Quadratic Equations

The Story of Mathematics

Philosophy of Mathematics

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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

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1. Activity No #2 ]

Making a rectangular garden

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 Identifying pure and adfected ouadratic equations- Activity No1

1. 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.
   

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1. Solving quadratic equation graphically.
   To find the sum and product of the roots of the quadratic equations.

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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.

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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. [4]

1. Activity 2:[5]

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 formformof 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,

Following explains the steps and gives examples of solving by completing the square. It also shows how the Quadratic Formula is generated by this process. So I'll just do just one example of the process in this lesson. If you need further feel free to reach me.
PROBLEM 2: Solve x2 – 4x – 8 = 0. By completing the square.
Interpretation of the problem:
Is it a quadratic equation?
Knowledge about coefficients of the variable.
Knowledge of steps for completing the given equation as square.
Knowledge of root.
Different approaches to solve:
Factorization Method (sometimes in few ex as x2+6x-7=0)
Completing the square.
As noted above, this quadratic does not factor, so I can't solve the equation by factoring. And they haven't given me the quadratic in a form that is ready to square-root. But there is a way for me to manipulate the quadratic to put it into that form, and then solve. It works like this:
1) First, I put the loose number “8” on the other side of the equation:
x2 – 4x – 8 = 0
x2 – 4x = 8
2) Then I look at the coefficient of the x-term, which is –4 in this case. I take half of this number (including the sign), giving me –2. Then I square this value to get +4, and add this squared value to both sides of the equation:
x2 – 4x + 4 = 8 + 4
x2 – 4x + 4 = 12
3) This process creates a quadratic that is a perfect square, and factoring gives me:
(x – 2)2 = 12
Tip : I know it's a "minus two" inside the parentheses because half of –4 is –2. If you note the sign when you're finding one-half of the coefficient, then you won't mess up the sign when you're converting to squared-binomial form.

4) Now I can square-root both sides of the equation, simplify, and solve:
(x – 2)2 = 12

Then the solution is
For each approach:
Prior knowledge:
About quadratic equation

About co-efficient’s

Comparing the equation with standard form

Dividing and squaring the value of ‘b’

Gap identification:
Recalling the standard equation.

Identifying the values of a,b,c in given equation

Dividing and squaring

Algorithm :
1. Equating the equation to zero or standard form.
2. Translating loose number “ c” i.e. constant on the other side of the equation
3. Making half and squaring the co-efficient of variable (x).
4. Add the square on both sides of equation.
5. Put in the complete square form. (a+b)2 or (a-b) 2
6. Find the value of variable ‘x’ /square-roots of the variable

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
  

Project Ideas

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Math Fun

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