Difference between revisions of "Quadratic Equations"

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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]
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| style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; " |[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
 
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
 
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While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
 
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
  
= Concept Map =
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== Concept Map ==
<mm>[[Quadratic_Equations.mm|Flash]]</mm>
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[[File:Quadratic_Equations.mm|Flash]]
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__FORCETOC__
 
__FORCETOC__
  
= Textbook =
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== Textbook ==
 
Please click here for Karnataka and other text books.
 
Please click here for Karnataka and other text books.
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#[http://ktbs.kar.nic.in/New/Textbooks/class-x/english/maths/class-x-english-maths-chapter09.pdf Karnataka text book for Class 10, Chapter 09 - Quadratic Equations]
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#[http://nimsdxb.com/wp-content/uploads/Unit-4_Quadratic_Equations_Core.pdf/ cbse text book]
  
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==Additional Information==
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{{#widget:Iframe |url=http://www.slideshare.net/slideshow/embed_code/10549763|width=450 |height=360 |border=1 }}
  
=Additional Information=
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===Useful websites===
==Useful websites==
 
==Reference Books==
 
  
= Teaching Outlines =
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[https://in.ixl.com/search?q=quadratic+equation/ For more information about quadratic equation]
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===Reference Books===
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[[Text_Books| relevent references]]
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=== Resources ===
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==== Resource Title ====
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[http://www.mathopenref.com/quadraticexplorer.html Quadratic Function Explorer]
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==== Description ====
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This interactive activity helps you realize the action of the three coefficients a, b, c in a quadratic equation.
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== Teaching Outlines ==
  
 
==Concept #1 - Introduction to quadratic equations==
 
==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.
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An equation of the form  <math>ax^2+bx+c = 0</math> where a ≠ 0 and a, b, c belongs to R.
  
 
===Learning objectives===
 
===Learning objectives===
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===Notes for teachers===
 
===Notes for teachers===
 
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#Basic knowledge of equations, linear equations,general form of linear equation, finding the rooots of equation, graphical representation of linear equations.<br>
''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.''
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#More importance to be given for signs while transforming the equations.
  
 
===Activities===
 
===Activities===
  
#Activity No #1 '''[[Quadratic_equations_introduction_to_quadratic_equation_activity_1|Introduction to quadratic equation]]'''
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#Activity No 1 '''[[Quadratic_equations_introduction_to_quadratic_equation_activity_1|Introduction to quadratic equation]]'''
 
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#Activity No 2 '''[[Quadratic_equations_introduction_to_quadratic_equation_actvity 2| Making a rectangular garden]]'''
----
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#Activity No 3 '''[[Quadratic_equations_introduction_to_quadratic_equation_activity 3| Understanding<math> ax^2+bx+c=0</math> geometrically]]'''
 
 
 
 
#Activity No #2 ]
 
'''[[Quadratic_equations_introduction_to_quadratic_equation_actvity 2| Making a rectangular garden]]'''
 
  
 
==Concept #2 - Types of equations==
 
==Concept #2 - Types of equations==
 
===Pure Quadratic Equation & Adfected Quadratic Equation===
 
===Pure Quadratic Equation & Adfected Quadratic Equation===
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Quadratic equation,in the form <math>ax^2+bx+c = 0</math>, is termed as quadratic expression and the equation of the form <math>ax^2+bx+c = 0</math>, a≠0 is called quadratic equation in x. This equation is also known to be pure quadratic equation if the value of b is zero i.e. b=0. Otherwise it is said to be adfected. The letters a, b, and c are called coefficients: and c is the constant coefficient.
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===Learning objectives===
 
===Learning objectives===
 
#To distinguish between pure & adfected equations among the given equations.
 
#To distinguish between pure & adfected equations among the given equations.
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===Notes for teachers===
 
===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.''
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#Knowledge of general form of quadratic equations<br>
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#roots of equation<br>
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#proper use of signs.
  
 
===Activities===
 
===Activities===
#Activity No #1 '''[[Identifying pure and adfected ouadratic equations- Activity No1]]'''
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'''[[Identifying pure and adfected ouadratic equations- Activity No1]]'''
  
#Activity No #2 '''Concept Name - Activity No.'''
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'''[http://mathworksheets4kids.com/equations/quadratic.html/ work sheet Activity No2]'''
  
 
==Concept #3 What is the solution of a quadratic equation==  
 
==Concept #3 What is the solution of a quadratic equation==  
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===Notes for teachers===
 
===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.''
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#different methods of solving quadratic equation
 
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#knowledge of suitable formula to be used to solve specific problem.
===Activities===
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#identify the type of quadratic equation.
#Activity No #1 '''Concept Name - Activity No.'''
 
