Factorising Monic Quadratic Expressions
Find the Factor Pair
Draw a line from each quadratic to its factorised form.
Factor Pairs Table
Find two integers that multiply to give c and add to give b, then write the factorised form.
x^2 + 8x + 15: factors of 15 that add to 8 -> factorised form:
x^2 + 2x - 8: factors of -8 that add to 2 -> factorised form:
x^2 - 9x + 18: factors of 18 that add to -9 -> factorised form:
x^2 - 4x - 21: factors of -21 that add to -4 -> factorised form:
Check the Factorisation
Circle the correct factorised form. Expand your chosen option to verify.
x^2 + 6x + 8
x^2 - 5x - 14
x^2 - 10x + 25
Factorise These Quadratics
Factorise each expression fully. Check by expanding.
x^2 + 11x + 28 =
x^2 - 6x + 9 =
x^2 + 3x - 18 =
x^2 - 13x + 42 =
Difference of Two Squares -- Factorised
Factorise each expression using a^2 - b^2 = (a + b)(a - b).
x^2 - 49 =
x^2 - 100 =
x^2 - 36 =
4x^2 - 9 =
Identify the Error
Each factorisation below contains a mistake. Find and correct it.
Student wrote: x^2 + 7x + 10 = (x + 5)(x + 3). What is wrong? Correct factorisation:
Student wrote: x^2 - 4x - 12 = (x - 6)(x + 2). What is wrong? Correct factorisation:
Solving by Factorising
Solve each equation by factorising, then use the zero product property.
x^2 + 5x + 4 = 0 -> solutions: x = ___, x = ___
x^2 - 7x + 6 = 0 -> solutions: x = ___, x = ___
x^2 - x - 20 = 0 -> solutions: x = ___, x = ___
Word Problem: Area
Solve this real-world problem using factorisation.
A rectangular garden has area (x^2 + 8x + 15) m^2. If one side is (x + 3) m, what is the length of the other side? Show all working.
Finding Factor Pairs That Add and Multiply
Find two integers p and q such that p x q = c and p + q = b for each quadratic x^2 + bx + c.
x^2 + 9x + 20: Find p and q so that p x q = 20 and p + q = 9. p = ___, q = ___
x^2 - 3x - 18: Find p and q so that p x q = -18 and p + q = -3. p = ___, q = ___
x^2 - 10x + 24: Find p and q so that p x q = 24 and p + q = -10. p = ___, q = ___
Factorising Quadratics with a Negative Constant
When c is negative, one factor is positive and one is negative. Factorise each expression.
x^2 + 5x - 6 =
x^2 - 2x - 15 =
x^2 + x - 12 =
x^2 - 3x - 40 =
Difference of Two Squares -- Factorising
Recognise and factorise each expression using a^2 - b^2 = (a + b)(a - b).
x^2 - 64 =
x^2 - 81 =
9x^2 - 16 =
25x^2 - 4 =
Check Factorisation by Expanding Back
For each factorised form, expand back to verify it matches the original expression.
Claim: x^2 - 7x + 12 = (x - 3)(x - 4). Verify by expanding (x - 3)(x - 4):
Claim: x^2 + 2x - 24 = (x + 6)(x - 4). Verify by expanding:
Claim: x^2 - 25 = (x + 5)(x - 5). Verify by expanding:
Spot the Sign Error
Circle the correct factorisation for each quadratic.
x^2 - 8x + 15
x^2 + 4x - 12
x^2 - 16
Factorise Trinomials — Foundational Set
Factorise each monic quadratic. Write the factor pair that works.
x^2 + 8x + 15 =
x^2 + 6x + 8 =
x^2 - 7x + 10 =
x^2 - 9x + 18 =
x^2 + 11x + 30 =
x^2 - 13x + 36 =
Factorise Mixed Positive and Negative
Factorise these expressions involving mixed signs.
x^2 + 3x - 18 =
x^2 - x - 30 =
x^2 + 5x - 24 =
x^2 - 4x - 21 =
x^2 + 2x - 35 =
x^2 - 6x - 27 =
Match Quadratic to Its Factors
Draw a line from each quadratic to its correct factorised form.
