Study Notes
Quadratics involve solving equations of the form ax² + bx + c = 0 using methods like factorisation, completing the square, and the quadratic formula. Understanding functions, sketching quadratic graphs, and interpreting the discriminant are also key aspects.
- Factorisation — solving by expressing the equation as a product of factors. Example: (2x-3)(x+1) = 0 gives x = 3/2 or x = -1.
- Completing the Square — rewriting the equation in the form p(x+q)² + r = 0. Example: x² + 2x - 7 = 0 becomes (x+1)² - 8 = 0.
- Quadratic Formula — a formula to find roots of ax² + bx + c = 0. Example: x = [-b ± √(b²-4ac)] / 2a.
- Function — a relation where each input has a single output. Example: f(x) = x² + 1.
- Quadratic Graphs — parabolas that can open upwards or downwards. Example: y = x² - 8x + 13.
- Discriminant — b² - 4ac, determines the nature of roots. Example: b² - 4ac > 0 means 2 distinct real roots.
Exam Tips
Key Definitions to Remember
- Factorisation: Expressing a quadratic as a product of linear factors.
- Completing the Square: Rewriting a quadratic in the form (x+p)² + q.
- Quadratic Formula: x = [-b ± √(b²-4ac)] / 2a.
- Discriminant: b² - 4ac, determines the number and type of roots.
Common Confusions
- Forgetting to set the equation to zero before factorising.
- Mixing up the signs when using the quadratic formula.
- Misinterpreting the discriminant's value.
Typical Exam Questions
- How do you solve 2x² - 3x - 5 = 0 by factorisation? Factor to find x = 5/2 or x = -1.
- What is the turning point of y = x² - 8x + 13? Completing the square gives turning point (4, -3).
- How many real roots does x² + 4x + 5 = 0 have? Discriminant is negative, so no real roots.
What Examiners Usually Test
- Ability to solve quadratics using different methods.
- Understanding and application of the discriminant.
- Sketching and interpreting quadratic graphs.