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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
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
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
