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ANAND CLASSES 9463138669
Quadratic Equation- Session1
A
A
UE-II
JALAN DH AR
H ELPLINE
9 4 6 3 1 3 8 6 6 9
ANAND CLASSES 9463138669
Session Objective
1. Definition of important terms
(equation,expression,polynomial,
identity,quadratic etc.)
2. Finding roots by factorization
method
3. General solution of roots .
4. Nature of roots
ANAND CLASSES 9463138669
Quadratic Equation - Definitions
(Expression & Equation)
Equation : Statement of equality
between two expression
0
Expression:
Representation of relationship
between two (or more) variables
Y= ax2+bx+c,
ax2 + bx + c =
_H001
Root:-value(s) for which a equation satisfies
x2-4x+3 = 0  (x-3)(x-1) = 0
 x = 3 or 1 satisfies x2-4x+3 = 0
Roots of x2-4x+3 = 0
Example:
ANAND CLASSES 9463138669
Quadratic Equation
Definitions (Polynomial)
Polynomial :
P(x) = a0 + a1x + a2x2 + … + anxn,
A polynomial equation of degree n always have n roots
Real or non-real
highest power of the variable
where a0, a1, a2, … an are coefficients ,
and n is positive integer
Degree of the polynomial :
_H001
 
n
a 0

ANAND CLASSES 9463138669
Quadratic Equation
Definitions (Polynomial)
Equation  2 roots (say 1,2)
(x-1)(x-2)=0  x2 - 3x+2 = 0
2nd degree equation  2 roots
2nd degree
equation
• Roots are 1,2
(x- 1 )(x- 2 )=0 x2-(1+2)x+ 12= 0
 ax2 + bx + c=0
_H001
ANAND CLASSES 9463138669
Quadratic Equation
Definitions (Polynomial)
• Roots are 1,2,3
(x- 1 )(x- 2 ) (x- 3) =0
 ax3 +bx2+cx+d = 0
3rd degree equation
3rd degree equation  3 roots
• Roots are 1,2, 3,……. n
(x- 1 )(x- 2 ) (x- 3)….. (x- n) =0
 anxn+an-1xn-1+…….+ a0 =0 nth degree equation
nth degree equation  n roots
_H001
ANAND CLASSES 9463138669
Quadratic Equation
Definitions (Quadratic & Roots)
Quadratic: A polynomial of degree=2
A quadratic equation always has two roots
y= ax2+bx+c
ax2+bx+c = 0 is a quadratic equation. (a  0 )
_H001
ANAND CLASSES 9463138669
Roots
x=-a ?
Where is the 2nd
root of quadratic
equation?
Then what is its
difference from
x+a=0
(x+a)2=0
(x+a)(x+a) =0
 x= -a, -a
two roots
Also satisfies condition for
quadratic equation
What are the roots of the equation
(x+a)2=0
_H001
ANAND CLASSES 9463138669
Identity
Identity : Equation true for all
values of the variable
(x+1)2 = x2+2x+1
Equation holds true for all real x
_H001
ANAND CLASSES 9463138669
Polynomial identity
If a polynomial equation of
degree n satisfies for the values
more than n it is an identity
Example: (x-1)2 = x2-2x+1
Is a 2nd degree polynomial
Satisfies for x=0 (0-1)2=0-0+1
Satisfies for x=1 (1-1)2=1-2+1
Satisfies for x=-1 (-1-1)2=1+2+1
2nd degree polynomial cannot have more than 2 roots
(x-1)2 = x2-2x+1 is an identity
_H001
ANAND CLASSES 9463138669
Polynomial identity
If P(x)=Q(x) is an identity
Polynomial of x
Co-efficient of like terms is same on both the side
Illustrative example
If (x+1)2=(a2)x2+2ax+a is an identity then find a?
LO-H01
ANAND CLASSES 9463138669
Illustrative Problem
(x+1)2=(a2)x2+2ax+a
x2+2x+1 =(a2)x2+2ax+a
is an identity
Equating co-efficient
x2 : a2=1
x : 2a=2
constant: a=1
 a= 1 a=1
satisfies all
equation
If (x+1)2=(a2)x2+2ax+a is an identity
then find a?
Solution
_H001
ANAND CLASSES 9463138669
Illustrative problem
Find the roots of the following
equation
(x a)(x b) (x a)(x c)
(a c)(b c) (a b)(c b)
(x c)(x b)
1
(c a)(b a)
   
 
   
 

 
Solution: By observation
For x=-a L.H.S= 0+0+1=1 = R.H.S
For x=-b L.H.S= 0+1+0=1
For x=-c L.H.S= 1+0+0=1
= R.H.S
= R.H.S
_H001
ANAND CLASSES 9463138669
Illustrative problem
2nd degree polynomial is satisfying for more than 2 values
Its an identity
Find the roots of the following
equation
(x a)(x b) (x a)(x c) (x c)(x b)
1
(a c)(b c) (a b)(c b) (c a)(b a)
     
  
     
Satisfies for all values of x
i.e. on simplification the given equation becomes
0x2+0x+0=0

