Maths

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Quadratic Equation

Friends these are the topics that we are going to deal with in this lesson:

• Quadratic Equation and its Solution
• Solution by Factorization
• Solution by Completing Square method
• Derivation of formula
• Finding nature of roots

So lets get started!

Quadratic Equation and its Solution

"Quadratic Equation is an equation in which the highest power of x is 2". By this statement I mean to say that the quadratic equations look like this ax²+bx+c=0.
Here x has the highest power 2.
Finding the solution of the equation means that the value of x for which it satisfies the equation. So when the value of x gives zero as solution then it is called Zero or Root of the equation. For example, if we find the solution of this equation then the value will be called the zero or root of the equation ax²+bx+c=0.

Solution by Factorization

It is process by which we can find zero of any equation. Here, we have to split the equation in such a way that product of a and c will be equal to two parts of b and the parts form b on adding. For example if x²-11x+30=0 then,

Hence we can easily see the find the of x.

Solution by completing square method

It is a very interesting as well as very challenging method. Here we have to break the equation in the form of (a+b) whole square or (a-b) whole square. So I have given one example of this in the below image:

So I think its clear for you to understand this method. This method help us in forming a direct rule for the value of roots which is derived in the next paragraph.

Derivation of formula

The formula is also known as Sridharacharya's Rule. This gives the direct value of x from any equation. This formula is written as:

 The derivation of this formula is very though and also very lengthy. So I have shown a different way to derive this formula in the picture below:

So now you may not have any problem in the derivation part of this rule. Now, the b2-4ac part of this equation is also termed as Discriminant and is written as D. This D helps us in finding the nature of roots.

Finding nature of roots

Basically, there are three types of roots:

• Real and Distinct 
• Real and Unequal
• Not Real (Imaginary)

For every type the value of D is different. So different value of D for Different types are given below:

• D>0 for Real and distinct roots (ie. if D=4 then the roots will be different) 
• D=0 for Real and Equal roots
• D<0 for Imaginary roots

Hence while solving problems, you should keep in mind that what is the nature of roots.

These were some of the important topics that are important for your basics of quadratic equation.

If you have got any question, you can post in the comment box below.

Arithmetic Progression 

Now it's time for you to know about Arithmetic Progression commonly known as AP.
These are the topics which are very much important in this chapter:

• Introduction
• T_(n)of an AP
• S_(n)of an AP
• Points that should be kept

Introduction 

SEQUENCE
A collection of numbers arranged in a definite order according to some definite rule (rules) is 
called a sequence.
Each number of the sequence is called a term of the sequence. The sequence is called finite or 
infinite according as the number of terms in it is finite or
infinite.
ARITHMETIC PROGRESSION
A sequence is called an arithmetic progression (abbreviated A.P.) if and only if the difference 
of any term from its preceding term is constant.
A sequence in which the common difference between successors and predecessors will be 
constant. i.e. a, a+d,a+2d
This constant is usually denoted by ‘d’ and is called common difference.
NOTE : The common difference ‘d’ can be positive, negative or zero.
SOME MORE EXAMPLES OF A PARE
(a) The heights (in cm) of some students of a school standing in a queue in the morning 
assembly are 147, 148, 149, ….. , 157.
(b) The minimum temperatures (in degree celsius) recorded for a week in the month of January 
in a city, arranged in ascending order are 3. 1, — 3. 0, — 2. 9, — 2. 8, — 2.7, — 2. 6, — 2. 5
(c) The balance money (in ) after paying 5% of the total loan of Z 1000 every month is 950, 
900, 850, 800, ….50.
(d) The cash prizes (in ₹) given by a school to the toppers of Classes Ito XII are, respectively, 
200, 250, 300, 350„ 750.
(e) The total savings (in ₹) after every month for 10 months when Z 50 are saved each month 
are 50, 100, 150, 200, 250, 300, 350, 400, 450, 500.
n
th TERM OF AN A.P. : It is denoted by tn and is given by the formula, tn = a + (n —1)d
where ‘a’ is first term of the series, n is the number of terms of the series and ‘d’ is the common 
difference of the series.
NOTE : An A.P which consists only finite number of terms is called a finite A.P. and which 
contains infinite number of terms is called infinite A.P.
REMARK : Each finite A.P has a last term and infinite A.Ps do not have a last term.
RESULT: In general, for an A.P a1 , a2, , an, we have d= ak + 1 — ak where ak + 1 and ak are the 
(k+ 1)th and the kth terms respectively.
SUM OF FIRST N TERMS OF AN A.P.
It is represented by symbol Sn and is given by the formula,
Sn= n/2{ 2a + (n — 1)d} or, Sn = n/2 { a + l} ; where ‘l’ denotes last term of the series and l= 
a+(n-1)d
REMARK : The nth term of an A.P is the difference of the sum to first n terms and the sum to 
first (n — 1) terms of it. — ie — an = Sn— Sn – 1.
TO FIND nth TERM FROM END OF AN A.P. :
n
th term from end is given by formula l – (n – 1)d
nth term from end of an A.P. = nth term of (l, l — d, l – 2d,…….)
=l+(n-1)(—d)=l—(n-1)d.
PROPERTY OF AN A.P. :
If ‘a’ , b, c are in A.P., then
b — a= c — b or 2b= a + c
THREE TERMS IN A.P. :
Three terms of an A. P. if their sum and product is given, then consider
a—d,a,a+d.
FOUR TERMS IN A.P. :
Consider a —3d, a — d, a+ d, a +3d.
NOTE :

The sum of first n positive integers is given by Sn= n(n + 1) / 2