This document discusses two methods for solving quadratic equations: 1) The formula method, which can be used to solve quadratic equations that cannot be solved through factorization. It provides an example of using the quadratic formula to solve the equation 5x^2 + 6x + 1 = 0. 2) Definitions of arithmetic progressions and geometric progressions. It states that an arithmetic progression has a constant difference between terms, while a geometric progression has a constant ratio between terms. Examples of sequences that fit each progression type are provided.

1. 3) Formula Method of Finding Roots
The quadratic equations which cannot
be solved through the method of
factorization can be solved with the
help of a formula

2. Example: Solve 5x2 + 6x + 1 = 0
Soln: Here Coefficients are: a = 5, b = 6, c =
1
Quadratic Formula: x =
−b ± (b2 − 4ac)
2𝑎
Putting the values of a, b and c, we get:
x =
−6 ± (62 − 4x5x1)
2∗5

3. x =
−6 ± (36− 20)
10
x =
−6 ± (16)
10
x =
−6 ± 4
10
x = −0.2 or −1

4. Arithmetic Progression ( A.P.) & Geometric
progression (G.P.)
Arithmetic Progression ( A.P.): In a sequence if the
difference between any term and its preceding
term (tn+1−tn) is constant, then the sequence is
called an Arithmetic Progression ( A.P.)
Consider the following sequences
1) 2, 5, 8, 11, 14, …
2) 4, 10, 16, 22, 28, …
are in Arithmetic Progression ( A.P.)

5. Geometric progression (G.P.):
In a sequence if the ratio between any term
and its preceding term (tn+1/tn) is constant,
then the sequence is called an Geometric
Progression ( G.P.).
Hence a G.P. can also be written as a, ar,
a𝑟2
, a𝑟3
, ....
where a is first term and r is common
ratio.

6. Examples :
i) 2, 4, 8, 16, .... [here a = 2, r = 2]
ii) 3, 9, 27, .... [a = 3, r =3]
are in G.P.

7. THANK YOU