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Solving quadratic equations

Easy Steps to Solve Quadratic Equations!

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Easy Steps to Solve Quadratic Equations!
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Safwan Master

@safwanmaster

·

23 Followers

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A comprehensive guide to solving quadratic equations step by step and analyzing their properties.

• The document covers various aspects of quadratic equations including completing the square, finding discriminants, and solving disguised quadratics

• Key topics include expressing quadratic in vertex form, finding real and repeated roots, and graphing quadratic functions

• Advanced concepts covered include working with equal roots, analyzing equations with parameters, and determining the nature of solutions

• Special attention is given to sketching quadratic curves and finding points of intersection with coordinate axes

• The material demonstrates how to determine maximum and minimum values of quadratic functions and analyze their behavior

05/04/2023

280

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

View

Page 2: Graphical Representations

This page focuses on the visual interpretation of quadratic equations through graphs and diagrams. It presents different forms of quadratic expressions and their corresponding graphical representations.

Example: Four different quadratic curves are shown, demonstrating how different forms of quadratic equations affect their graphical appearance

Vocabulary: Axis of symmetry - the vertical line that divides a parabola into two identical halves

Highlight: The relationship between the algebraic form and graphical representation is emphasized

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

View

Page 3: Advanced Quadratic Concepts

The page delves into more complex quadratic equations involving parameters and special conditions like equal roots.

Definition: Equal roots occur when the discriminant of a quadratic equation equals zero

Example: The equation (k-4)x² - 4x + k-2 = 0 is analyzed for equal roots

Highlight: The importance of working with exact values rather than decimal approximations

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

View

Page 4: Solving Methods

This section demonstrates various methods for solving quadratic equations step by step.

Example: Detailed solution of x⁴ - 4x² - 5 = 0 using substitution method

Highlight: The importance of verifying solutions and checking for extraneous roots

Definition: Discriminant formula: b² - 4ac determines the nature of roots

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

View

Page 5: Applications and Problem Solving

Focuses on practical applications and problem-solving strategies for quadratic equations.

Example: Finding points of intersection between a line and a quadratic curve

Highlight: The relationship between the discriminant and the number of real solutions

Definition: The vertex of a parabola represents either the maximum or minimum point

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

View

Page 6: Graphical Analysis

Detailed examination of different forms of quadratic expressions and their graphs.

Example: Analysis of y = (3x)², y = x² + 9, and y = (3x)(x + 3)

Highlight: The relationship between factored form and x-intercepts

Definition: X-intercepts are the points where a curve crosses the x-axis

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

View

Page 7: Advanced Problem Solving

Complex problem-solving involving parameters and conditions for real roots.

Example: Finding conditions for no real roots using the discriminant

Highlight: The importance of algebraic manipulation in solving parametric equations

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

View

Page 8: Special Cases and Applications

Explores special cases of quadratic equations and their applications.

Example: Solving equations with fractional powers and analyzing their solutions

Highlight: The significance of domain restrictions when solving equations

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

View

Page 9: Vertex Form and Applications

Focuses on the vertex form of quadratic equations and its applications.

Example: Finding the vertex form of a quadratic equation given its minimum point

Definition: Vertex form: y = a(x - h)² + k, where (h,k) is the vertex

Highlight: The relationship between the vertex form and the graph's transformations

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

View

Page 1: Introduction to Quadratic Equations

This page introduces fundamental concepts for solving quadratic equations and presents various problem types. The questions focus on expressing quadratics in standard forms and finding roots.

Example: Express 2x² + 12x + 23 in the form a(x + b)² + c

Highlight: The page emphasizes the importance of showing complete working steps when solving equations

Definition: A quadratic equation has repeated roots when its discriminant equals zero

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Subjects

/

Maths

/

Algebra

/

Solving quadratic equations

Easy Steps to Solve Quadratic Equations!

user profile picture

Safwan Master

@safwanmaster

·

23 Followers

Follow

A comprehensive guide to solving quadratic equations step by step and analyzing their properties.