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ANAND CLASSES-Math Science CBSE ICSE Coaching Center In Jalandhar for class 5 6 7 8 9 10-Best Tuition Center

  • 1. ANAND CLASSES 9463138669 Quadratic Equation- Session1 A A UE-II JALAN DH AR H ELPLINE 9 4 6 3 1 3 8 6 6 9
  • 2. ANAND CLASSES 9463138669 Session Objective 1. Definition of important terms (equation,expression,polynomial, identity,quadratic etc.) 2. Finding roots by factorization method 3. General solution of roots . 4. Nature of roots
  • 3. ANAND CLASSES 9463138669 Quadratic Equation - Definitions (Expression & Equation) Equation : Statement of equality between two expression 0 Expression: Representation of relationship between two (or more) variables Y= ax2+bx+c, ax2 + bx + c = _H001 Root:-value(s) for which a equation satisfies x2-4x+3 = 0  (x-3)(x-1) = 0  x = 3 or 1 satisfies x2-4x+3 = 0 Roots of x2-4x+3 = 0 Example:
  • 4. ANAND CLASSES 9463138669 Quadratic Equation Definitions (Polynomial) Polynomial : P(x) = a0 + a1x + a2x2 + … + anxn, A polynomial equation of degree n always have n roots Real or non-real highest power of the variable where a0, a1, a2, … an are coefficients , and n is positive integer Degree of the polynomial : _H001   n a 0 
  • 5. ANAND CLASSES 9463138669 Quadratic Equation Definitions (Polynomial) Equation  2 roots (say 1,2) (x-1)(x-2)=0  x2 - 3x+2 = 0 2nd degree equation  2 roots 2nd degree equation • Roots are 1,2 (x- 1 )(x- 2 )=0 x2-(1+2)x+ 12= 0  ax2 + bx + c=0 _H001
  • 6. ANAND CLASSES 9463138669 Quadratic Equation Definitions (Polynomial) • Roots are 1,2,3 (x- 1 )(x- 2 ) (x- 3) =0  ax3 +bx2+cx+d = 0 3rd degree equation 3rd degree equation  3 roots • Roots are 1,2, 3,……. n (x- 1 )(x- 2 ) (x- 3)….. (x- n) =0  anxn+an-1xn-1+…….+ a0 =0 nth degree equation nth degree equation  n roots _H001
  • 7. ANAND CLASSES 9463138669 Quadratic Equation Definitions (Quadratic & Roots) Quadratic: A polynomial of degree=2 A quadratic equation always has two roots y= ax2+bx+c ax2+bx+c = 0 is a quadratic equation. (a  0 ) _H001
  • 8. ANAND CLASSES 9463138669 Roots x=-a ? Where is the 2nd root of quadratic equation? Then what is its difference from x+a=0 (x+a)2=0 (x+a)(x+a) =0  x= -a, -a two roots Also satisfies condition for quadratic equation What are the roots of the equation (x+a)2=0 _H001
  • 9. ANAND CLASSES 9463138669 Identity Identity : Equation true for all values of the variable (x+1)2 = x2+2x+1 Equation holds true for all real x _H001
  • 10. ANAND CLASSES 9463138669 Polynomial identity If a polynomial equation of degree n satisfies for the values more than n it is an identity Example: (x-1)2 = x2-2x+1 Is a 2nd degree polynomial Satisfies for x=0 (0-1)2=0-0+1 Satisfies for x=1 (1-1)2=1-2+1 Satisfies for x=-1 (-1-1)2=1+2+1 2nd degree polynomial cannot have more than 2 roots (x-1)2 = x2-2x+1 is an identity _H001
  • 11. ANAND CLASSES 9463138669 Polynomial identity If P(x)=Q(x) is an identity Polynomial of x Co-efficient of like terms is same on both the side Illustrative example If (x+1)2=(a2)x2+2ax+a is an identity then find a? LO-H01
  • 12. ANAND CLASSES 9463138669 Illustrative Problem (x+1)2=(a2)x2+2ax+a x2+2x+1 =(a2)x2+2ax+a is an identity Equating co-efficient x2 : a2=1 x : 2a=2 constant: a=1  a= 1 a=1 satisfies all equation If (x+1)2=(a2)x2+2ax+a is an identity then find a? Solution _H001
  • 13. ANAND CLASSES 9463138669 Illustrative problem Find the roots of the following equation (x a)(x b) (x a)(x c) (a c)(b c) (a b)(c b) (x c)(x b) 1 (c a)(b a)                Solution: By observation For x=-a L.H.S= 0+0+1=1 = R.H.S For x=-b L.H.S= 0+1+0=1 For x=-c L.H.S= 1+0+0=1 = R.H.S = R.H.S _H001
  • 14. ANAND CLASSES 9463138669 Illustrative problem 2nd degree polynomial is satisfying for more than 2 values Its an identity Find the roots of the following equation (x a)(x b) (x a)(x c) (x c)(x b) 1 (a c)(b c) (a b)(c b) (c a)(b a)                Satisfies for all values of x i.e. on simplification the given equation becomes 0x2+0x+0=0