SlideShare a Scribd company logo
©Copyrighted by Rishabh & Group
Quadratic Equations
Group 1
FA-IV Project
Serial No. Topic
1 Certificate
2 Acknowledgement
3 Introduction
4 Sri Dharacharya Ji
5 Identification of Quadratic
Equations
6 Factorisation
Method
7 Completing the square
method
8 Discriminant & Nature of
roots
9 Problems based on
Quadratic Equations
10 Summary
11 Bibliography
Certificate
Thisistocertifythattheprojectworksatisfactorily carried out
inthegroupforFA-II.And hence, recorded in this file is the
bonafied work of group no. ___ consisting of members:
.......................................... of class X during the Academic year
2012-13.
Teacher in charge Date Principal
Acknowledgement
SpecialthankstoMrs.................madam for her guidance and
hersupport.Shehelpedthroughouttheproject.We thank our
Principalforhersupport. We thank our group members for
theircontribution and cooperation for making this project.
ThisProjectwasmadefrom the support and contribution of
our group members. So we will thank each one of us.
Sridhara (c. 870, India – c. 930 India) was an Indian
mathematician. He was born in Hooghly district in the 10th
Century AD. His father's name was Baladevacharya and mother's
name was Acchoka.
Works:-
 He was known for two treatises: Trisatika (sometimes called the
Patiganitasara) and the Patiganita. His major work Patiganitasara was
named Trisatika because it was witten in three hundred slokas.
 The book discusses counting of numbers,measures,natural
number,multiplication,division,zero,squares,cubes,fraction,rule of
three,interest-calculation,joint business or partnership and mensuration.
 Of all the Hindu Acharyas the exposition of Sridharacharya on zero is the
most explicit. He has written, "If 0(zero) is added to any number,the sum is
the same number; If 0(zero) is subtracted from any number,the number
remains unchanged; If 0(zero) is multiplied by any number,the product is
0(zero)". He has said nothing about division of any number by 0(zero).
 In the case of dividing a fraction he has found out the method of
mutiplying the fraction by the reciprocal of the divisor.
 He wrote on practical applicationsof algebra separated algebra
from arithmetic
Introduction and standard form
of Quadratic Equations
In mathematics, a quadratic equation is a
univariate polynomial equation of the second
degree. A general quadratic equation can be
written in the form
ax2
+ bx + c = 0
where x represents a variable or an unknown, and
a, b, and c are constants with a ≠ 0. (If a = 0, the
equation is a linear equation.)
The constants a, b, and c are called respectively,
the quadratic coefficient, the linear coefficient and
the constant term or free term. The term
"quadratic" comes from quadratus, which is the
Latin word for "square". Quadratic equations can
be solved by factorizing, completing the square,
graphing, Newton's method, and using the
quadratic formula
Identification of Quadratic Equations
Examples : Check whether the following are
quadratic equations:
(i) (x – 2)2
+ 1 = 2x – 3
(ii) x(x + 1) + 8 = (x + 2) (x – 2)
Solution :
(i) LHS = (x – 2)2
+ 1 = x2
– 4x + 4 + 1 = x2
– 4x + 5
Therefore, (x – 2)2
+ 1 = 2x – 3 can be rewritten as
= x2
– 4x + 5 = 2x – 3
i.e., x2
– 6x + 8 = 0
It is of the form ax2
+ bx + c = 0.
Therefore, the given equation is a quadratic equation.
(ii) Since x(x + 1) + 8 = x2
+ x + 8 and (x + 2)(x – 2) = x2 – 4
Therefore, x2
+ x + 8 = x2 – 4
i.e., x + 12 = 0
It is not of the form ax2
+ bx + c = 0.
Therefore, the given equation is not a quadratic equation.
Solution of a Quadratic Equation by Factorisation
Consider the quadraticequation 2x2
– 3x + 1 = 0. If we replace x
by 1 on the LHS of this equation, we get (2 × 12) – (3 × 1) + 1 = 0
= RHS of the equation. We say that 1 is a root of the quadratic
equation 2x2
– 3x + 1 = 0. This also means that 1 is a zero of the
quadraticpolynomial 2x2
– 3x + 1.
In general, a real number α is called a root of the
quadraticequation ax2
+ bx + c = 0, a ≠ 0 if a α2
+ bα + c = 0. We
also say that x = α is a solution of the quadraticequation,or that α
satisfies the quadraticequation.Note that the zeroes of the
quadraticpolynomial ax2
+ bx + c and the roots of the quadratic
equation ax2
+ bx + c = 0 are the same.
Example:
(i)Find the roots of the quadratic equation 6x2
– x – 2 = 0.
Sol. We have 6x2
– x – 2 = 6x2
+ 3x – 4x – 2
=3x (2x + 1) – 2 (2x + 1)
=(3x – 2)(2x + 1)
The roots of 6x2 – x – 2 = 0 are the values of x for which (3x – 2)(2x + 1) = 0
Therefore, 3x – 2 = 0 or 2x + 1 = 0,
i.e., x =
𝟐
𝟑
or x =−
𝟏
𝟐
Therefore, the roots of 6x2
– x – 2 = 0 are
𝟐
𝟑
and −
𝟏
𝟐
Completing the square and Derivation :-
The quadratic formula can be derived by the method of completing the square, so as to make use of the
algebraic identity:
Dividing the quadratic equation
by a (which is allowed because a is non-zero), gives:
The quadratic equation is now in a form to which the method of completing the square can be applied. To
"complete the square" is to add a constant to both sides of the equation such that the left hand side becomes a
complete square:
which produces
The right side can be written as a single fraction, with common denominator 4a2. This gives