• The document covers various aspects of quadratic equations including completing the square, finding discriminants, and solving disguised quadratics

• Key topics include expressing quadratic in vertex form, finding real and repeated roots, and graphing quadratic functions

• Advanced concepts covered include working with equal roots, analyzing equations with parameters, and determining the nature of solutions

• Special attention is given to sketching quadratic curves and finding points of intersection with coordinate axes

• The material demonstrates how to determine maximum and minimum values of quadratic functions and analyze their behavior

05/04/2023

280

 

11/12

 

Maths

5

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

Page 2: Graphical Representations

This page focuses on the visual interpretation of quadratic equations through graphs and diagrams. It presents different forms of quadratic expressions and their corresponding graphical representations.

Example: Four different quadratic curves are shown, demonstrating how different forms of quadratic equations affect their graphical appearance

Vocabulary: Axis of symmetry - the vertical line that divides a parabola into two identical halves

Highlight: The relationship between the algebraic form and graphical representation is emphasized

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

Page 3: Advanced Quadratic Concepts

The page delves into more complex quadratic equations involving parameters and special conditions like equal roots.

Definition: Equal roots occur when the discriminant of a quadratic equation equals zero

Example: The equation (k-4)x² - 4x + k-2 = 0 is analyzed for equal roots

Highlight: The importance of working with exact values rather than decimal approximations

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

Page 4: Solving Methods

This section demonstrates various methods for solving quadratic equations step by step.

Example: Detailed solution of x⁴ - 4x² - 5 = 0 using substitution method

Highlight: The importance of verifying solutions and checking for extraneous roots

Definition: Discriminant formula: b² - 4ac determines the nature of roots

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

Page 5: Applications and Problem Solving

Focuses on practical applications and problem-solving strategies for quadratic equations.

Example: Finding points of intersection between a line and a quadratic curve

Highlight: The relationship between the discriminant and the number of real solutions

Definition: The vertex of a parabola represents either the maximum or minimum point

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

Page 6: Graphical Analysis

Detailed examination of different forms of quadratic expressions and their graphs.

Example: Analysis of y = (3x)², y = x² + 9, and y = (3x)(x + 3)

Highlight: The relationship between factored form and x-intercepts

Definition: X-intercepts are the points where a curve crosses the x-axis

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

Page 7: Advanced Problem Solving

Complex problem-solving involving parameters and conditions for real roots.

Example: Finding conditions for no real roots using the discriminant

Highlight: The importance of algebraic manipulation in solving parametric equations

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

Page 8: Special Cases and Applications

Explores special cases of quadratic equations and their applications.

Example: Solving equations with fractional powers and analyzing their solutions

Highlight: The significance of domain restrictions when solving equations

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

Page 9: Vertex Form and Applications

Focuses on the vertex form of quadratic equations and its applications.

Example: Finding the vertex form of a quadratic equation given its minimum point

Definition: Vertex form: y = a(x - h)² + k, where (h,k) is the vertex

Highlight: The relationship between the vertex form and the graph's transformations

1. 2. 3. 4. 5. Name: Score: 51/57 Strengths: Completing the square Discriminant Disguised Quadratics Solving Sketching Any others? (i) (ii)

Page 1: Introduction to Quadratic Equations

This page introduces fundamental concepts for solving quadratic equations and presents various problem types. The questions focus on expressing quadratics in standard forms and finding roots.

Example: Express 2x² + 12x + 23 in the form a(x + b)² + c

Highlight: The page emphasizes the importance of showing complete working steps when solving equations

Definition: A quadratic equation has repeated roots when its discriminant equals zero

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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.

Ranked #1 Education App

Download in

Google Play

Download in

App Store

Knowunity is the #1 education app in five European countries

4.9+

Average app rating

15 M

Pupils love Knowunity

#1

In education app charts in 12 countries

950 K+

Students have uploaded notes

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

iOS User

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

Philip, iOS User

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

Lena, iOS user

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