Knowledge Organiser: Completing the Square
Part of Completing the Square · GCSE GCSE Mathematics revision
This topic summary covers Knowledge Organiser: Completing the Square within Completing the Square for GCSE Mathematics. Revise Completing the Square in Algebra for GCSE Mathematics with 10 exam-style questions and 3 flashcards. This topic appears regularly enough that it should still be part of a steady revision cycle. It is section 6 of 6 in this topic. Use this topic summary to connect the idea to the wider topic before moving on to questions and flashcards.
Topic position
Section 6 of 6
Practice
10 questions
Recall
3 flashcards
Knowledge Organiser: Completing the Square
Key Terms
- Completing the square: Rewriting ax² + bx + c in the form a(x + p)² + q
- Perfect square: An expression of the form (x + p)² = x² + 2px + p²
- Turning point: The minimum (or maximum) point of a parabola — found from the completed square form
- Vertex: Another name for the turning point of a parabola
- Minimum: The lowest point of a ∪-shaped parabola (when a > 0)
Must-Know Facts
- x² + bx + c = (x + b/2)² − (b/2)² + c
- The number inside the bracket is always HALF the coefficient of x
- Turning point of y = (x + p)² + q is at (−p, q)
- The y-coordinate of the turning point equals q (the constant outside the bracket)
- Use to solve: (x + p)² = k → x + p = ±√k → x = −p ± √k
- For ax² + bx + c (a≠1): factorise out a first
Key Formulas
- x² + bx + c = (x + b/2)² − (b/2)² + c
- Turning point: (−p, q) from y = (x + p)² + q
- Axis of symmetry: x = −p
- Solutions: x = −p ± √(−q) when (x + p)² + q = 0
Common Mistakes
- Halving b incorrectly: x² + 6x → (x + 3)², not (x + 6)² — always halve the coefficient of x
- Forgetting to subtract (b/2)²: x² + 6x = (x+3)² − 9, not just (x+3)²
- Turning point sign: y = (x + 3)² − 9 has turning point (−3, −9) — sign of p is reversed
- Leading coefficient ≠ 1: For 2x² + 8x, factorise out the 2 first: 2(x² + 4x) before completing the square
- Solving errors: From (x+3)² = 9, take square root of both sides: x+3 = ±3, so x = 0 or x = −6
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Practice Questions for Completing the Square
To write x² + 10x + 3 in the form (x + p)² + q, what is the value of p?
Explain how you can tell from the completed square form whether a quadratic has a minimum or a maximum turning point.
Quick Recall Flashcards
10 questions on Completing the Square — practise free
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