#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===
 
===Activities===
#Activity No #1 '''Concept Name - Activity No.'''
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#Activity No #1 #Activity No 3-[http://www.projectmaths.ie/students/strand4JC/student-activity-quadratic-formula.pdf| quadratic formula]<br>
#Activity No #2 '''Concept Name - Activity No.'''
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#Activity No #2 '''Concept Name - Activity No'''
  
 
==Concept #4Methods of solution==
 
==Concept #4Methods of solution==
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#Deriving formula to find the roots of quadratic equation.
 
#Deriving formula to find the roots of quadratic equation.
 
#Solving quadratic equation by using formula.<br>
 
#Solving quadratic equation by using formula.<br>
[http://en.wikipedia.org/wiki/Quadratic_equation]
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#Solving quadratic equation graphically.<br>
#Solving quadratic equation graphically.<br>To find the sum and product of the roots of the quadratic equations.
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#To find the sum and product of the roots of the quadratic equations.
[https://www.geogebratube.org/student/m8358]
 
  
 
===Notes for teachers===
 
===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.''
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*Students need to know factorisation
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*substitution of values and simplification
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*Identifying suitable method
  
 
===Activities===
 
===Activities===
#Activity No #1 '''Concept Name - Activity No.'''
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#Activity No 1 -[https://www.geogebratube.org/material/iframe/id/8357/width/968/height/487/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/preferhtml5| geogebra]      
<iframe scrolling="no" src=["https://www.geogebratube.org/material/iframe/id/8357/width/968/height/487/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/preferhtml]5"width="968px" height="487px" style="border:0px;"> </iframe>
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#Activity No 2-[http://www.wikihow.com/Solve-Quadratic-Equations/ learn more how to solve Q.E]
#Activity No #2 '''Concept Name - Activity No.'''
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#Activity 3-[http://www.learnnc.org/lp/pages/2981| learn quadratics]
[http://www.projectmaths.ie/students/strand4JC/student-activity-quadratic-formula.pdf]
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#Activity 4- [[Quadratic Equation solution activity1|Quadratic Equation solution]]  
  
 
==Concept #5'''Nature of roots'''==
 
==Concept #5'''Nature of roots'''==
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===Notes for teachers===
 
===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.''
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Guiding in Identifying the nature based on the value of discriminant
  
 
===Activities===
 
===Activities===
#Activity No #1 '''Concept Name - Activity No.'''
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#Activity No #1 '''Concept Name - Activity No.'''[http://interpret the nature of roots/ interpret the nature of the roots]
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#Activity No #2 '''Concept Name - Activity No.'''
 
#Activity No #2 '''Concept Name - Activity No.'''
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===Learning objectives===
 
===Learning objectives===
 
By applying the methods of solving quadratic equations, finding the solutions to real life situations.
 
By applying the methods of solving quadratic equations, finding the solutions to real life situations.
[https://www.youtube.com/watch?v=IGGnn9oa4QY]
 
#Activity 2:[http://www.ehow.com/info_8502727_applications-quadratic-equations.html]
 
  
 
===Notes for teachers===
 
===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.''
+
Help the students in Identifying parameters and suitable methods for solving application problems.
  
 
===Activities===
 
===Activities===
#Activity No #1 '''applications - .'''
+
#Activity No #1 [https://www.youtube.com/watch?v=IGGnn9oa4QYz| more word problems]
__FORCETOC__
+
#Activity 2:[http://www.ehow.com/info_8502727_applications-quadratic-equations.html| quadratics in real life]
 
 
=Activity - Name of Activity=
 
 
 
==Estimated Time==
 
 
 
==Materials/ Resources needed==
 
==Prerequisites/Instructions, if any==
 
==Multimedia resources==
 
==Website interactives/ links/ simulations/ Geogebra Applets==
 
==Process (How to do the activity)==
 
==Developmental Questions (What discussion questions)==
 
==Evaluation (Questions for assessment of the child)==
 
==Question Corner==
 
==Activity Keywords==
 
 
 
'''To link back to the concept page'''
 
[[Topic Page Link]]
 
#Activity No #2 '''Concept Name - Activity No.'''
 
