amazon.co.ukQuadratic Equations
A-Level Mathematics revision: Quadratic Equations. Learning objectives, key points, worked examples and practice questions across AQA, Edexcel, OCR, WJEC and CCEA.
📌 Key Points
Key Fact: Standard form: ax^2 + bx + c = 0 (a != 0)
Key Fact: Factorising when a=1: find two numbers with product c and sum b
Key Fact: Factorising when a!=1: use AC method or trial and error
Key Fact: Completing the square: x^2 + bx = (x + b/2)^2 - (b/2)^2
Key Fact: Quadratic formula: x = (-b +/- sqrt(b^2-4ac)) / 2a
Key Fact: Discriminant Delta = b^2 - 4ac: Delta > 0 -> two distinct real roots; Delta = 0 -> one repeated root; Delta < 0 -> no real roots
Key Fact: Quadratic inequalities: sketch parabola or use sign analysis of factors
Key Fact: Sum of roots = -b/a, product of roots = c/a
Key Fact: Hidden quadratics: substitute u = x^2, u = eˣ, etc.
🎯 Learning Objectives
- Solve quadratic equations by factorising
- Solve by completing the square
- Use the quadratic formula for any quadratic
- Interpret the discriminant to determine number and nature of roots
- Solve quadratic inequalities graphically and algebraically
- Form and solve quadratic equations from word problems
- Understand the relationship between roots and coefficients (sum/product)
💡 Worked Example
Exam-Style Question
Question: Solve 2x^2 - 5x - 3 = 0 and hence solve 2sin^2θ - 5sinθ - 3 = 0 for 0 <= θ < 2pi
Model Answer:
2x^2 - 5x - 3 = (2x+1)(x-3) = 0 -> x = -½ or x = 3. Since sinθ ∈ [-1,1], sinθ = -½ -> θ = 7pi/6, 11pi/6
❓ Practice Questions
Model answers are being added progressively - questions marked ✗ don't have one yet. Cross-check with your teacher or the official mark scheme.
Questions:
- Solve 3x^2 + 10x - 8 = 0 by factorising✗ answer coming soon
- Solve x^2 - 6x + 2 = 0 by completing the square✗ answer coming soon
- Use the quadratic formula for 5x^2 - 3x - 1 = 0, leave in surd form✗ answer coming soon
- Find the range of k for which kx^2 + 4x + k = 0 has real roots✗ answer coming soon
- Solve the inequality x^2 - 5x + 6 > 0✗ answer coming soon
- A rectangle has area 24 cm^2 and perimeter 20 cm. Find its dimensions✗ answer coming soon
🎬 Video Resources
- Quadratic Equations - Overview -- YouTube Search
- Quadratic Equations - Exam Questions -- YouTube Search
📄 Past Papers & Exam Resources
🔗 Further Reading & Resources
- Physics & Maths Tutor - Quadratic Equations -- Free notes, worksheets, and past paper questions by topic
- Dr Frost Maths - Quadratic Equations -- Free resources, slides, and interactive questions
- BBC Bitesize - Quadratic Equations -- BBC Bitesize revision resources
📚 Lesson Plan (50 minutes)
- Starter (5 min): Recall prior knowledge of quadratic equations with quick questions.
- Teaching (15 min): Work through each of the learning objectives, explaining principles step by step.
- Key points review (5 min): Revisit the key points together, confirming understanding.
- Worked example (10 min): Model the example question: Solve 2x^2 - 5x - 3 = 0 and hence solve 2sin^2θ - 5sinθ - 3 = 0 for 0 <= θ < 2pi. Solution: 2x^2 - 5x - 3 = (2x+1)(x-3) = 0 -> x = -½ or x = 3. Since sinθ ∈ [-1,1], sinθ = -½ -> θ = 7pi/6, 11pi/6
- Practice (10 min): Students attempt the practice questions independently; circulate and support.
- Plenary (5 min): Review answers and address misconceptions.
🏠 Homework
- Solve 3x^2 + 10x - 8 = 0 by factorising
- Solve x^2 - 6x + 2 = 0 by completing the square
- Use the quadratic formula for 5x^2 - 3x - 1 = 0, leave in surd form
- Find the range of k for which kx^2 + 4x + k = 0 has real roots
- Solve the inequality x^2 - 5x + 6 > 0
- A rectangle has area 24 cm^2 and perimeter 20 cm. Find its dimensions
🧾 Assessment
Check practice answers against the model answer; use the built-in practice questions as formative assessment.