Factorise Difference of Two Squares
Recognise and factorise using a^2 - b^2 = (a+b)(a-b).
x^2 - 49 =
x^2 - 100 =
4x^2 - 25 =
9x^2 - 4 =
x^2 - 144 =
16x^2 - 1 =
Perfect Square Trinomials — Recognise and Factorise
Recognise each as a perfect square trinomial and factorise.
x^2 + 10x + 25 =
x^2 - 16x + 64 =
x^2 + 20x + 100 =
x^2 - 8x + 16 =
4x^2 + 12x + 9 =
Factorise Correctly
Circle the correct factorisation.
x^2 - 5x + 4
x^2 + 7x - 8
x^2 - 14x + 49
x^2 - 9
Sort Quadratics by Factorisation Type
Sort each expression into the correct factorisation type.
Factorise and Verify — Medium Difficulty
Factorise, then verify by expanding.
x^2 + 14x + 33 = ___. Verify:
x^2 - 12x + 27 = ___. Verify:
x^2 + x - 42 = ___. Verify:
x^2 - 5x - 36 = ___. Verify:
Using Factorisation to Find Roots
Factorise each quadratic and write its roots (zeros).
x^2 - 7x + 6 = ___. Roots: x = ___ or x = ___
x^2 + 3x - 28 = ___. Roots: x = ___ or x = ___
x^2 - 25 = ___. Roots: x = ___ or x = ___
x^2 + 10x + 25 = ___. How many distinct roots does this have? Why?
Factorise — Harder Cases
Factorise these more challenging trinomials.
x^2 + 17x + 72 =
x^2 - 19x + 88 =
x^2 + 3x - 108 =
x^2 - 22x + 121 =
x^2 - 169 =
Factorisation in Architecture
Explore where quadratics appear in real-world areas and dimensions.
- 1A rectangular room has area x^2 + 8x + 15 square metres. Factorise to find expressions for its length and width.
- 2Research 'the quadratic formula' — it solves quadratics that cannot be factorised by inspection. Write the formula from memory after reading it.
- 3Find three objects at home whose dimensions involve a squared measurement (area of carpet, volume of a box, etc.). Write the area as a quadratic expression.
Factorise with Common Factor First
First extract the common factor, then factorise the remaining trinomial.
2x^2 + 10x + 12 = 2(___) =
3x^2 - 9x - 30 = 3(___) =
5x^2 - 20 = 5(___) =
4x^2 + 16x + 16 = 4(___) =
Geometric Applications of Factorisation
Use factorisation to solve these geometry problems.
A rectangle has area 80 cm^2. Its length is 2 cm more than its width. Write a quadratic equation for the width x: x(x+2) = 80. Rearrange, factorise and solve.
A square has area that is 9 less than 6 times its side length. Write and factorise the quadratic to find the side length.
A triangle has base (x+4) cm and height x cm. Its area is 30 cm^2. Find x.
Error Analysis — Factorisation
Each factorisation below contains an error. Find and correct it.
Student writes: x^2 + 7x + 12 = (x+3)(x+3). Error and correction:
Student writes: x^2 - 8x - 16 = (x-4)^2. Error and correction:
Student writes: x^2 - 9 = (x-3)^2. Error and correction:
Student writes: x^2 + 5x + 4 = (x+2)(x+2). Error and correction:
Factorise a Range of Expressions — Mixed Set
Factorise each expression. Identify the type before factorising.
x^2 - 121 (type: ___) =
x^2 + 16x + 64 (type: ___) =
x^2 - 9x + 14 (type: ___) =
x^2 + 4x - 45 (type: ___) =
3x^2 + 6x - 24 (type: ___) =
x^2 - 2x + 1 (type: ___) =
Connecting Factorisation and Graphing
Use factorisation to find key features of each parabola.
y = x^2 - x - 6. Factorise: y = ___. x-intercepts: x = ___ and x = ___
y = x^2 + 2x - 8. Factorise and find x-intercepts. What is the y-intercept?
y = x^2 - 4. Find x-intercepts and describe the symmetry of the parabola.
