, then write it as (x + p)(x + q). For a difference of two squares, use a² - b² = (a + b)(a - b). For example, x² + 5x + 6 = (x + 2)(x + 3) because 2 × 3 = 6 and 2 + 3 = 5.

Generate a free maths worksheet

Six questions across Foundation, Standard and Extension, with full worked solutions, written to the Australian Curriculum.

Want more than a worksheet?

Practice built from mistakes.

A worksheet on Year 10 quadratic factorising is a good start. Tuterly goes further: it turns any question students get wrong into fresh practice, at the right level, on what’s likely to be in the next test or exam.

~90% of Bayside Academics students using it weekly improved their test marks

Of the Bayside Academics students who used Tuterly at least once a week, about 90% improved their school test marks.

Based on the actual school test marks of 100+ Bayside Academics students (in person and online) in Years 7 to 10, before and after using Tuterly at least once a week. How we measured this

Free worksheet Tuterly

The questionsFree worksheetThe same questions for every student on this topic.TuterlyWritten from the questions they got wrong in homework and school tests, at their level.

How manyFree worksheetOne sheet at a time.TuterlyAs many fresh questions as they need, each one different, never just the numbers swapped.*

Before a testFree worksheetPick a topic and hope it’s the right one.TuterlyA practice test covering every part of the topic. Upload the term planner and it focuses on what’s coming up.

MarkingFree worksheetCheck the answer key yourself.TuterlyMarked as they go, with the exact step that went wrong explained.

ProgressFree worksheetNothing is saved.TuterlyEvery question tracked until it’s mastered, and a weekly review for parents.

For Years 7 to 10: $8.99 a week or $29.99 a month at the launch price (from 1 April 2027, $17.99 a week or $59.99 a month), with a 7-day free trial on the monthly plan only. Better marks in 6 months, or your money back.

Or try it free now on one of their questions →

Coming soon on iPhone. Android soon after.

Free to download · 7-day free trial on monthly · Years 7 to 10 maths · On iPhone. Android coming soon

* With a subscription. Fair use applies: up to 80 snapped questions and 120 new practice questions a day per student.

Worked examples

Year 10 quadratic factorising: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Factorising x² + bx + c
Factorise $x^2 + 7x + 12$ .
Show worked solution
Key idea: Find two integers that multiply to $12$ and add to $7$ .
1. List factor pairs of $12$ : $(1, 12),\ (2, 6),\ (3, 4)$ .
2. Check which pair adds to $7$ : $3 + 4 = 7$ . ✓
3. Write the factorised form: $(x + 3)(x + 4)$ .
4. Verify by expanding: $(x + 3)(x + 4) = x^2 + 4x + 3x + 12 = x^2 + 7x + 12$ . ✓
Answer: $(x + 3)(x + 4)$
Example 2 · Difference of two squares and perfect-square trinomials
Factorise each expression.

(a) $x^2 - 49$

(b) $x^2 - 10x + 25$
Show worked solution
Key idea: Recognise special quadratic forms: $a^2 - b^2 = (a-b)(a+b)$ and $a^2 - 2ab + b^2 = (a - b)^2$ .

Part (a)
1. Write $x^2 - 49$ as $x^2 - 7^2$ .
2. Apply the difference of two squares rule: $(x - 7)(x + 7)$ .
Part (b)
1. Check: $\sqrt{25} = 5$ and the middle term $-10x = -2 \times x \times 5$ . ✓
2. This is a perfect-square trinomial: $(x - 5)^2$ .
3. Verify: $(x-5)^2 = x^2 - 10x + 25$ . ✓
Answer: (a) $(x - 7)(x + 7)$ (b) $(x - 5)^2$
Example 3 · Real-life area problem
A rectangular garden has an area of $x^2 + 11x + 28$ square metres, where $x$ is a positive number. The length of the garden is $(x + 7)$ metres. Write an expression for the width of the garden by factorising the area expression, then find the width when $x = 3$ .
Show worked solution
Key idea: Area of a rectangle equals length multiplied by width, so width equals area divided by length. Factorise the area expression first.
1. Find two integers that multiply to $28$ and add to $11$ : $4 + 7 = 11$ and $4 \times 7 = 28$ . ✓
2. Factorise: $x^2 + 11x + 28 = (x + 4)(x + 7)$ .
3. Width $= \dfrac{(x + 4)(x + 7)}{(x + 7)} = x + 4$ metres.
4. Substitute $x = 3$ : width $= 3 + 4 = 7$ metres.
Answer: Width $= (x + 4)$ metres; when $x = 3$ , the width is $7$ metres.

Common mistakes

Where students usually lose marks

Wrong signs when factorising

$x^2 - 5x + 6 = (x + 2)(x + 3)$ (ignoring the negative signs)

Always check both the product and the sum with their signs. Here you need two numbers that multiply to $+6$ and add to $-5$ , which are $-2$ and $-3$ , giving $(x - 2)(x - 3)$ . Expand your answer to check.