Taking the square root of both sides yields
Isolating x, gives
Example:-
(i)Solve the equation 2x2
– 5x + 3 = 0 by the method
of completing the square.
Sol. The equation 2x2
– 5x + 3 = 0 is the same as 𝑥2
−
5
2
𝑥 + 3
2
= 0
Now, 𝑥2
−
5
2
𝑥 +
3
2
= (𝑥 −
5
4
)
2
− (
5
4
)
2
+
3
4
= (𝑥 −
5
4
)
2
−
1
16
Therefore, 2x2
– 5x + 3 = 0 can be written as (𝑥 −
5
4
)
2
−
1
16
= 0.
So, the roots of the equation 2x2
– 5x + 3 = 0 are exactly the same
as those of
(𝑥 −
5
4
)
2
−
1
16
= 0. Now, (𝑥 −
5
4
)
2
−
1
16
= 0 is the same as
(𝑥 −
5
4
)
2
=
1
16
Therefore, 𝑥 −
5
4
= ±
1
4
i.e., 𝑥 =
5
4
±
1
4
i.e., 𝑥 =
5
4
+
1
4
or 𝑥 =
5
4
−
1
4
i.e., 𝑥 =
3
2
or 𝑥 = 1
Therefore, the solutions of the equations are 𝑥 =
3
2
or 𝑥 = 1
Discriminant and Nature of roots:-
The roots of the equation ax2
+ bx + c = 0 are given by
𝑥 =
−𝑏±√ 𝑏2−4𝑎𝑐
2𝑎
If b2
– 4ac > 0, we get two distinct real roots −
𝑏
2𝑎
+
√𝑏2−4𝑎𝑐
2𝑎
and −
𝑏
2𝑎
−
√𝑏2−4𝑎𝑐
2𝑎
If b2
– 4ac = 0, then 𝑥 = −
𝑏
2𝑎
± 0, 𝑖. 𝑒. , 𝑥 = −
𝑏
2𝑎
𝑜𝑟 −
𝑏
2𝑎
So, the roots of the equation ax2
+ bx + c = 0 are both
−𝑏
2𝑎
.
Therefore, we say that the quadraticequation ax2
+ bx + c = 0 has
two equal real roots in this case.
If b2
– 4ac < 0, then there is no real number whose square is b2
–
4ac. Therefore, there are no real roots for the given quadratic
equation in this case.
Since b2
– 4ac determines whether the quadratic equation ax2
+
bx + c = 0 has real roots or not, b2
– 4ac is called the
discriminant of this quadratic equation.
So, a quadratic equation ax2 + bx + c = 0 has
(i) two distinct real roots, if b2
– 4ac > 0,
(ii) two equal real roots, if b2
– 4ac = 0,
(iii) no real roots, if b2
– 4ac < 0.
Example1:
(i)Find the discriminant of the equation 3x2
– 2x +
𝟏
𝟑
= 0 and hence find
the nature of its roots. Find them, if they are real.
Solution :
Here 𝑎 = 3, 𝑏 = −2 𝑎𝑛𝑑 𝑐 =
1
3
= 3.
Therefore, discriminant 𝑏2
− 4𝑎𝑐 = (−2) − 4 × 3 ×
1
3
= 4 − 4 = 0
Hence, the given quadratic equation has two equal real roots.
The roots are
−𝑏
2𝑎
,
−𝑏
2𝑎
, 𝑖. 𝑒. ,
2
6
,
2
6
, 𝑖. 𝑒.,
1
3
,
1
3
Example2:
Find the discriminant of the quadratic equation 2x2 –
4x + 3 = 0, and hence find the nature of its roots.
Solution :
The given equation is of the form ax2
+ bx + c = 0, where
a = 2, b = – 4 and c = 3.
Therefore, the discriminant b2
– 4ac = (– 4)2 – (4 × 2 × 3)
= 16 – 24 = – 8 < 0
So, the given equation has no real roots.
Problems based on Quadratic Equations:-
Example 1 : Find the roots of the equation 2x2
– 5x + 3 = 0, by factorisation.
Solution : Let us first split the middle term – 5x as –2x –3x [because (–2x) × (–3x)
= 6x2 = (2x2) × 3].
So, 2x2 – 5x + 3 = 2x2 – 2x – 3x + 3 = 2x (x – 1) –3(x – 1) = (2x – 3)(x – 1)
Now, 2x2 – 5x + 3 = 0 can be rewritten as (2x – 3)(x – 1) = 0.
So, the values of x for which 2x2 – 5x + 3 = 0 are the same for which (2x – 3)(x – 1) = 0,
i.e., either 2x – 3 = 0 or x – 1 = 0.
Now, 2x – 3 = 0 gives 𝑥 =
3
2
𝑎𝑛𝑑 𝑥 − 1 = 0 𝑔𝑖𝑣𝑒𝑠 𝑥 = 1
So, 𝑥 =
3
2
𝑎𝑛𝑑 𝑥 = 1 are the solutions of the equation.
Example 2 : Find the roots of the quadratic equation 𝟑𝒙 𝟐
− 𝟐√ 𝟔𝒙 + 𝟐 = 𝟎.
Solution : 𝟑𝒙 𝟐
− 𝟐√ 𝟔𝒙 + 𝟐 = 𝟑𝒙 𝟐
− √ 𝟔𝒙 + √ 𝟔𝒙 + 𝟐
= √ 𝟑𝒙(√ 𝟑𝒙 − √ 𝟐) − √ 𝟐 (√ 𝟑𝒙 − √ 𝟐)
= (√ 𝟑𝒙 − √ 𝟐)(√ 𝟑𝒙 − √ 𝟐)
So, the roots of the equation are the values of x for which
(√ 𝟑𝒙 − √ 𝟐)(√ 𝟑𝒙 − √ 𝟐) = 𝟎
Now, √ 𝟑𝒙 − √ 𝟐 = 𝟎 𝒇𝒐𝒓 𝒙 = √
𝟐
𝟑
So, this root is repeated twice, one for each repeated factor √ 𝟑𝒙 − √ 𝟐.
Therefore, the roots of 𝟑𝒙 𝟐
− 𝟐√ 𝟔𝒙 + 𝟐 = 𝟎 are √
𝟐
𝟑
, √
𝟐
𝟑
Example 3 : Find two consecutive odd positive integers, sum of whose squares
is 290.
Solution : Let the smaller of the two consecutive odd positive integers be x.
Then, the second integer will be x + 2. According to the question,
=x2 + (x + 2)2 = 290
i.e., x2 + x2 + 4x + 4 = 290
i.e., 2x2 + 4x – 286 = 0
i.e., x2 + 2x – 143 = 0
which is a quadratic equation in x.
Using the quadratic formula, we get
𝑥 =
−2±√4+572
2
=
−2±√576
2
=
−2±24
2
𝑥 = 11 𝑜𝑟 𝑥 = −13
But x is given to be an odd positive integer. Therefore, x ≠ – 13, x = 11.
Thus, the two consecutive odd integers are 11 and 13
Example 4 : Find the roots of the quadratic equation 3x2
– 5x + 2 = 0, if they
exist, using the quadratic formula.
Solution :
3x2 – 5x + 2 = 0. Here, a = 3, b = – 5, c = 2.
So, b2 – 4ac = 25 – 24 = 1 > 0.
Therefore, x =
5±√1
6
=
5±1
6
, 𝑖. 𝑒. 𝑥 = 1 𝑜𝑟 𝑥 =
2
3
So, the roots are
2
3
𝑎𝑛𝑑 1.
Summary
1. A quadratic equation in the variable x is of the form ax2
+
bx + c = 0, where a, b, c are real numbers and a ≠ zero.
2. A real number αis said to be a root of the quadratic
equation ax2
+ bx + c = 0, if aα
+ bα+ c = 0. The zeroes of
the quadratic polynomial ax2
+ bx + c and the roots of the
quadratic equation ax2
+ bx + c = 0 are the same.
3. If we can factorise ax2
+ bx + c into a product of two
linear factors, then the roots of the quadratic equation ax2
+
bx + c = 0 can be found by equating each factor to zero.
4. A quadratic equation can also be solved by the method of
completing the square.
5. Quadratic formula: The roots of a quadratic equation ax2
+
bx + c = 0 are given by
6. A quadratic equation ax2
+ bx + c = 0 has
(i) two distinct real roots, if b2
– 4ac > 0,
(ii) two equal roots (i.e., coincident roots), if b2
– 4ac = 0, and
(iii) no real roots, if b2
– 4ac < 0.



