Functions and the Discriminant

This section explores functions, their properties, and the discriminant of quadratic equations.

Definition: A function is a mathematical relationship that assigns each input to a unique output.

The chapter covers:

• Domain and range of functions
• Quadratic graphs and their features
• The discriminant and its role in determining the nature of roots

Vocabulary: Discriminant - the expression b² - 4ac in a quadratic equation ax² + bx + c = 0.

The discriminant is used to determine the number and nature of roots:

• If b² - 4ac > 0, there are two distinct real roots
• If b² - 4ac = 0, there is one repeated real root
• If b² - 4ac < 0, there are no real roots

Example: Solving a hidden quadratic equation: x⁴ - 13x² + 36 = 0 Let y = x², then solve y² - 13y + 36 = 0

This guide provides a comprehensive overview of AS level pure maths algebraic expressions, offering clear explanations and examples to help students master these crucial concepts.

Page 4: Quadratic Equations

This page introduces quadratic equations and various methods for solving them, including factorisation, completing the square, and the quadratic formula.

Definition: A quadratic equation is defined as ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.

Example: The page demonstrates the process of completing the square for the equation x² + 6x + 2, resulting in (x + 3)² - 7.

Highlight: The quadratic formula is presented as x = [-b ± √(b² - 4ac)] / (2a).

Vocabulary: Factorisation, completing the square, and the quadratic formula are introduced as the three main methods for solving quadratic equations.