Simple random sampling

Previously we have discussed about application of differential calculus  and In today's session we are going to discuss about Simple random sampling  which is fronm cbse previous year question papers class 12.

Some points to understand Simple random sampling:-

• The set of total observations that can be calculated in the experiment is called population.

• Sample can be defined as a set of observations drawn from the population.

• Statistics is related to sample which is a measurable characteristic of the samples like standard deviation.

• Sampling method is the procedure of selecting sample elements from the population that were made.

• Random number is a matter of chance having no relationship with the occurrence of other number.

• Simple random sampling related to sample method having following properties

1.  In Simple random sampling population can have n number of objects.

2.  Simple random sample that is chosen can also have N number of objects, here n ≠ N.

3.  All possible samples of N objects have same probability to occur.

Samples chosen with the help of simple random sampling are refers to the population this is the measure profit of choosing simple random sampling, it means that the conclusion will be valid.

There are several methods to get simple random sample for example lottery ticket is the example of simple random sampling in which a unique number is assigned to each member of N population. These numbers are placed in a bowl and then thoroughly mixed then a blind folded researcher selects n numbers from the bowl. The selected n numbers are called the samples taken from the population.

There are two other terms too; sampling with replacements and sampling without replacements.

In the above example of lottery ticket when a number is picked from the bowl; if it putted aside by the blind folded researcher then the probability of occurrence of this number is only once and if this number is putted back in the bowl then the number can be selected further too.

This is sampling with or without replacements.

If a population element has the chance to be selected more than one time then it is called sampling with replacements.

If a population element only can be selected one time then this sampling is called sampling without replacements.

In the next session we will discuss about Representations of data and You can visit our website for getting help from online tutors.

Mode in Grade VIII

Previously we have discussed about calculus problem solver and In today's session we are going to discuss about Mode which comes under gujarat board textbooks online, Its mathematical term that comes in statistics. It can be defined as particular value that occurs most number of the times in a list. In simple words mode is a most frequent value in a data set or most common value in a group. Mode or statistical mode is same thing. Let us understand mode with an easy example:-

We have a list of numbers :- 3, 5, 7, 2, 8, 9, 2, 4, 2

The first step to find mode is to arrange the list in ascending order than we will get.

2, 2, 2, 3, 4, 5, 7, 8, 9

The next step to find out particular number that has occurred most number of times.

Here the particular number is 2.

So the mode will be 2.

Mode is used to collect information about non-deterministic numbers, also called random numbers in a single quantity.

Now let us discuss how calculate mode when list is fractional with an example:-

Now we are talking a example of fraction list :- 3.2, 3.7, 4.1, 3.2, 5.6, 2.4, 3.2, 2.4

First of all arrange the list in an ascending order:- 2.4, 2.4, 3.2, 3.2, 3.2, 3.7, 4.1, 5.6

3.2 is a number that has occurred most number of times in the above list.

So the mode is 3.2

Advantage of mode

-provide support to mean,median to solve statistical problems.

-to identify wether an event has occurred more than one time or not.

-used for counting the number of times

Note:- A mode will not exist if there is no repetition of any number in a list. A list contain may be more than one mode if two numbers sharing same occurrence of time

The above described mode information will be truly helpful for grade VIII students.

In the next topic we are going to discuss Range in Grade VIII and You can visit our website for getting information about free online math help.

Math Blog on Median

In reference with data handling to be studied in Grade VIII  of CBSE math Syllabus today we are going to learn about median.
We have already learned about math questions on mean, which help us to find the average of the given collection of data.
As you have studied earlier that the raw data collected is to be arranged in the ascending or descending order in order to retrieve some information from it.
In this Blog we will discuss about median in math. Median is the mid value of the collected data. Suppose we have collected the age of 5 teachers in the school.
They are 35, 24, 45, 22 and 36.(Know more about Median in broad manner here,)
 To find the median, we will first arrange this data in ascending order:
We get:
  22, 24, 35, 36 45
We observe that 35 is the median.
Now we look at the data when the number of data is even, in such cases we select middle two terms of the data.
Then we find the average of these two values.
To sum up, we produce the formula for the same:
To find the median, the following steps are to be followed:
1. Arrange the data in ascending order; let the number of entries be n
2. Then we will find if the number of entries is even or odd
3 If the number of entries is odd then
    Median = (n+1) / 2 th term
And if the number of terms is even then
    Median = [ (n/2) th term + (n/2) +1th term ] /2
 So children we first check the number of entries in the given set of raw data and then accordingly proceed to find the median.