Thank You
Bibliography
 www.google.com
 www.wikipedia.com
 Google Images
 www.ncert.nic
By Rishabh
Kartik
Hriday
Himanshu
Abhijith

More Related Content

What's hot

Shubhanshu math project work , polynomial
Shubhanshu math project work ,  polynomialShubhanshu math project work ,  polynomial
Shubhanshu math project work , polynomial
Shubhanshu Bhargava
 
PPT on Trigonometric Functions. Class 11
PPT on Trigonometric Functions. Class 11PPT on Trigonometric Functions. Class 11
PPT on Trigonometric Functions. Class 11
Rushikesh Reddy
 
X ch 1 real numbers
X  ch 1  real numbersX  ch 1  real numbers
X ch 1 real numbers
AmruthaKB2
 
Probability 10th class
Probability 10th classProbability 10th class
Probability 10th class
manjunathindira
 
Trigonometry Presentation For Class 10 Students
Trigonometry Presentation For Class 10 StudentsTrigonometry Presentation For Class 10 Students
Trigonometry Presentation For Class 10 Students
Abhishek Yadav
 
LINEAR EQUATION IN TWO VARIABLES PPT
LINEAR EQUATION  IN  TWO VARIABLES PPTLINEAR EQUATION  IN  TWO VARIABLES PPT
LINEAR EQUATION IN TWO VARIABLES PPT
Abhishek Dev
 
Polynomials CLASS 10
Polynomials CLASS 10Polynomials CLASS 10
Polynomials CLASS 10
Nihas Nichu
 
Arithmetic progression
Arithmetic progressionArithmetic progression
Arithmetic progression
Mayank Devnani
 
Trigonometry project
Trigonometry projectTrigonometry project
Trigonometry project
Kajal Soni
 
Maths ppt on some applications of trignometry
Maths ppt on some applications of trignometryMaths ppt on some applications of trignometry
Maths ppt on some applications of trignometry
Harsh Mahajan
 
polynomials of class 10th
polynomials of class 10thpolynomials of class 10th
polynomials of class 10th
Ashish Pradhan
 
surface area and volume ppt for class 10
surface area and volume ppt for class 10surface area and volume ppt for class 10
surface area and volume ppt for class 10
7232
 
CLASS X MATHS Coordinate geometry
CLASS X MATHS Coordinate geometryCLASS X MATHS Coordinate geometry
CLASS X MATHS Coordinate geometry
Rc Os
 
Social Issues class- X
Social Issues class- XSocial Issues class- X
Social Issues class- X
Mayanksingh760
 
Circles IX
Circles IXCircles IX
Circles IX
Vaibhav Goel
 
Presentation of Polynomial
Presentation of PolynomialPresentation of Polynomial
Presentation of Polynomial
RajatUpadhyay20
 
CLASS X MATHS LINEAR EQUATIONS
CLASS X MATHS LINEAR EQUATIONSCLASS X MATHS LINEAR EQUATIONS
CLASS X MATHS LINEAR EQUATIONS
Rc Os
 
Quadratic equation
Quadratic equation   Quadratic equation
Quadratic equation
HOME!
 
PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C
PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION CPPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C
PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C
RAMBABU SIRIPURAPU
 
Pair of linear equation in two variable
Pair of linear equation in two variable Pair of linear equation in two variable
Pair of linear equation in two variable
Vineet Mathur
 

What's hot (20)

Shubhanshu math project work , polynomial
Shubhanshu math project work ,  polynomialShubhanshu math project work ,  polynomial
Shubhanshu math project work , polynomial
 
PPT on Trigonometric Functions. Class 11
PPT on Trigonometric Functions. Class 11PPT on Trigonometric Functions. Class 11
PPT on Trigonometric Functions. Class 11
 
X ch 1 real numbers
X  ch 1  real numbersX  ch 1  real numbers
X ch 1 real numbers
 
Probability 10th class
Probability 10th classProbability 10th class
Probability 10th class
 
Trigonometry Presentation For Class 10 Students
Trigonometry Presentation For Class 10 StudentsTrigonometry Presentation For Class 10 Students
Trigonometry Presentation For Class 10 Students
 
LINEAR EQUATION IN TWO VARIABLES PPT
LINEAR EQUATION  IN  TWO VARIABLES PPTLINEAR EQUATION  IN  TWO VARIABLES PPT
LINEAR EQUATION IN TWO VARIABLES PPT
 
Polynomials CLASS 10
Polynomials CLASS 10Polynomials CLASS 10
Polynomials CLASS 10
 
Arithmetic progression
Arithmetic progressionArithmetic progression
Arithmetic progression
 
Trigonometry project
Trigonometry projectTrigonometry project
Trigonometry project
 
Maths ppt on some applications of trignometry
Maths ppt on some applications of trignometryMaths ppt on some applications of trignometry
Maths ppt on some applications of trignometry
 
polynomials of class 10th
polynomials of class 10thpolynomials of class 10th
polynomials of class 10th
 
surface area and volume ppt for class 10
surface area and volume ppt for class 10surface area and volume ppt for class 10
surface area and volume ppt for class 10
 
CLASS X MATHS Coordinate geometry
CLASS X MATHS Coordinate geometryCLASS X MATHS Coordinate geometry
CLASS X MATHS Coordinate geometry
 
Social Issues class- X
Social Issues class- XSocial Issues class- X
Social Issues class- X
 
Circles IX
Circles IXCircles IX
Circles IX
 
Presentation of Polynomial
Presentation of PolynomialPresentation of Polynomial
Presentation of Polynomial
 
CLASS X MATHS LINEAR EQUATIONS
CLASS X MATHS LINEAR EQUATIONSCLASS X MATHS LINEAR EQUATIONS
CLASS X MATHS LINEAR EQUATIONS
 
Quadratic equation
Quadratic equation   Quadratic equation
Quadratic equation
 
PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C
PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION CPPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C
PPTs FOR 9TH CLASS COORDINATE GEOMETRY INTRODUCTION C
 
Pair of linear equation in two variable
Pair of linear equation in two variable Pair of linear equation in two variable
Pair of linear equation in two variable
 

Viewers also liked

Quadratic equations
Quadratic equationsQuadratic equations
Quadratic equations
A M
 
Quadratic Equation
Quadratic EquationQuadratic Equation
Quadratic Equation
itutor
 
Quadratic Function Presentation
Quadratic Function PresentationQuadratic Function Presentation
Quadratic Function Presentation
RyanWatt
 
Quadratic functions my maths presentation
Quadratic functions my maths presentationQuadratic functions my maths presentation
Quadratic functions my maths presentation
University of Johannesburg
 
Quadratic Formula Presentation
Quadratic Formula PresentationQuadratic Formula Presentation
Quadratic Formula Presentation
anjuli1580
 
