Fun Study Notes for Algebra with Easy Examples and Cool Worksheets

Open

Milkshakemi

10/11/2022

Maths

A level Maths: Pure Y1 - Algebraic expression and Quadratics

Subjects

Maths

Algebra

Algebraic expressions

Fun Study Notes for Algebra with Easy Examples and Cool Worksheets

The document provides comprehensive algebraic expressions edexcel study notes covering key topics in algebra and functions. It explains surds, quadratic equations, functions, and related concepts in detail, offering clear explanations and examples for students.

Key points:

Covers index laws, surds, and rationalizing denominators
Explains quadratic equations, including solving methods and graphing
Discusses functions, domains, and ranges
Explores the discriminant and its applications
Addresses hidden quadratics and solving techniques

10/11/2022

Chp 1. Algebraic Expressions
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ac b c

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:

Factorization
Completing the square
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.

View

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.

View

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.

View

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.

View

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.

View

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.

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Subjects

Maths

Algebra

Algebraic expressions

Maths

•

2,225

•

10 Nov 2022

•

7 pages

Fun Study Notes for Algebra with Easy Examples and Cool Worksheets

Milkshakemi

@milkshakemi

Key points:

Covers index laws, surds,... Show more

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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:

Factorization
Completing the square
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.

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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.

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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.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

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.

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By signing up you accept Terms of Service and Privacy Policy

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.

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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.

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

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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:

Rationalizing a single surd in the denominator
Rationalizing a denominator with two surds $difference$
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.

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Fun Study Notes for Algebra with Easy Examples and Cool Worksheets

Chapter 2: Quadratics

Functions and the Discriminant

Similar content

Can't find what you're looking for? Explore other subjects.

Knowunity is the #1 education app in five European countries

Knowunity has been named a featured story on Apple and has regularly topped the app store charts in the education category in Germany, Italy, Poland, Switzerland, and the United Kingdom. Join Knowunity today and help millions of students around the world.

4.9+

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#1

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Still not convinced? See what other students are saying...

I love this app so much, I also use it daily. I recommend Knowunity to everyone!!! I went from a D to an A with it :D

The app is very simple and well designed. So far I have always found everything I was looking for :D

I love this app ❤️ I actually use it every time I study.

Fun Study Notes for Algebra with Easy Examples and Cool Worksheets

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Chapter 2: Quadratics

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Functions and the Discriminant

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Chapter 1: Algebraic Expressions

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