Factorisation Practice Games
Practise factorisation skills through these activities.
- 1Write ten quadratics on slips of paper and their factorisations on separate slips. Shuffle and race to match them correctly. Time each round and try to beat your previous time.
- 2Challenge a family member: you write a product of two binomials (don't show them), they expand it, then factorise back to check. Swap roles.
- 3Look up 'quadratic factorisation online quiz' and complete 20 problems timed. Record your accuracy.
Factorisation — Exam-Style Questions
Work independently for 20 minutes. Show all steps including verification.
Fully factorise: x^2 - 15x + 56
Fully factorise: x^2 + x - 72
Fully factorise: 2x^2 + 14x + 24
Fully factorise: 9x^2 - 25
Solve: x^2 - 2x - 63 = 0 by factorising.
A rectangle has area (x^2 + 8x + 15) cm^2. Write its dimensions as binomial factors.
Factorisation Unit Review
Comprehensive review of all factorisation types. Allow 20 minutes.
State the sum-product rule for factorising x^2 + bx + c in your own words.
Factorise: x^2 - 23x + 132
Factorise: x^2 + 14x + 49 - y^2 (hint: group the first three terms first).
Describe in words how to identify: (a) a perfect square trinomial, (b) a difference of two squares, (c) a general monic trinomial.
Factorise Larger Quadratics
These have larger factor pairs. List factor pairs systematically.
x^2 + 18x + 77 =
x^2 - 16x + 63 =
x^2 + 5x - 84 =
x^2 - 3x - 70 =
x^2 - 24x + 143 =
Factorise and Solve — Practice Set
Factorise, then solve the equation.
x^2 - 9x + 14 = 0. Factorise and solve.
x^2 + 8x + 12 = 0. Factorise and solve.
x^2 - 16 = 0. Factorise and solve.
x^2 + 6x = 0. Factorise (common factor) and solve.
x^2 - 64 = 0. Factorise and solve.
Correct Factorisation and Roots
Circle the correct factorisation and its roots.
x^2 - 5x - 24 and its roots
x^2 + 11x + 24 and its roots
Applications of Factorisation
Use factorisation to solve these real-world problems.
A ball is thrown upward. Its height h metres after t seconds is h = -t^2 + 6t. When is the ball at ground level? Factorise and solve.
A rectangular plot has width w m and length (w+4) m. Its area is 96 m^2. Write and solve the quadratic equation for w.
The product of two consecutive positive integers is 90. Write and solve the quadratic equation to find the integers.
Factorisation — Mixed Final Practice
Factorise each expression completely. Show all working.
x^2 + 13x + 40 =
x^2 - 20x + 99 =
x^2 - 6x - 91 =
6x^2 + 24x + 18 =
x^2 - 225 =
x^2 + 22x + 121 =
x^2 - 30x + 225 =
2x^2 - 18 =
Link to Graphing: x-Intercepts
Factorise each quadratic, then state the x-intercepts and sketch the parabola.
y = x^2 - x - 6. Factorise: ___. x-intercepts: (___, 0) and (___, 0). Sketch the parabola and label the x-intercepts.
y = x^2 - 9. Factorise: ___. x-intercepts: ___. What type of parabola is this?
y = x^2 - 4x + 4. Factorise: ___. How many x-intercepts does this parabola have? Why?
Teaching Factorisation
The best way to consolidate understanding is to teach it to someone else.
- 1Explain the sum-product method to a family member using three examples. Let them check your working.
- 2Create a 'Factorisation Fact Sheet' listing all three types (general, perfect square, DOTS) with an example and a tip for each.
- 3Make a set of flashcards: write a quadratic on one side and its factorised form on the other. Quiz yourself until you can do 20 in a row correctly.
- 4Research 'completing the square' — the technique you will use in Year 10 when factorisation by inspection fails.