Quadratic Equations (Quadratic Formula) Using PowerPoint
Quadratic Equations (Quadratic Formula) Using PowerPointQuadratic Equations (Quadratic Formula) Using PowerPoint
Quadratic Equations (Quadratic Formula) Using PowerPoint
richrollo
 
93639430 additional-mathematics-project-work-2013
93639430 additional-mathematics-project-work-201393639430 additional-mathematics-project-work-2013
93639430 additional-mathematics-project-work-2013
Rashmi R Rashmi R
 
Quadratic equations lesson 3
Quadratic equations lesson 3Quadratic equations lesson 3
Quadratic equations lesson 3
KathManarang
 
Number System
Number SystemNumber System
Number System
samarthagrawal
 
Sridharacharya[1]
Sridharacharya[1]Sridharacharya[1]
Sridharacharya[1]
Poonam Singh
 
Quadratic And Roots
Quadratic And RootsQuadratic And Roots
Quadratic And Roots
Peking
 
Mathematics Project Slides
Mathematics Project SlidesMathematics Project Slides
Mathematics Project Slides
mkulawat
 
Module in solving quadratic equation
Module in solving quadratic equationModule in solving quadratic equation
Module in solving quadratic equation
aleli ariola
 
Quadratic functions power point
Quadratic functions power pointQuadratic functions power point
Quadratic functions power point
tongj
 
Strategic intervention materials on mathematics 2.0
Strategic intervention materials on mathematics 2.0Strategic intervention materials on mathematics 2.0
Strategic intervention materials on mathematics 2.0
Brian Mary
 
Module 10 Topic 4 solving quadratic equations part 1
Module 10 Topic 4   solving quadratic equations part 1Module 10 Topic 4   solving quadratic equations part 1
Module 10 Topic 4 solving quadratic equations part 1
Lori Rapp
 
Arithmetic progression
Arithmetic progressionArithmetic progression
Arithmetic progression
Rajaram Narasimhan
 
Quadratic Equations Graphing
Quadratic Equations   GraphingQuadratic Equations   Graphing
Quadratic Equations Graphing
kliegey524
 
ARITHMETIC PROGRESSIONS
ARITHMETIC PROGRESSIONS ARITHMETIC PROGRESSIONS
ARITHMETIC PROGRESSIONS
Vamsi Krishna
 
ppt on circles
ppt on circlesppt on circles
ppt on circles
Ravi Kant
 

Viewers also liked (20)

Quadratic equations
Quadratic equationsQuadratic equations
Quadratic equations
 
Quadratic Equation
Quadratic EquationQuadratic Equation
Quadratic Equation
 
Quadratic Function Presentation
Quadratic Function PresentationQuadratic Function Presentation
Quadratic Function Presentation
 
Quadratic functions my maths presentation
Quadratic functions my maths presentationQuadratic functions my maths presentation
Quadratic functions my maths presentation
 
Quadratic Formula Presentation
Quadratic Formula PresentationQuadratic Formula Presentation
Quadratic Formula Presentation
 
Quadratic Equations (Quadratic Formula) Using PowerPoint
Quadratic Equations (Quadratic Formula) Using PowerPointQuadratic Equations (Quadratic Formula) Using PowerPoint
Quadratic Equations (Quadratic Formula) Using PowerPoint
 
93639430 additional-mathematics-project-work-2013
93639430 additional-mathematics-project-work-201393639430 additional-mathematics-project-work-2013
93639430 additional-mathematics-project-work-2013
 
Quadratic equations lesson 3
Quadratic equations lesson 3Quadratic equations lesson 3
Quadratic equations lesson 3
 
Number System
Number SystemNumber System
Number System
 
Sridharacharya[1]
Sridharacharya[1]Sridharacharya[1]
Sridharacharya[1]
 
Quadratic And Roots
Quadratic And RootsQuadratic And Roots
Quadratic And Roots
 
Mathematics Project Slides
Mathematics Project SlidesMathematics Project Slides
Mathematics Project Slides
 
Module in solving quadratic equation
Module in solving quadratic equationModule in solving quadratic equation
Module in solving quadratic equation
 
Quadratic functions power point
Quadratic functions power pointQuadratic functions power point
Quadratic functions power point
 
Strategic intervention materials on mathematics 2.0
Strategic intervention materials on mathematics 2.0Strategic intervention materials on mathematics 2.0
Strategic intervention materials on mathematics 2.0
 
Module 10 Topic 4 solving quadratic equations part 1
Module 10 Topic 4   solving quadratic equations part 1Module 10 Topic 4   solving quadratic equations part 1
Module 10 Topic 4 solving quadratic equations part 1
 
Arithmetic progression
Arithmetic progressionArithmetic progression
Arithmetic progression
 
Quadratic Equations Graphing
Quadratic Equations   GraphingQuadratic Equations   Graphing
Quadratic Equations Graphing
 
ARITHMETIC PROGRESSIONS
ARITHMETIC PROGRESSIONS ARITHMETIC PROGRESSIONS
ARITHMETIC PROGRESSIONS
 
ppt on circles
ppt on circlesppt on circles
ppt on circles
 

Similar to Maths Project Quadratic Equations

MATHS PRESENTATION OF CH 4.pptx
MATHS PRESENTATION OF CH 4.pptxMATHS PRESENTATION OF CH 4.pptx
MATHS PRESENTATION OF CH 4.pptx
ShubhamVishwakarma959872
 
Mayank and Srishti presentation on gyandeep public school
Mayank  and Srishti presentation on gyandeep public schoolMayank  and Srishti presentation on gyandeep public school
Mayank and Srishti presentation on gyandeep public school
MayankYadav777500
 
Sreeku
SreekuSreeku
Sreeku
achukichu
 
Quadraticequation
QuadraticequationQuadraticequation
Quadraticequation
Allanna Unias
 
Quadratic equation
Quadratic equation Quadratic equation
Quadratic equation
Shivangi Tidke
 
Quadratic Equation
Quadratic EquationQuadratic Equation
Quadratic Equation
NayanKohare
 
QUADRATIC EQUATIONS WITH MATHS PROPER VERIFY
QUADRATIC EQUATIONS WITH MATHS PROPER VERIFYQUADRATIC EQUATIONS WITH MATHS PROPER VERIFY
QUADRATIC EQUATIONS WITH MATHS PROPER VERIFY
ssuser2e348b
 
Quadratic equations
Quadratic equationsQuadratic equations
Quadratic equations
Mervin Dayrit
 
Quadratic Equations
Quadratic EquationsQuadratic Equations
Quadratic Equations
Wenslette Rosique
 
Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHS
Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHSTricks to remember the quadratic equation.ACTION RESEARCH ON MATHS
Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHS
angelbindusingh
 
Rbsc class-9-maths-chapter-4
Rbsc class-9-maths-chapter-4 Rbsc class-9-maths-chapter-4
Rbsc class-9-maths-chapter-4
Arvind Saini
 