Difference of two squares applied to a sum

$x^2 + 49 = (x - 7)(x + 7)$ (treating a sum as a difference of squares)

The difference of two squares rule only works when there is a subtraction: $a^2 - b^2$ . The expression $x^2 + 49$ has no real factors and cannot be factorised over the integers.

Forgetting to check the middle term for a perfect square

$x^2 + 12x + 36$ written as $(x + 6)(x + 7)$ because $6 \times 7 = 42$ , not $36$

Before using the standard method, check whether the expression is a perfect-square trinomial: is the last term a perfect square, and does the middle term equal $2 \times x \times \sqrt{\text{last term}}$ ? Here $2 \times 6 = 12$ ✓, so the answer is $(x + 6)^2$ .

Not fully factorising

$2x^2 + 6x + 4$ factorised as $(x + 1)(2x + 4)$ , leaving a common factor inside

Always take out any common numerical factor first. Here, take out $2$ : $2(x^2 + 3x + 2) = 2(x + 1)(x + 2)$ . Check every bracket to make sure no further factorisation is possible.

Try one yourself

Your turn

Factorise $x^2 - 3x - 28$ and find the positive value of $x$ for which the expression equals zero.

Got a different quadratic factorising question wrong? Upload it to Tuterly and get the worked solution and a fresh one to practise, free.

About this worksheet

Why we built it

Year 10 expects students to fluently expand and factorise quadratics. These worksheets give the structured repetition needed to make the pattern recognition automatic - critical for senior Methods maths.

What's covered

Sub-skills your student will practise

✓Expanding binomial products (FOIL)
✓Factorising x^2 + bx + c
✓Difference of two squares
✓Perfect-square trinomials
✓Mixed-format expand and factorise problems

Other free Year 10 worksheet generators

Year 10

Simultaneous equations

Year 10

Quadratic equations

Year 10

Trigonometry

Year 10

Surface area and volume

Year 10

Linear equations

Year 10

Linear inequalities

Year 10

Algebraic fractions

Year 10

Exponential equations

Year 10

Coordinate geometry

Year 10

Compound interest

Year 10

Statistics

Year 10

Probability

Year 10

Factorising by common factor

Year 10

Index laws with algebra

Year 10

Substituting and rearranging formulas

Year 10

Linear equations with algebraic fractions

Year 10

Graphs of parabolas, circles, hyperbolas and exponentials

Year 10

Direct and inverse proportion

Year 10

Geometric reasoning and proofs

Year 10

Networks

Year 10

Scatterplots and lines of good fit

Year 10

Two-way tables

Year 10

Multi-step probability and tree diagrams

Year 10

Logarithmic scales

Or generate any topic from the full worksheet builder.

From our families

Parents on the practice questions.

“Tuterly has been exactly what we needed since Tom started secondary school. Once the topic tests began, he was struggling to keep up. Now he has unlimited practice, all matched to what his class is actually covering, and his Maths results have improved noticeably.”

Rebecca M.

Year 8 · Maths

“My daughter used to leave revision until the night before. Having a platform she can dip into for fifteen minutes after dinner has completely changed that. She's calmer going into assessments, and honestly, so am I.”

Priya S.

Year 9 · Maths

“I like that I can just keep doing questions until it clicks. If I get something wrong, it shows me why, and then gives me another one like it.”

Ella, student

Year 10 · Maths

Verified parent

Frequently asked

Questions parents ask about quadratic factorising

What's the cleanest way to factorise a monic quadratic?

Find two numbers that multiply to the constant term and add to the x coefficient. For x^2 - 7x + 12: factors of 12 that add to -7 are -3 and -4. So (x - 3)(x - 4).

How do students factorise a non-monic quadratic?

Use the 'ac method' (or 'splitting the middle term'): for ax^2 + bx + c, find factors of ac that sum to b, then split bx into those two terms and factorise by grouping. Year 10A goes deeper into this.

What's perfect-square factorisation?

A specific pattern: a^2 + 2ab + b^2 = (a + b)^2 and a^2 - 2ab + b^2 = (a - b)^2. So x^2 + 6x + 9 factorises as (x + 3)^2. Recognising the pattern saves time on exam questions.

Why is fluent factorising so important in Year 10?

Because every Methods topic from now on assumes it - solving equations, sketching parabolas, finding intercepts, completing the square. Students who can't factorise fluently spend twice as long on every Methods question.

Better maths marks start with the next wrong answer.

Tuterly turns every missed maths question into practice tailored towards the student, not just a generic worksheet.

Coming soon on iPhone. Android soon after.

Free to download · 7-day free trial on monthly · Years 7 to 10 maths · On iPhone. Android coming soon
Or try it free now with your own question →