Completing the Square: Mastery Pack
This lesson covers the algebraic technique of completing the square for monic and non-monic quadratics. Students will learn to manipulate expressions, solve equations, and identify turning points, moving from basic fluency to complex problem-solving.
Learning objectives
- Express quadratic expressions in the form (x + p)² + q by completing the square.
- Solve quadratic equations by completing the square, leaving answers in surd form where necessary.
- Identify the coordinates of the turning point of a quadratic graph from its completed square form.
Lesson plan (60 minutes)
Starter · 10 min
Retrieval practice: Expanding double brackets and identifying perfect squares (e.g., (x+3)^2).
Introduction · 15 min
Direct instruction using the 'Explainer' steps and worked example. Focus on the visual area model if helpful.
Main Activity · 20 min
Students work through differentiated worksheets. Teacher circulates to address misconceptions.
Assessment · 10 min
Multiple-choice quiz to check for understanding and identify remaining gaps.
Plenary · 5 min
Reviewing the most difficult quiz question and previewing the turning point application.
Key vocabulary
- Quadratic
- A polynomial where the highest power of the variable is 2 (e.g., ax² + bx + c).
- Monic Quadratic
- A quadratic where the coefficient of x² is 1.
- Completing the Square
- The process of writing a quadratic in the form a(x+p)² + q.
- Turning Point
- The maximum or minimum point on a parabola.
- Coefficient
- The number multiplying a variable (e.g., 4 is the coefficient in 4x).
Core worksheet (first questions)
- Complete the square for x² + 8x + 9
- Complete the square for x² - 10x - 5
- Complete the square for x² + 3x - 4 (use decimals)
- Solve x² + 6x + 7 = 0 by completing the square. Leave your answer in surd form.
Quiz (sample)
1. What is x² + 8x expressed in completed square form?
- A. (x + 8)² - 64
- B. (x + 4)² - 16
- C. (x + 4)² + 16
- D. (x + 4)² - 8
2. Complete the square for x² - 6x + 5.
- A. (x - 3)² - 9
- B. (x - 3)² + 5
- C. (x - 3)² - 4
- D. (x + 3)² - 4
3. Complete the square for x² + 10x - 1.
- A. (x + 5)² - 26
- B. (x + 5)² - 24
- C. (x + 5)² - 25
- D. (x + 10)² - 26
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