MIT Math Syllabus 10-3 Lesson 7: Quadratic equations
MIT Math Syllabus 10-3 Lesson 7: Quadratic equationsMIT Math Syllabus 10-3 Lesson 7: Quadratic equations
MIT Math Syllabus 10-3 Lesson 7: Quadratic equations
Lawrence De Vera
 
Bonus math project
Bonus math projectBonus math project
Bonus math project
Kenton Hemsing
 
Chapter 2
Chapter  2Chapter  2
Chapter 2
jennytuazon01630
 
DISCRIMINANT.ppt
DISCRIMINANT.pptDISCRIMINANT.ppt
DISCRIMINANT.ppt
NelsonNelson56
 
Solving quadratic equations[1]
Solving quadratic equations[1]Solving quadratic equations[1]
Solving quadratic equations[1]
RobinFilter
 
Solving quadratic equations
Solving quadratic equationsSolving quadratic equations
Solving quadratic equations
srobbins4
 
Linear equations in two variables
Linear equations in two variablesLinear equations in two variables
Linear equations in two variables
Vinisha Pathak
 
Pair of linear equation in two variables
Pair of linear equation in two variables Pair of linear equation in two variables
Pair of linear equation in two variables
shivangi gupta
 
C2 st lecture 2 handout
C2 st lecture 2 handoutC2 st lecture 2 handout
C2 st lecture 2 handout
fatima d
 

Similar to Maths Project Quadratic Equations (20)

MATHS PRESENTATION OF CH 4.pptx
MATHS PRESENTATION OF CH 4.pptxMATHS PRESENTATION OF CH 4.pptx
MATHS PRESENTATION OF CH 4.pptx
 
Mayank and Srishti presentation on gyandeep public school
Mayank  and Srishti presentation on gyandeep public schoolMayank  and Srishti presentation on gyandeep public school
Mayank and Srishti presentation on gyandeep public school
 
Sreeku
SreekuSreeku
Sreeku
 
Quadraticequation
QuadraticequationQuadraticequation
Quadraticequation
 
Quadratic equation
Quadratic equation Quadratic equation
Quadratic equation
 
Quadratic Equation
Quadratic EquationQuadratic Equation
Quadratic Equation
 
QUADRATIC EQUATIONS WITH MATHS PROPER VERIFY
QUADRATIC EQUATIONS WITH MATHS PROPER VERIFYQUADRATIC EQUATIONS WITH MATHS PROPER VERIFY
QUADRATIC EQUATIONS WITH MATHS PROPER VERIFY
 
Quadratic equations
Quadratic equationsQuadratic equations
Quadratic equations
 
Quadratic Equations
Quadratic EquationsQuadratic Equations
Quadratic Equations
 
Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHS
Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHSTricks to remember the quadratic equation.ACTION RESEARCH ON MATHS
Tricks to remember the quadratic equation.ACTION RESEARCH ON MATHS
 
Rbsc class-9-maths-chapter-4
Rbsc class-9-maths-chapter-4 Rbsc class-9-maths-chapter-4
Rbsc class-9-maths-chapter-4
 
MIT Math Syllabus 10-3 Lesson 7: Quadratic equations
MIT Math Syllabus 10-3 Lesson 7: Quadratic equationsMIT Math Syllabus 10-3 Lesson 7: Quadratic equations
MIT Math Syllabus 10-3 Lesson 7: Quadratic equations
 
Bonus math project
Bonus math projectBonus math project
Bonus math project
 
Chapter 2
Chapter  2Chapter  2
Chapter 2
 
DISCRIMINANT.ppt
DISCRIMINANT.pptDISCRIMINANT.ppt
DISCRIMINANT.ppt
 
Solving quadratic equations[1]
Solving quadratic equations[1]Solving quadratic equations[1]
Solving quadratic equations[1]
 
Solving quadratic equations
Solving quadratic equationsSolving quadratic equations
Solving quadratic equations
 
Linear equations in two variables
Linear equations in two variablesLinear equations in two variables
Linear equations in two variables
 
Pair of linear equation in two variables
Pair of linear equation in two variables Pair of linear equation in two variables
Pair of linear equation in two variables
 
C2 st lecture 2 handout
C2 st lecture 2 handoutC2 st lecture 2 handout
C2 st lecture 2 handout
 

Recently uploaded

PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdfPRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
nservice241
 
Lecture Notes Unit4 Chapter13 users , roles and privileges
Lecture Notes Unit4 Chapter13 users , roles and privilegesLecture Notes Unit4 Chapter13 users , roles and privileges
Lecture Notes Unit4 Chapter13 users , roles and privileges
Murugan146644
 
3. Maturity_indices_of_fruits_and_vegetable.pptx
3. Maturity_indices_of_fruits_and_vegetable.pptx3. Maturity_indices_of_fruits_and_vegetable.pptx
3. Maturity_indices_of_fruits_and_vegetable.pptx
UmeshTimilsina1
 
Introduction to Banking System in India.ppt
Introduction to Banking System in India.pptIntroduction to Banking System in India.ppt
Introduction to Banking System in India.ppt
Dr. S. Bulomine Regi
 
SQL Server Interview Questions PDF By ScholarHat
SQL Server Interview Questions PDF By ScholarHatSQL Server Interview Questions PDF By ScholarHat
SQL Server Interview Questions PDF By ScholarHat
Scholarhat
 
SD_Integrating 21st Century Skills in Classroom-based Assessment.pptx
SD_Integrating 21st Century Skills in Classroom-based Assessment.pptxSD_Integrating 21st Century Skills in Classroom-based Assessment.pptx
SD_Integrating 21st Century Skills in Classroom-based Assessment.pptx
elwoodprias1
 
MATATAG CURRICULUM sample lesson exemplar.docx
MATATAG CURRICULUM sample lesson exemplar.docxMATATAG CURRICULUM sample lesson exemplar.docx
MATATAG CURRICULUM sample lesson exemplar.docx
yardenmendoza
 
E-learning Odoo 17 New features - Odoo 17 Slides
E-learning Odoo 17  New features - Odoo 17 SlidesE-learning Odoo 17  New features - Odoo 17 Slides
E-learning Odoo 17 New features - Odoo 17 Slides
Celine George
 
JavaScript Interview Questions PDF By ScholarHat
JavaScript Interview  Questions PDF By ScholarHatJavaScript Interview  Questions PDF By ScholarHat
JavaScript Interview Questions PDF By ScholarHat
Scholarhat
 
1. Importance_of_reducing_postharvest_loss.pptx
1. Importance_of_reducing_postharvest_loss.pptx1. Importance_of_reducing_postharvest_loss.pptx
1. Importance_of_reducing_postharvest_loss.pptx
UmeshTimilsina1
 
Java MCQ Questions and Answers PDF By ScholarHat
Java MCQ Questions and Answers PDF By ScholarHatJava MCQ Questions and Answers PDF By ScholarHat
Java MCQ Questions and Answers PDF By ScholarHat
Scholarhat
 
Our Guide to the July 2024 USPS® Rate Change
Our Guide to the July 2024 USPS® Rate ChangeOur Guide to the July 2024 USPS® Rate Change
Our Guide to the July 2024 USPS® Rate Change
Postal Advocate Inc.
 
