Free Grade 3 Completing the Square Flashcards
Free Grade 3 flashcards 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
Foundation
35 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 − 6x + 9.[1]
- 2.Factorise x² − 9x + 18.[1]
- 3.Factorise x² + 0x − 81.[1]
- 4.Factorise x² + 0x − 25.[1]
- 5.Solve x² + 1x − 12 = 0.[2]
- 6.Solve x² − 10x + 21 = 0.[2]
- 7.Solve x² + 8x + 12 = 0.[2]
- 8.Solve x² − 1x − 72 = 0.[2]
- 9.Solve x² − 13x + 42 = 0.[3]
- 10.Solve x² + 2x − 63 = 0.[3]
- 11.Solve x² + 4x − 5 = 0.[3]
- 12.Solve x² + 3x − 40 = 0.[3]
Show answers and working
1. (x − 3)(x − 3)
Find two numbers that multiply to 9 and add to -6: -3 and -3. So x² − 6x + 9 = (x − 3)(x − 3).
2. (x − 6)(x − 3)
Find two numbers that multiply to 18 and add to -9: -6 and -3. So x² − 9x + 18 = (x − 6)(x − 3).
3. (x + 9)(x − 9)
Find two numbers that multiply to -81 and add to 0: 9 and -9. So x² + 0x − 81 = (x + 9)(x − 9).
4. (x + 5)(x − 5)
Find two numbers that multiply to -25 and add to 0: 5 and -5. So x² + 0x − 25 = (x + 5)(x − 5).
5. x = -4 or x = 3
Factorise: (x + 4)(x − 3) = 0. One bracket must be zero: x = -4 or x = 3.
6. x = 3 or x = 7
Factorise: (x − 3)(x − 7) = 0. One bracket must be zero: x = 3 or x = 7.
7. x = -2 or x = -6
Factorise: (x + 2)(x + 6) = 0. One bracket must be zero: x = -2 or x = -6.
8. x = -8 or x = 9
Factorise: (x + 8)(x − 9) = 0. One bracket must be zero: x = -8 or x = 9.
9. x = 6 or x = 7
Factorise: (x − 6)(x − 7) = 0. One bracket must be zero: x = 6 or x = 7.
10. x = -9 or x = 7
Factorise: (x + 9)(x − 7) = 0. One bracket must be zero: x = -9 or x = 7.
11. x = 1 or x = -5
Factorise: (x − 1)(x + 5) = 0. One bracket must be zero: x = 1 or x = -5.
12. x = 5 or x = -8
Factorise: (x − 5)(x + 8) = 0. One bracket must be zero: x = 5 or x = -8.
Worked examples
Factorise x² − 7x + 10.
- Find two numbers that multiply to 10 and add to -7: -2 and -5.
- So x² − 7x + 10 = (x − 2)(x − 5).
- Answer: (x − 2)(x − 5)
Solve x² − 1x − 12 = 0.
- Factorise: (x − 4)(x + 3) = 0.
- One bracket must be zero: x = 4 or x = -3.
- Answer: x = 4 or x = -3
Solve x² − 10x + 25 = 0.
- Factorise: (x − 5)(x − 5) = 0.
- One bracket must be zero: x = 5 or x = 5.
- Answer: x = 5 or x = 5
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.
Every free resource for Completing the Square
One concept, one ecosystem — all of it free.
Goes deeper
The entities below this one, each with the same complete free resource set.
Sibling concepts
Related concepts
Lateral links, including across subjects.
Next topics
All free formats for this concept
Every kind below is a real page, generated from this entity.
Learn — Meet the idea.
Practice — Build fluency.
Revise — Keep it in memory.
Assess — Prove mastery.
Apply — Use it in context.
Teach — Deliver it to a class.
Frequently asked
- Is this Grade 3 Completing the Square flashcards really free?
- Yes. Every flashcards 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 flashcards 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.
Related resources
Stuck on Completing the Square? The AI Tutor already knows this page, its prerequisites and its common mistakes.
Open the free AI Tutor