Free Grade 3 Working with Completing the Square Worksheets
Free Grade 3 worksheets for Working with Completing the Square: 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
Core
20 min
24
Learning objectives
Every objective is checkable, and each one maps to an activity and an assessment.
- UnderstandExplain working with completing the square
Explain working with completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
- ApplyCalculate working with completing the square
Calculate working with completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
- AnalyseCompare working with completing the square
Compare working with completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
- EvaluateJustify working with completing the square
Justify working with completing the square 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 Working with Completing the Square 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 + 5x − 6.[1]
- 2.Factorise x² − 4x + 3.[1]
- 3.Factorise x² + 2x − 15.[1]
- 4.Factorise x² − 8x + 15.[1]
- 5.Solve x² + 16x + 63 = 0.[2]
- 6.Solve x² + 9x + 18 = 0.[2]
- 7.Solve x² − 12x + 35 = 0.[2]
- 8.Solve x² − 7x + 6 = 0.[2]
- 9.Solve x² − 10x + 21 = 0.[3]
- 10.Solve x² − 6x − 7 = 0.[3]
- 11.Solve x² + 0x − 49 = 0.[3]
- 12.Solve x² + 0x − 4 = 0.[3]
Show answers and working
1. (x + 6)(x − 1)
Find two numbers that multiply to -6 and add to 5: 6 and -1. So x² + 5x − 6 = (x + 6)(x − 1).
2. (x − 3)(x − 1)
Find two numbers that multiply to 3 and add to -4: -3 and -1. So x² − 4x + 3 = (x − 3)(x − 1).
3. (x + 5)(x − 3)
Find two numbers that multiply to -15 and add to 2: 5 and -3. So x² + 2x − 15 = (x + 5)(x − 3).
4. (x − 5)(x − 3)
Find two numbers that multiply to 15 and add to -8: -5 and -3. So x² − 8x + 15 = (x − 5)(x − 3).
5. x = -7 or x = -9
Factorise: (x + 7)(x + 9) = 0. One bracket must be zero: x = -7 or x = -9.
6. x = -6 or x = -3
Factorise: (x + 6)(x + 3) = 0. One bracket must be zero: x = -6 or x = -3.
7. x = 7 or x = 5
Factorise: (x − 7)(x − 5) = 0. One bracket must be zero: x = 7 or x = 5.
8. x = 1 or x = 6
Factorise: (x − 1)(x − 6) = 0. One bracket must be zero: x = 1 or x = 6.
9. x = 3 or x = 7
Factorise: (x − 3)(x − 7) = 0. One bracket must be zero: x = 3 or x = 7.
10. x = 7 or x = -1
Factorise: (x − 7)(x + 1) = 0. One bracket must be zero: x = 7 or x = -1.
11. x = 7 or x = -7
Factorise: (x − 7)(x + 7) = 0. One bracket must be zero: x = 7 or x = -7.
12. x = -2 or x = 2
Factorise: (x + 2)(x − 2) = 0. One bracket must be zero: x = -2 or x = 2.
Worked examples
Factorise x² − 9x + 20.
- Find two numbers that multiply to 20 and add to -9: -5 and -4.
- So x² − 9x + 20 = (x − 5)(x − 4).
- Answer: (x − 5)(x − 4)
Solve x² + 2x − 3 = 0.
- Factorise: (x − 1)(x + 3) = 0.
- One bracket must be zero: x = 1 or x = -3.
- Answer: x = 1 or x = -3
Solve x² + 9x + 8 = 0.
- Factorise: (x + 8)(x + 1) = 0.
- One bracket must be zero: x = -8 or x = -1.
- Answer: x = -8 or x = -1
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 Working with Completing the Square worksheet really free?
- Yes. Every worksheets 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 Introducing Completing the Square, Factorising Quadratics. Each one has its own free lesson, worksheet and quiz.
- How is the worksheet 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 Working with Completing the Square?
- Move on to Completing the Square in Context, Expanding Double Brackets, Factorising Quadratics, The Quadratic Formula, which build directly on this idea.
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