FIRST AID PRESENTATION ON INDUSTRIAL SAFETY by dr lal.ppt
FIRST AID PRESENTATION ON INDUSTRIAL SAFETY by dr lal.pptFIRST AID PRESENTATION ON INDUSTRIAL SAFETY by dr lal.ppt
FIRST AID PRESENTATION ON INDUSTRIAL SAFETY by dr lal.ppt
ashutoshklal29
 
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
Nguyen Thanh Tu Collection
 
11. Post harvest quality, Quality criteria and Judgement.pptx
11. Post harvest quality, Quality criteria and Judgement.pptx11. Post harvest quality, Quality criteria and Judgement.pptx
11. Post harvest quality, Quality criteria and Judgement.pptx
UmeshTimilsina1
 
Odoo 17 Events - Attendees List Scanning
Odoo 17 Events - Attendees List ScanningOdoo 17 Events - Attendees List Scanning
Odoo 17 Events - Attendees List Scanning
Celine George
 
How to Manage Shipping Connectors & Shipping Methods in Odoo 17
How to Manage Shipping Connectors & Shipping Methods in Odoo 17How to Manage Shipping Connectors & Shipping Methods in Odoo 17
How to Manage Shipping Connectors & Shipping Methods in Odoo 17
Celine George
 
QCE – Unpacking the syllabus Implications for Senior School practices and ass...
QCE – Unpacking the syllabus Implications for Senior School practices and ass...QCE – Unpacking the syllabus Implications for Senior School practices and ass...
QCE – Unpacking the syllabus Implications for Senior School practices and ass...
mansk2
 
FINAL MATATAG Science CG 2023 Grades 3-10.pdf
FINAL MATATAG Science CG 2023 Grades 3-10.pdfFINAL MATATAG Science CG 2023 Grades 3-10.pdf
FINAL MATATAG Science CG 2023 Grades 3-10.pdf
maritescanete2
 
Imagination in Computer Science Research
Imagination in Computer Science ResearchImagination in Computer Science Research
Imagination in Computer Science Research
Abhik Roychoudhury
 

Recently uploaded (20)

PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdfPRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
PRESS RELEASE - UNIVERSITY OF GHANA, JULY 16, 2024.pdf
 
Lecture Notes Unit4 Chapter13 users , roles and privileges
Lecture Notes Unit4 Chapter13 users , roles and privilegesLecture Notes Unit4 Chapter13 users , roles and privileges
Lecture Notes Unit4 Chapter13 users , roles and privileges
 
3. Maturity_indices_of_fruits_and_vegetable.pptx
3. Maturity_indices_of_fruits_and_vegetable.pptx3. Maturity_indices_of_fruits_and_vegetable.pptx
3. Maturity_indices_of_fruits_and_vegetable.pptx
 
Introduction to Banking System in India.ppt
Introduction to Banking System in India.pptIntroduction to Banking System in India.ppt
Introduction to Banking System in India.ppt
 
SQL Server Interview Questions PDF By ScholarHat
SQL Server Interview Questions PDF By ScholarHatSQL Server Interview Questions PDF By ScholarHat
SQL Server Interview Questions PDF By ScholarHat
 
SD_Integrating 21st Century Skills in Classroom-based Assessment.pptx
SD_Integrating 21st Century Skills in Classroom-based Assessment.pptxSD_Integrating 21st Century Skills in Classroom-based Assessment.pptx
SD_Integrating 21st Century Skills in Classroom-based Assessment.pptx
 
MATATAG CURRICULUM sample lesson exemplar.docx
MATATAG CURRICULUM sample lesson exemplar.docxMATATAG CURRICULUM sample lesson exemplar.docx
MATATAG CURRICULUM sample lesson exemplar.docx
 
E-learning Odoo 17 New features - Odoo 17 Slides
E-learning Odoo 17  New features - Odoo 17 SlidesE-learning Odoo 17  New features - Odoo 17 Slides
E-learning Odoo 17 New features - Odoo 17 Slides
 
JavaScript Interview Questions PDF By ScholarHat
JavaScript Interview  Questions PDF By ScholarHatJavaScript Interview  Questions PDF By ScholarHat
JavaScript Interview Questions PDF By ScholarHat
 
1. Importance_of_reducing_postharvest_loss.pptx
1. Importance_of_reducing_postharvest_loss.pptx1. Importance_of_reducing_postharvest_loss.pptx
1. Importance_of_reducing_postharvest_loss.pptx
 
Java MCQ Questions and Answers PDF By ScholarHat
Java MCQ Questions and Answers PDF By ScholarHatJava MCQ Questions and Answers PDF By ScholarHat
Java MCQ Questions and Answers PDF By ScholarHat
 
Our Guide to the July 2024 USPS® Rate Change
Our Guide to the July 2024 USPS® Rate ChangeOur Guide to the July 2024 USPS® Rate Change
Our Guide to the July 2024 USPS® Rate Change
 
FIRST AID PRESENTATION ON INDUSTRIAL SAFETY by dr lal.ppt
FIRST AID PRESENTATION ON INDUSTRIAL SAFETY by dr lal.pptFIRST AID PRESENTATION ON INDUSTRIAL SAFETY by dr lal.ppt
FIRST AID PRESENTATION ON INDUSTRIAL SAFETY by dr lal.ppt
 
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
BỘ ĐỀ THI HỌC SINH GIỎI CÁC TỈNH MÔN TIẾNG ANH LỚP 9 NĂM HỌC 2023-2024 (CÓ FI...
 
11. Post harvest quality, Quality criteria and Judgement.pptx
11. Post harvest quality, Quality criteria and Judgement.pptx11. Post harvest quality, Quality criteria and Judgement.pptx
11. Post harvest quality, Quality criteria and Judgement.pptx
 
Odoo 17 Events - Attendees List Scanning
Odoo 17 Events - Attendees List ScanningOdoo 17 Events - Attendees List Scanning
Odoo 17 Events - Attendees List Scanning
 
How to Manage Shipping Connectors & Shipping Methods in Odoo 17
How to Manage Shipping Connectors & Shipping Methods in Odoo 17How to Manage Shipping Connectors & Shipping Methods in Odoo 17
How to Manage Shipping Connectors & Shipping Methods in Odoo 17
 
QCE – Unpacking the syllabus Implications for Senior School practices and ass...
QCE – Unpacking the syllabus Implications for Senior School practices and ass...QCE – Unpacking the syllabus Implications for Senior School practices and ass...
QCE – Unpacking the syllabus Implications for Senior School practices and ass...
 