Factorisation — Comprehensive Final Exam Practice
Work for 25 minutes without notes. Show all working including verification.
Factorise: x^2 + 15x + 56
Factorise: x^2 - 13x + 42
Factorise completely: 4x^2 - 100
Factorise: x^2 + 4x - 77
Solve by factorising: x^2 - 7x - 18 = 0
A rectangle has area (x^2 + 14x + 48) cm^2. Find its dimensions.
Show that x^2 - 6x + 9 = (x-3)^2 and explain why this quadratic has only one root.
Discriminant Check Before Factorising
For each quadratic, calculate b^2 - 4c. If it is a perfect square, factorise; if not, write 'cannot factorise over integers'.
x^2 + 5x + 4: b^2 - 4c = ___. Factorise or state why not:
x^2 + 5x + 5: b^2 - 4c = ___. Factorise or state why not:
x^2 - 7x + 6: b^2 - 4c = ___. Factorise or state why not:
x^2 - 3x + 3: b^2 - 4c = ___. Factorise or state why not:
Creative Factorisation Challenges
These require careful thought — take your time.
Find all integer values of k for which x^2 + kx + 12 can be factorised over integers. List all possibilities.
Find all integer values of k for which x^2 + 6x + k can be factorised over integers.
Factorise x^2 - y^2 - 2y - 1 by grouping (hint: the last three terms form a perfect square).
Fully Factorised or Not?
Circle the answer that is FULLY factorised.
x^2 - 16
2x^2 - 8
x^2 + 6x + 9
Link to Simultaneous Equations
Use factorisation to find the intersections of these parabola and line pairs.
Find where y = x^2 - 4 and y = 0 intersect. Factorise and state the x-values.
Find where y = x^2 + 2x and y = 0 intersect. Factorise and state the x-values.
Find where y = x^2 - 3x - 10 and y = 0 intersect. Factorise and state both x-values.
Sort Expressions by Factorisation Type
Classify each expression.
Proof and Generalisation
Use factorisation to prove these algebraic results.
Show algebraically that the difference of two squares a^2 - b^2 always equals (a+b)(a-b). Start from the right-hand side and expand.
Show that x^2 - (p+q)x + pq = (x-p)(x-q) by expanding the right side.
Use factorisation to show that 50^2 - 49^2 = 99. (Hint: difference of two squares.)
Without a calculator, evaluate 1001^2 - 999^2. Show your method.
Factorisation — Self-Assessment and Next Steps
Reflect on your factorisation skills.
Rate your confidence for: (a) general monic trinomials ___/5, (b) perfect squares ___/5, (c) DOTS ___/5, (d) with common factor first ___/5.
Which factorisation type took you the longest to master? What strategy helped you understand it?
Write three factorisations from memory that you got wrong earlier, now done correctly.
What are you most looking forward to in the next topic (Solving Quadratic Equations)?
Solve Quadratics by Factorisation — Extension
Rearrange each equation to standard form, factorise, and solve.
x^2 = 7x - 12. Rearrange, factorise and solve.
x^2 + 3x = 18. Rearrange, factorise and solve.
x(x - 5) = 24. Expand, rearrange, factorise and solve.
(x + 2)^2 = 2x + 13. Expand, rearrange, factorise and solve.
Factorisation Fluency Sprint
Factorise all 12 expressions as quickly as possible. Aim for under 12 minutes.
1. x^2 + 7x + 6 = 2. x^2 - 7x + 6 = 3. x^2 + 5x - 6 =
4. x^2 - 36 = 5. x^2 - 10x + 25 = 6. x^2 + 14x + 48 =
7. x^2 - 3x - 28 = 8. x^2 + 16x + 64 = 9. x^2 - 19x + 90 =
10. x^2 + 4x - 96 = 11. x^2 - 144 = 12. 2x^2 + 10x + 8 =
Investigate: Products of Binomials from Roots
Construct quadratic expressions that have the given roots, by working backwards.
Write a quadratic with roots x = 3 and x = 5. Expand to verify.