 
 
 
=Assessment activities for CCE=
 
=Assessment activities for CCE=
 +
.[Http://Tube.geogebra.org/m/105393c|quadratic quiz]
  
 
=Hints for difficult problems =
 
=Hints for difficult problems =
#If P & q are the roots of the equation 2a^-4a+1=0 find the value of
+
1.If P & q are the roots of the equation <math>2a^2-4a+1=0</math> find the value of  
p^3+q^3<br>
+
<math>p^3+q^3</math><br>
Pre requisites:
+
[[solution]]<br>
#Standard form of quadratic equation
+
2.The altitude of a triangle is 6cm greter than its base. If its area is 108cmsq .Find its base.<br>
#Formula to find the sum & product of quadratic equation
+
[[solution]]<br>
#Knowledge of using appropriate identity
+
3.Solve <math>x^2-4x-8=0</math> By completing the square. <br>
Interpretation of the Problem:
+
[[solution]]
#Compare the equation with standard form and identify the values of a,b,c
 
#To find the sum formformof the roots of the quadratic equation using the formula
 
#To find the product of the roots of the equation
 
# Using the identity & rewriting p^3+q^3 as (p+q)^3-3pq(p+q)
 
#Substitute the values of m+n & mn in (p+q)^3-3pq(p+q)
 
#Simplification
 
Concepts:
 
#Formula to find the sum and product of the roots of the quadratic equation
 
#Identity (a+b)^3=a^3+b^3+3ab(a+b)
 
Algorithm: <br>
 
Consider the equation 2a^2-4a+1=0<br>
 
Here a=2,b=-4 & c=1<br>
 
If p & q are the roots of the quadratic equation then<br>
 
p+q=-b/a=-(-4)/2=2<br>
 
pq=c/a=1/2<br>
 
Therefore,<br>
 
p^3+q^3=(p+q)^3-3pq(p+q)<br> =(2)^3-3(1/2)(2)<br>=
 
8-3<br>=5,<br>
 
 
 
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.<br>
 
PROBLEM 2: Solve  x2 – 4x – 8 = 0. By completing the square.<br>
 
Interpretation of the problem:<br>
 
Is it  a quadratic equation?<br>
 
Knowledge about coefficients of the variable.<br>
 
Knowledge of steps for completing the given equation as square.<br>
 
Knowledge of root.<br>
 
Different approaches to solve:<br>
 
Factorization Method (sometimes in few ex as x2+6x-7=0)<br>
 
Completing the square.<br>
 
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:<br>
 
1) First, I put the loose number “8”  on the other side of the equation:<br>
 
x2 – 4x – 8 = 0 <br>
 
x2 – 4x = 8<br>
 
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:<br>
 
x2 – 4x + 4 = 8 + 4 <br>
 
x2 – 4x + 4 = 12 <br>
 
3) This process creates a quadratic that is a perfect square, and factoring gives me:<br>
 
(x – 2)2 = 12<br>
 
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.<br>
 
 
 
 
 
 
 
4) Now I can square-root both sides of the equation, simplify, and solve:<br>
 
(x – 2)2 = 12<br>
 
 
 
Then the solution is<br>
 
For each approach:<br>
 
Prior knowledge:<br>
 
About quadratic equation <br>
 
: About co-efficient’s <br>
 
: Comparing the equation with standard form<br>
 
: Dividing and squaring the  value of ‘b’<br>
 
Gap identification:<br>
 
Recalling the standard equation.<br>
 
: Identifying the values of a,b,c in given equation<br>
 
: Dividing and squaring<br>
 
Algorithm :<br>
 
1. Equating the equation to zero or standard form.<br>
 
2. Translating  loose number “ c” i.e. constant  on the other side of the equation<br>
 
3. Making half  and squaring  the co-efficient of variable (x).<br>
 
4. Add the square on both sides of equation.<br>
 
5. Put in the complete square form. (a+b)2 or (a-b) 2<br>
 
6. Find the value of variable ‘x’ /square-roots of the variable<br>
 
 
 
=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.<br>
 
Statement: Solving problem based on quadratic equations.<br>
 
*Interpretation  of the problem:<br> * Converting data in to eqn.<br> *Knowledge about area of a triangle.<br>*knowledge of the formula of area of triangle.<br>*Methods of finding the roots of the eqn.<br> *Methods of finding the roots of the
 
*Different approches to solve the problem: <br>*Factorisation
 
*Using formula
 
*using graph
 
*Concept used:Forming the eqn. 216=x(x+6)
 
216=x2+6x<br>
 
x2 +6x -216=0<br>
 
Substitution:  x 2 +18x-12x -216=0<br>
 
Simplification:  x(x+18)-12(x+18)=0<br>
 
(x+18)( x-12)=0<br>
 
(x+18)=0 (x-12)=0<br>
 
x=-18, x=12<br>.
 