In the next blog we are going to discuss Mode in Grade VIII and if anyone want to know about How to calculate Median in grade 9th then they can refer to Internet and text books for understanding it more precisely and also know about some interesting questions like is square root of 7 a rational number.

Mean in Grade VIII

Previously we have discussed about properties of numbers worksheets and Today we will discuss about mean in math which you need to study in grade VIII of indian certificate of secondary education board which can provide you huge help with math. Mean is the average value of any given series. Mean is widely used in statistics. You will be using application of mean in higher class. Mean is very useful whenever you are asked to find the average of any given series. Let's see some examples to have a good idea about mean.(want to Learn more about Mean ,click here),

Example1 find the mean of the given series?

4,5,6,7,8,9

solution : This series contains 6 elements, so number of element = 6

mean= sum of all terms of the series/number of elements

mean = 4+5+6+7+8+9/6

mean=39/6

mean=6.5

So 6.5 is the required mean for this expression.

Mean is very useful and whenever you are asked to calculate the average of any quantity, it will simply give you the average of that particular quantity. We will see an example:

Example 2: find the average marks of Sachin in grade VIII. His marks are as fallows

hindi -67, math-80, english-76, science-76,computer-60.

Solution:

there are 5 subjects, so the series contains 5 elements,

now as we know mean of the series is = sum of all terms of the series/number of elements

mean=67+80+76+76+60/5

mean=359/5

mean=71.8

mean of the all the subjects is 71.8.

As you are seeing that mean is 71.8, so we can say that mean can be an integer or can be a real number

Example 3:Find the mean of the series given below

2,5,7,9,11

Mean of the series will be sum of all the values divide by number of values

There are 5 elements in the series.

Now as we know mean of the series is = sum of all terms of the series/number of element

2+5+7+9+11/5

34/5

6.8 is the mean for given series.

This is all about the Mean of any series. In this article we have dealt briefly about mean. If you still feel any problem with this and other topics like How to tackle eighth standard Algebra you can visit different websites to solve your problems.

Graph and Slope of Lines in Grade VIII

Hello friends, Previously we have discussed about probability examples and today we are going to learn about Graph and Math problems related to it., slope of lines for grade VIII of icse board. The slope of a line is generally represented by m. Simply slope is the rate at which the path of a line rises or decreases or the slope is a number that tells how steep the line goes up and down or slope is a ratio of vertical to horizontal distances. From the equation of a straight line (want to Learn more about Slope ,click here),
y = mx + b
Here the slope of line m is multiplied by x and b is the y-intercept where line crosses the y-axis. This is the equation of line and sensibly named as slope-intercept form. The graphical form of this equation can be quite straightforward, particularly if the values of m and b are relatively simple numbers. In the slope of line if the value of y changes then the value of x also changes. For example in the line y = (2/5) x – 3, here the slope is m = 2/5. This means that, starting at any point on this line, we can get to another point on the line by going up 2 units and then going to right 5 units. In other words, for every unit that x moves to the right, y goes up by two-fifths of unit. Now the formula of the slope of the line, if two points (x_1,y₁) and (x₂,y₂) are given then the slope m of the line is
m=y₂-y₁/x₂-x₁, (x₁≠x₂)
For example the slope of the line segment joining the points (1, - 6) and (- 6, 2). Here x₁ =1, x₂=-6, y₁=-6 and y₂=2, now using above formula
Slope= m=y₂-y₁/x₂-x₁
          2+6/-6-1=- 8/7
So, m=- 8/7
The graphical form of slope is shown in the figure.

Now I am going to tell you the different types of slopes and their definitions. The different types of slopes are Slope of Parallel Lines, Slope of Perpendicular Lines and Negative Slope. Firstly the Parallel Lines, if the slopes of two lines are equal then these lines are parallel. The two parallel lines never intersect. If each line will cut the x axis at the same angle then slopes are
m₁=tan x and m₂=tan x
 m₁= m₂
This shows that if two lines are parallel then their slope must be same.
The next is Slope of Perpendicular Lines. Perpendicular lines are lines which makes right angle at their intersection. Slope of Perpendicular is defined as if there is change in y co-ordinate then there is also change in x co-ordinate. On other hand, when two lines are perpendicular then the slope of one is the negative reciprocal of the other one. That is if the slope of one line is m then slope of the other is -1/m. -1/m is the slope of the other line is the negative reciprocal of m and is called negative slope. For example if the slope m=0.342 then the negative reciprocal of 0.342 is -1/0.342.
From the above discussion I hope that it would help you to understand the slope of lines and if anyone want to know about Permutations and combinations then they can refer to internet and text books for understanding it more precisely. Read more maths topics of different grades such as Factors and Number Sequences in Grade VIII in the next session here.

