Study Notes
Quadratics involve solving equations of the form a퓍²+b퓍+c = 0 using various methods such as factorisation, completing the square, and the quadratic formula. They also include understanding the nature of roots and solving quadratic inequalities.
- Factorisation — a method to solve quadratic equations by expressing them as a product of linear factors.
Example: (2퓍-3)(퓍+1)=0 gives roots 퓍=3/2 or 퓍=-1. - Completing the Square — rewriting a quadratic equation in the form p(퓍+q)² + r=0 to find solutions.
Example: 2퓍² + 4퓍 - 14 = 0 becomes (퓍+1)² = 8. - Quadratic Formula — a formula derived from completing the square to solve any quadratic equation.
Example: 퓍 = [-b ± √(b²-4ac)]/(2a). - Discriminant — the expression b²-4ac used to determine the nature of the roots of a quadratic equation.
Example: If b²-4ac > 0, there are 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 p(퓍+q)² + r=0.
- Quadratic Formula: 퓍 = [-b ± √(b²-4ac)]/(2a).
- Discriminant: b²-4ac, used to determine the nature of roots.
Common Confusions
- Forgetting to check for common factors before factorising.
- Misplacing terms when completing the square.
Typical Exam Questions
- How do you solve 2퓍² + 5퓍 + 3 = 0 by factorisation? Use (2퓍+3)(퓍+1)=0 to find 퓍=-3/2 or 퓍=-1.
- What is the vertex form of 퓍² + 6퓍 + 8? Completing the square gives (퓍+3)² - 1.
- How many real roots does 퓍² - 4퓍 + 4 have? One real root, as b²-4ac=0.
What Examiners Usually Test
- Ability to solve quadratic equations using different methods.
- Understanding the nature of roots using the discriminant.
- Solving simultaneous equations involving one linear and one quadratic equation.