# Base=12cm,  <br> Altitude=x+6
 
=12+6=18cm.<br>
 
'''Prior Knowledge''' -<br>
 
*Methods of solving the Eqn<br>
 
*Factorisation<br>
 
*Using Formula<br>
 
*Using Graph<br>
 
 
 
= Project Ideas =
 
[http://calculus-geometry.hubpages.com/hub/Free-Online-Quadratic-Equations-Quiz]
 
  
= Math Fun =
+
[[Category:Class 10]]
[http://education.ti.com/en/timathnspired/us/algebra-1/quadratic-functions]
+
[[Category:Quadratic Equations]]

Latest revision as of 20:00, 19 December 2020

ಕನ್ನಡದಲ್ಲಿ ನೋಡಿ

The Story of Mathematics

Philosophy of Mathematics

Teaching of Mathematics

Curriculum and Syllabus

Topics in School Mathematics

Textbooks

Question Bank

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Concept Map

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Textbook

Please click here for Karnataka and other text books.

  1. Karnataka text book for Class 10, Chapter 09 - Quadratic Equations
  2. cbse text book

Additional Information

Useful websites

For more information about quadratic equation

Reference Books

relevent references

Resources

Resource Title

Quadratic Function Explorer

Description

This interactive activity helps you realize the action of the three coefficients a, b, c in a quadratic equation.

Teaching Outlines

Concept #1 - Introduction to quadratic equations

An equation of the form where a ≠ 0 and a, b, c belongs to R.

Learning objectives

converting verbal statement into equations.

Notes for teachers

  1. Basic knowledge of equations, linear equations,general form of linear equation, finding the rooots of equation, graphical representation of linear equations.
  2. More importance to be given for signs while transforming the equations.

Activities

  1. Activity No 1 Introduction to quadratic equation
  2. Activity No 2 Making a rectangular garden
  3. Activity No 3 Understanding geometrically

Concept #2 - Types of equations

Pure Quadratic Equation & Adfected Quadratic Equation

Quadratic equation,in the form , is termed as quadratic expression and the equation of the form , a≠0 is called quadratic equation in x. This equation is also known to be pure quadratic equation if the value of b is zero i.e. b=0. Otherwise it is said to be adfected. The letters a, b, and c are called coefficients: and c is the constant coefficient.

Learning objectives

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

Notes for teachers

  1. Knowledge of general form of quadratic equations
  2. roots of equation
  3. proper use of signs.

Activities

Identifying pure and adfected ouadratic equations- Activity No1

work sheet Activity No2

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

  1. different methods of solving quadratic equation
  2. knowledge of suitable formula to be used to solve specific problem.
  3. identify the type of quadratic equation.

Activities

  1. Activity No #1 #Activity No 3-quadratic formula
  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.
  6. To find the sum and product of the roots of the quadratic equations.

Notes for teachers

  • Students need to know factorisation
  • substitution of values and simplification
  • Identifying suitable method

Activities

  1. Activity No 1 -geogebra
  2. Activity No 2-learn more how to solve Q.E
  3. Activity 3-learn quadratics
  4. Activity 4- Quadratic Equation solution

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

Guiding in Identifying the nature based on the value of discriminant

Activities

  1. Activity No #1 Concept Name - Activity No.the nature of roots/ interpret the nature of the roots
  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

Help the students in Identifying parameters and suitable methods for solving application problems.

Activities

  1. Activity No #1 more word problems
  2. Activity 2:quadratics in real life

Assessment activities for CCE

.quiz

Hints for difficult problems

1.If P & q are the roots of the equation find the value of
solution
2.The altitude of a triangle is 6cm greter than its base. If its area is 108cmsq .Find its base.
solution
3.Solve By completing the square.
solution