Real Numbers in Grade VIII

In earlier sections we have discussed and practiced on rational numbers worksheet and Today we will discuss about the real numbers and  properties of real number for grade VIII of Maharashtra Board Syllabus.
So before starting, we will first try to find that what the real numbers actually are.
The real numbers can be defined as the set of numbers that consists of all the rational numbers(maximum used in rational expressions) together with all the irrational numbers. In general language we can say that all integers, small or large, whole number, decimal numbers are all real numbers
Except the imaginary numbers (the numbers which have negative terms under the roots are called imaginary numbers) all numbers are known as real numbers
For Example:

Given set A =  0, 2.9, -5, 4, -7, p , is a set which  consists of  natural numbers,  whole numbers,  integers,  rational numbers,  irrational numbers but all element of  set A represent the real number.

Now we will learn about some properties of real numbers which apply on all real number. These properties will be very helpful to solve algebraic problems. We will discuss each property in detail and will try to explain with some examples. Let's talk about  Commutative properties of real numbers
For addition:            a+b = b+a
For example: 1:     3+5 = 5+3 =8
     
                     2:      4 + 5 = 5 + 4 =9
   
For multiplication:       a*b =b*a
For example: 1:              3*7 =7*3 = 21
                    2:               5 × 3 = 3 × 5 =15
But this property does not satisfy for subtraction and division
                b-a ≠ a-b and   a/b ≠ b/a
  for example:    4 – 5 ≠ 5 – 4    and          4 ÷ 5 ≠ 5 ÷ 4
    Now let's move on other property that is Associative property of real numbers.
For addition : a+(b+c) =   (a+b)+c
For example:   (4x + 2x) + 7x = 4x + (2x + 7x)
                       (4 + 5) + 6 = 5 + (4 + 6)

For multiplication: a*(b*c) = (a*b)*c
For example:1:   (3x*4x)*3y = 3x * (4x*3y)
                   2:    (4 × 5) × 6 = 5 × (4 × 6)
The associative property of multiplication tells us that we can group numbers in a product in any way we want and still get the same answer.(want to Learn more about Real Numbers ,click here),
Associative property for real numbers does not satisfy in subtraction and division
                          (a-b)-c  ≠  a-(b-c)
For example:    (4-5)-6  ≠  4-(5-6)
                (a/b)/c ≠  a/(b/c)
For example (4 ÷ 5) ÷ 6 ≠ 4 ÷ (5÷ 6)
Next is Distributive property of real numbers. In this we see that
a*(b+c) = a*b + a*c
(a+b)*(c+d) = a*c + a*d + b*c + b*d

for example 1:  4(x + 5) = 4x + 20
                   2: 3(4 – x) =12 -3x
                   3:  (a – 3) (b + 4) = ab + 4a – 3b – 12
Now comes to Identity property of real numbers
For addition:              a+0 = a
For example:           5y + 0 = 5y

The identity property for addition tells us that zero added to any number is the number itself.
Zero is called the "additive identity."

For multiplication:     a*1 =a
For example:  2c × 1 = 2c
The identity property for multiplication tells us that the number 1 multiplied times any number gives the number itself as a result. The number 1 is called the "multiplicative identity."

So this is all about real numbers. If you want to more details on the topics like range and Probability in Grade VIII then you can visit various websites on the internet.

Math bolg on grade VIII

Friends, today i am going to teach you one of the interesting topic of mathematics, formulas for measurements . In  grade  VII  we  study  about  some  geometrical  figures  like  -  line, Triangle, circle  etc ,  but  in  grade VIII of maharashtra board we  study,  what  kind  of  formulas  are  we  use  to  measure  these  geometrical  elements and  learn  formulas for measurements which is  a very important part of online tutoring . Here  measurement means finding  distance , area  and  volume  of  geometric figures  and  so  we  divide  measurement  of  geometrical  figures  in  three  units –

• measuring   distance 
• measuring   area  of   geometrical  figures
• measuring  volume of  geometrical  figures

So,  first  of  all  we  discuss  formula  of  measurement of  distance  :
 1 )   if  there is  two  points P( x₁ , y₁ )  and   Q(x_{2 ,}y₂)  in  a  plane ,   then  formula  to   measuring distance  between  these  two  points   PQ  is  –
    sqrt ((x₂ – x₁ )²  +  ( y₂ –  y₁ )² )
    here   sqrt  means  square  root
2)  if  there  is  two  points  P( x₁ , y₁ , z₁  )  and   Q(x_{2 ,}y₂ , z₂ )  in  a  space ,  then formula  to  measuring  distance  between  these  two  points  PQ  is –
    sqrt ((x₂ – x₁ )²  +  ( y₂ –  y₁ )²  + (z₂   -  z₁ )² )
    here   sqrt  means  square  root and to know more about measurements click here,
Then  we  further  proceed  to   discuss   measuring  area  of   geometrical  figures  :

