Quadratic Equations

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Textbook

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Teaching Outlines

Concept #1 - Introduction to quadratic equations

An equation of the form ax^2+bx+c = 0 where a ≠ 0 and a, b, c belongs to R.

Learning objectives

converting verbal statement into equations.

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

1. Activity No #1 introduction to quadratic equation

Please use this link: [1]

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1. Activity No #2 Concept Name - Activity No.

A gardner wants his garden to have ageometrical shape. He decides on the following rules for building the flowerbeds.

1. They must all be rectangular
2. The perimeter and area must be the same.
   

How many different flower beds can the gardener make if one of the sides ia 3 units less than the other side.

1. How many different flower beds can the gardener make if both the sides are of same length.

1. notes for teacher-

The aim of this activity is to make a situation that leads to the quadratic equation-

x(x-3)=4x-6

x^2=4x

Students need to use their own strategies to solve the equations. Students may, for example establish a set of equivalent quadratic equations trough the balancing method which they are familiar in the context of linear equations.

1. .1.x^2-3x = 4x-6 => x^2-7x= -6
2. .2.x^2-4x = 0

However ,students will probably have no other methods available but to solve using numerical method. They might set up tables from original eqn.

They need to be encouraged to move through the numbers to find the solutions and to make sense of the solution. It also needs to be made explicit here that we are now dealing with an equation that involves a term with an unknown of second degree. This is one feature that distinguishes it from linear equation.

1. note- In using the balancing method pupil might very well divide both sides of equation
   

x^2 =4x by x 

x=4

This must be discussed.

Concept #2 - Types of equations

Learning objectives

1. To distinguish between pure & adfected equations among the given equations.
2. Standard forms of pure & adfected quadratic equations.

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

1. Activity No #1 Concept Name - Activity No.
2. Activity No #2 Concept Name - Activity No.

Concept #3 What is the solution of a quadratic equation

The roots of the Quadratic Equation which satisfy the equation

Learning objectives

1. x=k is a solution of the quadratic equation if k satisfies the quadratic equation
2. Any quadratic equation has at most two roots.
3. The roots form the solution set of quadratic equation.

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

1. Activity No #1 Concept Name - Activity No.
2. Activity No #2 Concept Name - Activity No.

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

1. Activity No #1 Concept Name - Activity No.
2. Activity No #2 Concept Name - Activity No.

Concept #4Methods of solution

Different methods of finding the solution to a quadratic equation

1. Factorisation method
2. Completing the square method
3. Formula method
4. Graphical method.

Learning objectives

1. Solving quadratic equation by factorisation method
2. Solving quadratic equation by completing the square method
3. Deriving formula to find the roots of quadratic equation.
4. Solving quadratic equation by using formula.
5. Solving quadratic equation graphically.

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

1. Activity No #1 Concept Name - Activity No.

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1. Activity No #2 Concept Name - Activity No.

Concept #5Nature of roots

The roots of a quadratic equation can be real & equal, real & distinct or imaginary. Nature of roots depends on the values of b^-4ac.

Learning objectives

1. To find the discriminant & interpret the nature of the roots of the given quadratic equation.

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

1. Activity No #1 Concept Name - Activity No.

1. Activity No #2 Concept Name - Activity No.

Concept #6applications

Solving problems based on quadratic equations.

Learning objectives

By applying the methods of solving quadratic equations, finding the solutions to real life situations.

Notes for teachers

These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.

Activities

1. Activity No #1 applications - .

Activity - Name of Activity

Activity Keywords

To link back to the concept page Topic Page Link

1. Activity No #2 Concept Name - Activity No.

Hints for difficult problems

1. If P & q are the roots of the equation 2a^-4a+1=0 find the value of

p^3+q^3
Pre requisites:

1. Standard form of quadratic equation
2. Formula to find the sum & product of quadratic equation
3. Knowledge of using appropriate identity

Interpretation of the Problem:

1. Compare the equation with standard form and identify the values of a,b,c
2. To find the sum of the roots of the quadratic equation using the formula
3. To find the product of the roots of the equation
4. Using the identity & rewriting p^3+q^3 as (p+q)^3-3pq(p+q)
5. Substitute the values of m+n & mn in (p+q)^3-3pq(p+q)
6. Simplification

Concepts:

1. Formula to find the sum and product of the roots of the quadratic equation
2. Identity (a+b)^3=a^3+b^3+3ab(a+b)

Algorithm:
Consider the equation 2a^2-4a+1=0
Here a=2,b=-4 & c=1
If p & q are the roots of the quadratic equation then
p+q=-b/a=-(-4)/2=2
pq=c/a=1/2
Therefore,
p^3+q^3=(p+q)^3-3pq(p+q)

      =(2)^3-3(1/2)(2)

      =8-3

      =5

Ex.no.9.11 /problem no.9

The altitude of a triangle is 6cm greter than its base. If its area is 108cmsq .Find its base.
Statement: Solving problem based on quadratic equations.

• Interpretation of the problem:
  * Converting data in to eqn.
  *Knowledge about area of a triangle.
  *knowledge of the formula of area of triangle.
  *Methods of finding the roots of the eqn.
  *Methods of finding the roots of the
• Different approches to solve the problem:
  *Factorisation
• Using formula
• using graph
• Concept used:Forming the eqn. 216=x(x+6)

216=x2+6x
x2 +6x -216=0
Substitution: x 2 +18x-12x -216=0
Simplification: x(x+18)-12(x+18)=0
(x+18)( x-12)=0
(x+18)=0 (x-12)=0
x=-18, x=12
.

1. Base=12cm,
   Altitude=x+6

=12+6=18cm.
Prior Knowledge -

• Methods of solving the Eqn
  
• Factorisation
  
• Using Formula
  
• Using Graph