Write a quadratic with roots x = -4 and x = 7. Expand to verify.
Write a quadratic with a double root at x = -2. Expand to verify.
Write a quadratic with roots x = 0 and x = 10. What kind of quadratic is this?
Factorise These Non-Obvious Expressions
Factorise. Some may require grouping or spotting a hidden structure.
x^2 - (a + b)x + ab =
x^2 - a^2 - 2a - 1. Hint: factor the last three terms first.
(x+1)^2 - (x-1)^2 =
x^4 - 16. Hint: treat x^2 as a variable, factorise twice.
Factorise or Cannot Be Factorised?
Circle 'Factorisable' if integer factors exist, or 'Not factorisable' if not.
x^2 + 7x + 11
x^2 - 8x + 12
x^2 + 6x + 12
x^2 - 100
Factorisation in Context — Word Problems
Model and solve each word problem using factorisation.
The product of two consecutive odd integers is 143. Find them using factorisation.
A square room has area x^2 + 18x + 81 m^2. What is the side length?
A quadratic path has height h = -t^2 + 8t metres at time t seconds. When does it reach the ground? Factorise to find t.
Factorisation — Final Ten Questions
Factorise each expression completely. Include common factors where needed.
1. x^2 + 11x + 30 = 2. x^2 - 17x + 70 =
3. x^2 + 6x - 40 = 4. x^2 - 8x - 65 =
5. x^2 - 289 = 6. x^2 + 24x + 144 =
7. x^2 - 26x + 169 = 8. 5x^2 + 15x - 20 =
9. 6x^2 - 6 = 10. 2x^2 - 16x + 32 =
Match Quadratic to Root Values
Match each quadratic equation (set = 0) to its solutions.
Factorisation Practice — 15-Question Final Sprint
Factorise all 15 expressions. Set a timer for 15 minutes.
1. x^2 + 9x + 18 2. x^2 - 9x + 18 3. x^2 - 4x - 32
4. x^2 + 4x - 32 5. x^2 - 64 6. x^2 + 18x + 81
7. x^2 - 20x + 100 8. x^2 + 7x - 18 9. x^2 - 7x - 18
10. x^2 + 11x - 26 11. x^2 - 11x - 26 12. 2x^2 - 8x - 24
13. 4x^2 - 9 14. x^2 - 23x + 132 15. x^2 + 23x + 132
Factorisation — Capstone Problem
Solve this multi-step problem combining factorisation, geometry and algebra.
A rectangular swimming pool is x m long and (x - 4) m wide. A concrete path of width 2 m surrounds the pool. The total area (pool + path) is 120 m^2. (a) Write an expression for the total area. (b) Form a quadratic equation. (c) Factorise and solve for x. (d) State the dimensions of the pool.
Factorisation — Proofs via Factorisation
Use factorisation to prove each algebraic identity.
Prove: n^2 - 1 = (n-1)(n+1). Expand right side to verify.
Use the difference of two squares to show 1000^2 - 999^2 = 1999 without a calculator.
Show that any odd perfect square minus 1 is divisible by 8. Let the odd number be (2k+1). Hint: use (2k+1)^2 - 1.
Factorisation Mastery — Unit Completion
Complete these final questions to demonstrate full mastery of factorisation.
Describe the three-step approach to factorising any monic quadratic.
Factorise: x^2 - 34x + 289 (notice it is a very large perfect square)
Factorise: 10x^2 + 5x - 15. Show the common factor step first.
Write one example of: (a) a quadratic that cannot be factorised over integers, (b) one that has a double root, (c) one that uses DOTS.
Factorisation Mastery — Extended Application
Apply factorisation to this multi-concept problem.
A projectile's height (in metres) at time t seconds is given by h = -t^2 + 10t. (a) Factorise h. (b) When is h = 0? (c) What is the maximum height and when does it occur? (Hint: the maximum occurs at the average of the two roots.)
Spot the Correct Full Factorisation
Circle the fully factorised form.
6x^2 - 24
x^2 + 5x + 4
x^2 - 2x + 1