1.  Area  of   triangle     =    1    *   b  *  h
                                          2

                       here   b  refers    base  length   of  triangle   and   h  refers  to  height  length  of  triangle
2.   Area  of   square       =     a²
               here  a  refers  to  one  side  of  square
3.    Area  of  rectangle   =      a* b
                here  a  refers  to  length  of  rectangle and  b  refers  to  width  of  rectangle
4.    Area   of   parallelogram   =   b  * h
                here  b  refers  to  one  side  length  and  h  refers  to  width  between  two  sides
5.    Area  of   circle   =   pi * r²
                here  pi = 3.14  and   r  refers  to  radius  of  circle
6.    Area  of  eclipse  =  pi * r₁  * r₂
                here  pi = 3.14  and  r₁  and  r₂ are  two side  radius  of  eclipse
7.     Area  of   sphere  =  4 * pi * r²
                here  pi = 3.14  and  r  refers  to  radius  of  sphere
These  are  standard  area  formulas  which  use  to  measure  area  of   geometric  figures  and    we  further  proceed  to  discuss  formula  of  measuring  volume  of  geometrical  figures –

1. Volume  of  cube  =  a³ 

         here  a  refers  to  each  side  of  cube
2.  Volume  of  Cuboid  = a * b * c
         here  a  shows  length , b  shows width  and  c  shows  height
3.  Volume  of  cylinder  =  pi *  r² * h
         here  pi = 3.14  , r  refers  to  radius  of  cylinder  and  h  refers  to  height  of  cylinder
4.  Volume  of  pyramid   =   1    *  b *  h                            
                                              3
         here  b  refers  to  width  of  pyramid  and  h  refers  to  height  of  pyramid
5.  Volume  of   sphere    =    4    *   pi  *  r² *  h
                                              3
         here  pi = 3.14  ,  r   refers  to  radius  of  sphere  and   h  refers  to  height  of  sphere


These  are   standard  formulas for measuring  volume  of   geometrical  figures and  You can also refer Grade IX blog for further reading on Basic constructions.Read more maths topics of different grades such as Range in Grade VIII  in the next session here. 

Learn Geometry and Transformation on a coordinate plane

Hello Friends, in today's class we all are going to discuss about one of the most interesting and a bit complex topic of mathematics, geometry. In VIII grade gujarat board students  are  familiar  with algebraic expressions  of  mathematics , but  geometry  is  new  subject  for  them  . First  of  all   one  question  arises  in  student’s  mind  what is  geometry ? Basically  when  we  study  about  shape like Triangle, size  and figure  of  space   than  this  study  is  called  as  a  geometry (Learn more about natural numbers here),

like when two points  meets  it  makes  line  and  four  lines  make  rectangle  , square  etc  .  For  understanding  the properties  of  geometry  what  grade VIII  student  supposed  to  do,  they  have  to  study  about  coordinate  system  and   Transformations on a coordinate plane . coordinate  system  is  basically  a  system   which tell   position  of   points  or  position  of  geometry element . We  can  understand coordinate  system  by  coordinate  plane  .  So,  first  of all  we  have  to  understand coordinate plane . coordinate  plane  is  basically   two  intersecting  lines (combination  of  horizontal  and  vertical  lines)  where  horizontal line  is  known  as  a  x  coordinate  and  vertical  line  is  known  as  a  y coordinate , for  example  we  have  a  line  on  a  plane   whose  one  point  coordinate  is  (0,1)  means  x  direction  coordinate  is  0  and  y  direction coordinate  is  1 .  But  when  we  want  to study  about  3D  coordinate  of  space,  we  have  to  include  z coordinate  on  plane  which  shown  outward  direction  form plane  and  in  3d ,  coordinate  plane  work  as  a  ( x, y, z )  coordinates .
                                       Now , when  we  move  some  point  with  same  distance  in  same  direction  on   plane ,  then  this  process  is  called  as  a  Transformations on a coordinate plane . We  can  understand  Transformations on a coordinate plane  with  an  example  like  when  we  move  table  from  one  place  to  another in  room ,  then  we  have   to   move  each  bottom legs  of  table    with  same  distance in  same  direction .  In  mathematical  point  of  view , when  we  move  one  object P (x , y)   with      a  units  in  horizontal  direction ( moves  right )  and  b  units  in  vertical   direction ( moves  upward)   on   coordinate  plane ,  then   after  this  Transformations on a  coordinate plane  process   coordinate  of  object  P( x +a , y +b) . Here  one  think  is  noted  that  in  horizontal  direction  moves  right  shows  positive  movement  and  moves  left  shows  negative  movement  of  object   and  in  vertical  direction   upward  movement  shows  positive  movement   and  downward  movement  shows  negative  movement  of  object .  So, we  take  some  example  to  understand  Transformations on a coordinate plane –
Let  one  point  A (2 , 3)  is  moves  right  with  2 unit  and  moves  up  with  5  unit  then  after  transformation  coordinate  of   A ( 2 + 2 ,  3 + 5 )  or   A ( 4 , 8 ) .
Similarly when  A( 2, 3 )  is  moves  left with  1 unit  and  move  up  with  3  unit  then  after   transformation   coordinate  of  A ( 2 – 1 ,  3 + 3)   or   A ( 1 ,  6)
So, I  hope  u  understand  about  geometry  portion  of Eighth Grade syllabus and Read more maths topics of different grades such as Basic constructions
in the next session here.

