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.
๐ 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 = eหฃ, etc.
๐ฏ 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)
๐ก Worked Example
Exam-Style Question
Question: Solve 2x^2 - 5x - 3 = 0 and hence solve 2sin^2ฮธ - 5sinฮธ - 3 = 0 for 0 <= ฮธ < 2pi
Model Answer:
2x^2 - 5x - 3 = (2x+1)(x-3) = 0 -> x = -ยฝ or x = 3. Since sinฮธ โ [-1,1], sinฮธ = -ยฝ -> ฮธ = 7pi/6, 11pi/6
โ Practice Questions
Questions:
- 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
๐ฌ Video Resources
- Quadratic Equations - Overview -- YouTube Search
- Quadratic Equations - Exam Questions -- YouTube Search
๐ Past Papers & Exam Resources
๐ 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