FINAL MATATAG Science CG 2023 Grades 3-10.pdf
FINAL MATATAG Science CG 2023 Grades 3-10.pdfFINAL MATATAG Science CG 2023 Grades 3-10.pdf
FINAL MATATAG Science CG 2023 Grades 3-10.pdf
 
Imagination in Computer Science Research
Imagination in Computer Science ResearchImagination in Computer Science Research
Imagination in Computer Science Research
 

Maths Project Quadratic Equations

  • 1. ©Copyrighted by Rishabh & Group Quadratic Equations Group 1 FA-IV Project
  • 2. Serial No. Topic 1 Certificate 2 Acknowledgement 3 Introduction 4 Sri Dharacharya Ji 5 Identification of Quadratic Equations 6 Factorisation Method 7 Completing the square method 8 Discriminant & Nature of roots 9 Problems based on Quadratic Equations 10 Summary 11 Bibliography
  • 3. Certificate Thisistocertifythattheprojectworksatisfactorily carried out inthegroupforFA-II.And hence, recorded in this file is the bonafied work of group no. ___ consisting of members: .......................................... of class X during the Academic year 2012-13. Teacher in charge Date Principal
  • 4. Acknowledgement SpecialthankstoMrs.................madam for her guidance and hersupport.Shehelpedthroughouttheproject.We thank our Principalforhersupport. We thank our group members for theircontribution and cooperation for making this project. ThisProjectwasmadefrom the support and contribution of our group members. So we will thank each one of us.
  • 5. Sridhara (c. 870, India – c. 930 India) was an Indian mathematician. He was born in Hooghly district in the 10th Century AD. His father's name was Baladevacharya and mother's name was Acchoka. Works:-  He was known for two treatises: Trisatika (sometimes called the Patiganitasara) and the Patiganita. His major work Patiganitasara was named Trisatika because it was witten in three hundred slokas.  The book discusses counting of numbers,measures,natural number,multiplication,division,zero,squares,cubes,fraction,rule of three,interest-calculation,joint business or partnership and mensuration.  Of all the Hindu Acharyas the exposition of Sridharacharya on zero is the most explicit. He has written, "If 0(zero) is added to any number,the sum is the same number; If 0(zero) is subtracted from any number,the number remains unchanged; If 0(zero) is multiplied by any number,the product is 0(zero)". He has said nothing about division of any number by 0(zero).  In the case of dividing a fraction he has found out the method of mutiplying the fraction by the reciprocal of the divisor.  He wrote on practical applicationsof algebra separated algebra from arithmetic
  • 6. Introduction and standard form of Quadratic Equations In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the form ax2 + bx + c = 0 where x represents a variable or an unknown, and a, b, and c are constants with a ≠ 0. (If a = 0, the equation is a linear equation.) The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square". Quadratic equations can be solved by factorizing, completing the square, graphing, Newton's method, and using the quadratic formula
  • 7. Identification of Quadratic Equations Examples : Check whether the following are quadratic equations: (i) (x – 2)2 + 1 = 2x – 3 (ii) x(x + 1) + 8 = (x + 2) (x – 2) Solution : (i) LHS = (x – 2)2 + 1 = x2 – 4x + 4 + 1 = x2 – 4x + 5 Therefore, (x – 2)2 + 1 = 2x – 3 can be rewritten as = x2 – 4x + 5 = 2x – 3 i.e., x2 – 6x + 8 = 0 It is of the form ax2 + bx + c = 0. Therefore, the given equation is a quadratic equation. (ii) Since x(x + 1) + 8 = x2 + x + 8 and (x + 2)(x – 2) = x2 – 4 Therefore, x2 + x + 8 = x2 – 4 i.e., x + 12 = 0 It is not of the form ax2 + bx + c = 0. Therefore, the given equation is not a quadratic equation.
  • 8. Solution of a Quadratic Equation by Factorisation Consider the quadraticequation 2x2 – 3x + 1 = 0. If we replace x by 1 on the LHS of this equation, we get (2 × 12) – (3 × 1) + 1 = 0 = RHS of the equation. We say that 1 is a root of the quadratic equation 2x2 – 3x + 1 = 0. This also means that 1 is a zero of the quadraticpolynomial 2x2 – 3x + 1. In general, a real number α is called a root of the quadraticequation ax2 + bx + c = 0, a ≠ 0 if a α2 + bα + c = 0. We also say that x = α is a solution of the quadraticequation,or that α satisfies the quadraticequation.Note that the zeroes of the quadraticpolynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same. Example: (i)Find the roots of the quadratic equation 6x2 – x – 2 = 0. Sol. We have 6x2 – x – 2 = 6x2 + 3x – 4x – 2 =3x (2x + 1) – 2 (2x + 1) =(3x – 2)(2x + 1) The roots of 6x2 – x – 2 = 0 are the values of x for which (3x – 2)(2x + 1) = 0 Therefore, 3x – 2 = 0 or 2x + 1 = 0, i.e., x = 𝟐 𝟑 or x =− 𝟏 𝟐 Therefore, the roots of 6x2 – x – 2 = 0 are 𝟐 𝟑 and − 𝟏 𝟐
  • 9. Completing the square and Derivation :- The quadratic formula can be derived by the method of completing the square, so as to make use of the algebraic identity: Dividing the quadratic equation by a (which is allowed because a is non-zero), gives: The quadratic equation is now in a form to which the method of completing the square can be applied. To "complete the square" is to add a constant to both sides of the equation such that the left hand side becomes a complete square: which produces The right side can be written as a single fraction, with common denominator 4a2. This gives Taking the square root of both sides yields Isolating x, gives
