The document discusses properties and identities of quadratic equations, focusing on conditions for identity and the nature of roots based on the discriminant. It explains that quadratic equations typically have two roots, which can be real or imaginary, and it provides methods for finding these roots. Additionally, it addresses problems related to common roots, ensuring equations maintain their solutions under certain conditions.

1. QUADRATIC EQUATION
Identities
An equation is called identity if and only if all real numbers are the solution of its.
Problem: Find all value of p for
( -3p+2) + ( -5p+4)x + p - = 0
is an identity.
Sol: -3p+2 = 0 -5p+4 = 0 p- = 0
⇒ (p-2) (p-1) = 0 ⇒ (p-4) (p-1) = 0 ⇒p (1-p)=0
⇒ P = 2, 1 ⇒ p=4, 1 ⇒ p=0, 1
∴p=1 ans
Problem: Number of solution of equation is
+ + = 1
a) 0 b) 2 c) 3 d) ∞
Ans: (d)
Sol: max power of x is 2 but it 3 roots can be easily seen x=a, x=b, x=c. i.e. it is not a quadratic equation, it is an identity thus all
real numbers are its solution.
QUADRATIC EQUATION
Where a≠0
is a quadratic equation and the value of x which is satisfied this equation is called its ‘Root’.
NOTE: - It has always two roots, may be real & imaginary; equal & unequal.
Quadratic Expression:
y =
Where a≠0 and x, y are variable and a, b, c are any constant

2. Sridharacharya Method for finding root of quadratic equation
⇒
D = is called its ‘Discriminant’
α = +
β = -
where α, β are its roots
Divided by a both side;
⇒
Problem: If & are roots of the quadratic equation then find the value of
(i) +
(ii) +
(iii) | – |
(iv) +
Ans:
(i) - 2
(ii) + 3
(iii)
(iv) +

3. Problem: If & are roots of the quadratic equation then find the quadratic
equation which have roots
i. +2, +2
ii. ,
iii. ,
iv. ,
Sol: (i)
Method 1
st
Required equation is
⇒
⇒
Method 2
nd
(using symmetry)
Suppose that p & q are the roots of required equation then
p=α+2 ⇒α=p-2
q=β+2 ⇒β=q-2
∵α & β are roots of
∴ &
p
⇒
q
Ans:
(i)
(ii)
(iii)
(iv)
Nature of Roots
NOTE: - It is fouls statement that if D≥0 then roots are real in all conduction. It is true if and only if D≥0
with a, b & c are real numbers
e.g. ℩ - 5x + ℩=0
Nature of Roots
D=0 D≠0
Both roots are equal & real Both roots are different
If a, b, c ∈ R

4. D>0 D<0
Roots are real Roots are imaginary
NOTE: - In this conduction roots are conjugates
If a, b, c ∈ Q (rational)
If D is a square of any rational number If D is not a square of any rational number
then roots are also rational then roots are also irrational
NOTE:- In this conduction, get irrational roots are conjugates
NOTE:- If a=1, b & c are integers and D is a square of any rational number then both roots of equation
are integers.
Problem: If a, b and c are real then prove that equation
have real and unequal roots.
Problem: If a, b and c are rational then prove that equation
(
have rational roots.
Condition for common roots of two given quadratic equation
# Both roots are common if and only if
# For one common root
Let α is a common root of given two equation
∴
From cross multiplication;
= =
⇒ α= =
NOTE:- If and are real and both equation has one common root then other root is
also common.

5. Problem: If and have a common root then find the value of k.
Ans: k=1
Problem: