Free Grade 3 Completing the Square in Context Lessons
Free Grade 3 lessons for Completing the Square in Context: 12 real questions with a full answer key and worked solutions. A quadratic has an x² term. Factorise by finding two numbers that multiply to the constant and add to the x-coefficient; each bracket gives a solution.
Free
Foundation
20 min
24
Learning objectives
Every objective is checkable, and each one maps to an activity and an assessment.
- RememberIdentify completing the square in context
Identify completing the square in context accurately in a familiar context.
Assessed by: Exit ticket of four short items
- UnderstandExplain completing the square in context
Explain completing the square in context accurately in a familiar context.
Assessed by: Exit ticket of four short items
- ApplyCalculate completing the square in context
Calculate completing the square in context accurately in a familiar context.
Assessed by: Exit ticket of four short items
- AnalyseCompare completing the square in context
Compare completing the square in context accurately in a familiar context.
Assessed by: Exit ticket of four short items
Prerequisites
Close these gaps first — each has its own free lesson, worksheet and quiz.
How Completing the Square in Context works
A quadratic has an x² term. Factorise by finding two numbers that multiply to the constant and add to the x-coefficient; each bracket gives a solution.
12 practice questions
Easy to hard, 24 marks in total. Try them first, then open the answers.
- 1.Factorise x² − 9x + 14.[1]
- 2.Factorise x² + 0x − 1.[1]
- 3.Factorise x² − 10x + 24.[1]
- 4.Factorise x² + 10x + 25.[1]
- 5.Solve x² + 16x + 64 = 0.[2]
- 6.Solve x² − 11x + 28 = 0.[2]
- 7.Solve x² − 7x + 10 = 0.[2]
- 8.Solve x² + 3x − 18 = 0.[2]
- 9.Solve x² − 11x + 24 = 0.[3]
- 10.Solve x² − 8x + 7 = 0.[3]
- 11.Solve x² + 2x − 8 = 0.[3]
- 12.Solve x² + 11x + 18 = 0.[3]
Show answers and working
1. (x − 7)(x − 2)
Find two numbers that multiply to 14 and add to -9: -7 and -2. So x² − 9x + 14 = (x − 7)(x − 2).
2. (x + 1)(x − 1)
Find two numbers that multiply to -1 and add to 0: 1 and -1. So x² + 0x − 1 = (x + 1)(x − 1).
3. (x − 4)(x − 6)
Find two numbers that multiply to 24 and add to -10: -4 and -6. So x² − 10x + 24 = (x − 4)(x − 6).
4. (x + 5)(x + 5)
Find two numbers that multiply to 25 and add to 10: 5 and 5. So x² + 10x + 25 = (x + 5)(x + 5).
5. x = -8 or x = -8
Factorise: (x + 8)(x + 8) = 0. One bracket must be zero: x = -8 or x = -8.
6. x = 4 or x = 7
Factorise: (x − 4)(x − 7) = 0. One bracket must be zero: x = 4 or x = 7.
7. x = 5 or x = 2
Factorise: (x − 5)(x − 2) = 0. One bracket must be zero: x = 5 or x = 2.
8. x = -6 or x = 3
Factorise: (x + 6)(x − 3) = 0. One bracket must be zero: x = -6 or x = 3.
9. x = 3 or x = 8
Factorise: (x − 3)(x − 8) = 0. One bracket must be zero: x = 3 or x = 8.
10. x = 7 or x = 1
Factorise: (x − 7)(x − 1) = 0. One bracket must be zero: x = 7 or x = 1.
11. x = 2 or x = -4
Factorise: (x − 2)(x + 4) = 0. One bracket must be zero: x = 2 or x = -4.
12. x = -9 or x = -2
Factorise: (x + 9)(x + 2) = 0. One bracket must be zero: x = -9 or x = -2.
Worked examples
Factorise x² − 12x + 36.
- Find two numbers that multiply to 36 and add to -12: -6 and -6.
- So x² − 12x + 36 = (x − 6)(x − 6).
- Answer: (x − 6)(x − 6)
Solve x² + 6x − 27 = 0.
- Factorise: (x − 3)(x + 9) = 0.
- One bracket must be zero: x = 3 or x = -9.
- Answer: x = 3 or x = -9
Solve x² + 6x − 27 = 0.
- Factorise: (x + 9)(x − 3) = 0.
- One bracket must be zero: x = -9 or x = 3.
- Answer: x = -9 or x = 3
Interactive practice
Free, no account required — answer on screen or print it.
Common mistakes
What learners typically get wrong, and how to fix it.
Gives x = 3 from (x + 3) = 0.
Correction: x + 3 = 0 means x = −3.
Gets the signs muddled when factorising.
Correction: Check by expanding the brackets.
Teacher tips
- · Use a product–sum grid for finding the pair.
Parent tips
- · Ask what two numbers multiply to 12 and add to 7.
Real-life applications
- · Projectile paths, maximising area.
Assessment objectives
How mastery is evidenced, and which standards it maps to.
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Frequently asked
- Is this Grade 3 Completing the Square in Context lesson really free?
- Yes. Every lessons on FreeMathWorksheet is free to use, print and share — no account, no paywall and no watermark.
- What should a learner already know before this?
- Start with Working with Completing the Square, Introducing Completing the Square, Factorising Quadratics. Each one has its own free lesson, worksheet and quiz.
- How is the lesson sequenced?
- Recognition first, then understanding, application, reasoning and finally mastery — the same ladder used across every free resource on the platform.
- What comes next after Completing the Square in Context?
- Move on to Expanding Double Brackets, Factorising Quadratics, The Quadratic Formula, Quadratic Graphs, which build directly on this idea.
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