  • 10. Example:- (i)Solve the equation 2x2 – 5x + 3 = 0 by the method of completing the square. Sol. The equation 2x2 – 5x + 3 = 0 is the same as 𝑥2 − 5 2 𝑥 + 3 2 = 0 Now, 𝑥2 − 5 2 𝑥 + 3 2 = (𝑥 − 5 4 ) 2 − ( 5 4 ) 2 + 3 4 = (𝑥 − 5 4 ) 2 − 1 16 Therefore, 2x2 – 5x + 3 = 0 can be written as (𝑥 − 5 4 ) 2 − 1 16 = 0. So, the roots of the equation 2x2 – 5x + 3 = 0 are exactly the same as those of (𝑥 − 5 4 ) 2 − 1 16 = 0. Now, (𝑥 − 5 4 ) 2 − 1 16 = 0 is the same as (𝑥 − 5 4 ) 2 = 1 16 Therefore, 𝑥 − 5 4 = ± 1 4 i.e., 𝑥 = 5 4 ± 1 4 i.e., 𝑥 = 5 4 + 1 4 or 𝑥 = 5 4 − 1 4 i.e., 𝑥 = 3 2 or 𝑥 = 1 Therefore, the solutions of the equations are 𝑥 = 3 2 or 𝑥 = 1
  • 11. Discriminant and Nature of roots:- The roots of the equation ax2 + bx + c = 0 are given by 𝑥 = −𝑏±√ 𝑏2−4𝑎𝑐 2𝑎 If b2 – 4ac > 0, we get two distinct real roots − 𝑏 2𝑎 + √𝑏2−4𝑎𝑐 2𝑎 and − 𝑏 2𝑎 − √𝑏2−4𝑎𝑐 2𝑎 If b2 – 4ac = 0, then 𝑥 = − 𝑏 2𝑎 ± 0, 𝑖. 𝑒. , 𝑥 = − 𝑏 2𝑎 𝑜𝑟 − 𝑏 2𝑎 So, the roots of the equation ax2 + bx + c = 0 are both −𝑏 2𝑎 . Therefore, we say that the quadraticequation ax2 + bx + c = 0 has two equal real roots in this case. If b2 – 4ac < 0, then there is no real number whose square is b2 – 4ac. Therefore, there are no real roots for the given quadratic equation in this case. Since b2 – 4ac determines whether the quadratic equation ax2 + bx + c = 0 has real roots or not, b2 – 4ac is called the discriminant of this quadratic equation. So, a quadratic equation ax2 + bx + c = 0 has (i) two distinct real roots, if b2 – 4ac > 0, (ii) two equal real roots, if b2 – 4ac = 0, (iii) no real roots, if b2 – 4ac < 0.
  • 12. Example1: (i)Find the discriminant of the equation 3x2 – 2x + 𝟏 𝟑 = 0 and hence find the nature of its roots. Find them, if they are real. Solution : Here 𝑎 = 3, 𝑏 = −2 𝑎𝑛𝑑 𝑐 = 1 3 = 3. Therefore, discriminant 𝑏2 − 4𝑎𝑐 = (−2) − 4 × 3 × 1 3 = 4 − 4 = 0 Hence, the given quadratic equation has two equal real roots. The roots are −𝑏 2𝑎 , −𝑏 2𝑎 , 𝑖. 𝑒. , 2 6 , 2 6 , 𝑖. 𝑒., 1 3 , 1 3 Example2: Find the discriminant of the quadratic equation 2x2 – 4x + 3 = 0, and hence find the nature of its roots. Solution : The given equation is of the form ax2 + bx + c = 0, where a = 2, b = – 4 and c = 3. Therefore, the discriminant b2 – 4ac = (– 4)2 – (4 × 2 × 3) = 16 – 24 = – 8 < 0 So, the given equation has no real roots.
  • 13. Problems based on Quadratic Equations:- Example 1 : Find the roots of the equation 2x2 – 5x + 3 = 0, by factorisation. Solution : Let us first split the middle term – 5x as –2x –3x [because (–2x) × (–3x) = 6x2 = (2x2) × 3]. So, 2x2 – 5x + 3 = 2x2 – 2x – 3x + 3 = 2x (x – 1) –3(x – 1) = (2x – 3)(x – 1) Now, 2x2 – 5x + 3 = 0 can be rewritten as (2x – 3)(x – 1) = 0. So, the values of x for which 2x2 – 5x + 3 = 0 are the same for which (2x – 3)(x – 1) = 0, i.e., either 2x – 3 = 0 or x – 1 = 0. Now, 2x – 3 = 0 gives 𝑥 = 3 2 𝑎𝑛𝑑 𝑥 − 1 = 0 𝑔𝑖𝑣𝑒𝑠 𝑥 = 1 So, 𝑥 = 3 2 𝑎𝑛𝑑 𝑥 = 1 are the solutions of the equation. Example 2 : Find the roots of the quadratic equation 𝟑𝒙 𝟐 − 𝟐√ 𝟔𝒙 + 𝟐 = 𝟎. Solution : 𝟑𝒙 𝟐 − 𝟐√ 𝟔𝒙 + 𝟐 = 𝟑𝒙 𝟐 − √ 𝟔𝒙 + √ 𝟔𝒙 + 𝟐 = √ 𝟑𝒙(√ 𝟑𝒙 − √ 𝟐) − √ 𝟐 (√ 𝟑𝒙 − √ 𝟐) = (√ 𝟑𝒙 − √ 𝟐)(√ 𝟑𝒙 − √ 𝟐) So, the roots of the equation are the values of x for which (√ 𝟑𝒙 − √ 𝟐)(√ 𝟑𝒙 − √ 𝟐) = 𝟎 Now, √ 𝟑𝒙 − √ 𝟐 = 𝟎 𝒇𝒐𝒓 𝒙 = √ 𝟐 𝟑 So, this root is repeated twice, one for each repeated factor √ 𝟑𝒙 − √ 𝟐. Therefore, the roots of 𝟑𝒙 𝟐 − 𝟐√ 𝟔𝒙 + 𝟐 = 𝟎 are √ 𝟐 𝟑 , √ 𝟐 𝟑
  • 14. Example 3 : Find two consecutive odd positive integers, sum of whose squares is 290. Solution : Let the smaller of the two consecutive odd positive integers be x. Then, the second integer will be x + 2. According to the question, =x2 + (x + 2)2 = 290 i.e., x2 + x2 + 4x + 4 = 290 i.e., 2x2 + 4x – 286 = 0 i.e., x2 + 2x – 143 = 0 which is a quadratic equation in x. Using the quadratic formula, we get 𝑥 = −2±√4+572 2 = −2±√576 2 = −2±24 2 𝑥 = 11 𝑜𝑟 𝑥 = −13 But x is given to be an odd positive integer. Therefore, x ≠ – 13, x = 11. Thus, the two consecutive odd integers are 11 and 13 Example 4 : Find the roots of the quadratic equation 3x2 – 5x + 2 = 0, if they exist, using the quadratic formula. Solution : 3x2 – 5x + 2 = 0. Here, a = 3, b = – 5, c = 2. So, b2 – 4ac = 25 – 24 = 1 > 0. Therefore, x = 5±√1 6 = 5±1 6 , 𝑖. 𝑒. 𝑥 = 1 𝑜𝑟 𝑥 = 2 3 So, the roots are 2 3 𝑎𝑛𝑑 1.
  • 15. Summary 1. A quadratic equation in the variable x is of the form ax2 + bx + c = 0, where a, b, c are real numbers and a ≠ zero. 2. A real number αis said to be a root of the quadratic equation ax2 + bx + c = 0, if aα + bα+ c = 0. The zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same. 3. If we can factorise ax2 + bx + c into a product of two linear factors, then the roots of the quadratic equation ax2 + bx + c = 0 can be found by equating each factor to zero. 4. A quadratic equation can also be solved by the method of completing the square. 5. Quadratic formula: The roots of a quadratic equation ax2 + bx + c = 0 are given by 6. A quadratic equation ax2 + bx + c = 0 has (i) two distinct real roots, if b2 – 4ac > 0, (ii) two equal roots (i.e., coincident roots), if b2 – 4ac = 0, and (iii) no real roots, if b2 – 4ac < 0.    Thank You
  • 16. Bibliography  www.google.com  www.wikipedia.com  Google Images  www.ncert.nic By Rishabh Kartik Hriday Himanshu Abhijith