Chapter 2: Quadratics

This chapter delves into quadratic equations and their solutions, covering essential concepts for AS level pure maths algebraic expressions.

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

The chapter outlines three main methods for solving quadratic equations:

1. Factorization
2. Completing the square
3. Quadratic formula

Example: Completing the square for x² + 6x + 2: $x + 3$² - 9 + 2 = $x + 3$² - 7

The quadratic formula is presented as:

x = $-b ± √(b² - 4ac)$ / (2a)

This chapter also introduces quadratic graphs, discussing their shape, key features, and how to find the turning point by completing the square.

Highlight: The direction of the parabola depends on the sign of 'a' in the quadratic equation ax² + bx + c.

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.

Page 5: Functions and Quadratic Graphs

This section explores the concept of functions and the graphical representation of quadratic equations.

Definition: A function is described as a machine that takes an input $x$, converts it mathematically, and produces an output [f$x$ or g$x$].

Example: The page illustrates the general form of a quadratic function as f$x$ = ax² + bx + c and its corresponding graph.

Highlight: The shape of the parabola depends on the sign of 'a': positive 'a' results in a U-shape, while negative 'a' produces an inverted U-shape.

Vocabulary: Domain (set of possible inputs) and range (set of possible outputs) are introduced as key terms in function analysis.

Page 6: The Discriminant

This page focuses on the discriminant of a quadratic equation and its role in determining the nature of the equation's roots.

Definition: The discriminant is defined as b² - 4ac for a quadratic function f$x$ = ax² + bx + c.

Highlight: The value of the discriminant indicates the number and nature of roots:

• If b² - 4ac > 0, the function has two distinct real roots.
• If b² - 4ac = 0, the function has one repeated real root.
• If b² - 4ac < 0, the function has no real roots.

Example: The page includes a problem to find the range of k for which x² + 2x + k = 0 has two distinct real roots.

Page 7: Hidden Quadratics

The final page addresses the concept of hidden quadratics and techniques for solving such equations.

Example: The page demonstrates how to solve the equation x⁴ - 13x² + 36 = 0 by substituting y = x² to reveal a hidden quadratic equation.

Highlight: The solution process involves factorising the resulting quadratic equation in y and then solving for x by taking square roots.

Vocabulary: Hidden quadratics are equations that can be transformed into standard quadratic form through substitution.

Chapter 1: Algebraic Expressions

This chapter introduces fundamental concepts in algebraic expressions, focusing on index laws and surds.

Definition: Surds are irrational numbers that cannot be expressed as a simple fraction.

The chapter emphasizes the importance of understanding index laws and their application to terms with the same base. It also covers the properties of surds, including:

• The product and quotient of surds
• Simplification of surds
• Rationalizing denominators

Highlight: When simplifying surds, remembering square numbers is crucial for efficient problem-solving.

Example: √(ab) = √a × √b, but √$a + b$ ≠ √a + √b

The section on rationalizing surds presents three key cases:

1. Rationalizing a single surd in the denominator
2. Rationalizing a denominator with two surds (difference)
3. Rationalizing a denominator with two surds (sum)

Vocabulary: Rational number - a number that can be expressed as a fraction a/b, where a and b are integers.