Free Grade 3 Completing the Square Worksheets
Free Grade 3 worksheets for 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
Stretch
25 min
24
Learning objectives
Every objective is checkable, and each one maps to an activity and an assessment.
- UnderstandExplain completing the square
Explain completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
- ApplyCalculate completing the square
Calculate completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
- AnalyseCompare completing the square
Compare completing the square accurately in a familiar context.
Assessed by: Exit ticket of four short items
- EvaluateJustify completing the square
Justify 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 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 + 3x − 4.[1]
- 2.Factorise x² + 12x + 27.[1]
- 3.Factorise x² + 13x + 42.[1]
- 4.Factorise x² − 8x + 7.[1]
- 5.Solve x² − 8x + 7 = 0.[2]
- 6.Solve x² + 12x + 35 = 0.[2]
- 7.Solve x² + 7x − 18 = 0.[2]
- 8.Solve x² + 3x − 10 = 0.[2]
- 9.Solve x² − 15x + 56 = 0.[3]
- 10.Solve x² − 4x + 3 = 0.[3]
- 11.Solve x² + 5x − 6 = 0.[3]
- 12.Solve x² − 2x − 8 = 0.[3]
Show answers and working
1. (x + 4)(x − 1)
Find two numbers that multiply to -4 and add to 3: 4 and -1. So x² + 3x − 4 = (x + 4)(x − 1).
2. (x + 9)(x + 3)
Find two numbers that multiply to 27 and add to 12: 9 and 3. So x² + 12x + 27 = (x + 9)(x + 3).
3. (x + 7)(x + 6)
Find two numbers that multiply to 42 and add to 13: 7 and 6. So x² + 13x + 42 = (x + 7)(x + 6).
4. (x − 1)(x − 7)
Find two numbers that multiply to 7 and add to -8: -1 and -7. So x² − 8x + 7 = (x − 1)(x − 7).
5. x = 7 or x = 1
Factorise: (x − 7)(x − 1) = 0. One bracket must be zero: x = 7 or x = 1.
6. x = -7 or x = -5
Factorise: (x + 7)(x + 5) = 0. One bracket must be zero: x = -7 or x = -5.
7. x = 2 or x = -9
Factorise: (x − 2)(x + 9) = 0. One bracket must be zero: x = 2 or x = -9.
8. x = -5 or x = 2
Factorise: (x + 5)(x − 2) = 0. One bracket must be zero: x = -5 or x = 2.
9. x = 8 or x = 7
Factorise: (x − 8)(x − 7) = 0. One bracket must be zero: x = 8 or x = 7.
10. x = 3 or x = 1
Factorise: (x − 3)(x − 1) = 0. One bracket must be zero: x = 3 or x = 1.
11. x = -6 or x = 1
Factorise: (x + 6)(x − 1) = 0. One bracket must be zero: x = -6 or x = 1.
12. x = -2 or x = 4
Factorise: (x + 2)(x − 4) = 0. One bracket must be zero: x = -2 or x = 4.
Worked examples
Factorise x² + 4x + 4.
- Find two numbers that multiply to 4 and add to 4: 2 and 2.
- So x² + 4x + 4 = (x + 2)(x + 2).
- Answer: (x + 2)(x + 2)
Solve x² + 11x + 18 = 0.
- Factorise: (x + 2)(x + 9) = 0.
- One bracket must be zero: x = -2 or x = -9.
- Answer: x = -2 or x = -9
Solve x² + 7x + 12 = 0.
- Factorise: (x + 4)(x + 3) = 0.
- One bracket must be zero: x = -4 or x = -3.
- Answer: x = -4 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 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 Factorising Quadratics, Expanding Double Brackets, Sequences. 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 Completing the Square?
- Move on to The Quadratic Formula, Quadratic Graphs, Algebraic Expressions, Equations, which build directly on this idea.
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