Factors and Number Sequences in Grade VIII

We have studied a lot about numbers till now. We have come across different types of numbers in various grades of different education board, which includes Natural Numbers (used for counting), whole numbers (used for measurements), Even and Odd Numbers. Now we will proceed and learn more about Math problems associated with it.
Let us first talk about Factors of a given number. Any number which exactly divides any number is called the factor of any number. Let us take 16.
 16 is exactly divisible by 1, 2, 4, 8, and 16 . So 1,2,4,8, and 16 itself are the factors of 16.
Also we know that 1 is the factor of all the numbers as any number multiplied by 1 gives the number itself.
Multiple of a number: A number is multiple of all its factors, which means if we multiply any factors we get the number. In the above mentioned example we find that 16 comes in the table of 1, 2, 4 ,8 and 16 , so 16 is the multiple of all its factors. To find the multiples of any number we simply write its table as
 multiples of 5 are 5, 10, 15, 20, 25, ------- and so on.
Primes : Any number is called a prime number if it has only 1 and itself as two of its factors. It means that a number is prime, if it is divisible by 1 and itself.
Composite Numbers: The number which has factors other than 1 and itself are called Composite Numbers. The smallest composite number is 4 as the factors of 4 are 1, 2 and 4 itself.
 Co- primes: If we have two numbers, not necessary primes, such that they do not have any common factor except 1 are called co- prime numbers. Ex 4 , 5.  Here common factors of 4 & 5 is only 1. So they are co- prime numbers.
Twin Prime Numbers: A pair of  numbers which have two consecutive odd numbers are called twin primes. Ex: 3, 5   and  11, 13 are consecutive odd as well as consecutive prime numbers.Know more about factors here,
We come across such number sequences in our day to day mathematical work.

We should remember the following facts related to numbers:
a. 1 (one) is a number which is neither prime, nor composite.
b. 2 is the only even number which is prime. All other prime numbers are odd numbers.
c. 2 is the smallest Prime number.
 Prime factorization: To find all the factors of any number which are all primes they are called prime factorization. Suppose we need to find the factors of 36.
 36 can be written as 36 = 2 * 18, but 18 is not a prime,
 further we write 18 as 2 * 9, but 9 is again not prime,
 again we write 9 as 3 * 3, which are primes,
so we can write 36 = 2 * 2 * 3 * 3, where we get all the numbers as prime factors.

We observe that factors can be written in different order, but the product remains the same. This is called unique property of factorization.  Factorization also helps in finding the L.C.M. and H.C.F. of any given numbers. Also various numerical sequences are formed using the factors.

So children, this is a brief article about prime numbers, factors, multiples, and number sequences and If you want to know about Variables in VIII Grade and also about Congruence in Grade VII then there are many websites available on the internet where you can get detailed knowledge about all the topics.

Probability in Grade VIII

When we talk about the possibility of occurrence of any event, it is called probability and its an important part of cbse syllabus.
Any occurrence of the experiment is called an Event. Events can be dependent events or independent events.
Children if we talk about a independent event, in any experiment if the event occurs independently then it is called independent event. For example, if we throw a dice, the possibility is of  getting any numbers 1, 2, 3, 4, 5 and 6. But we find the possibilities are independent that is it does not depend on anything else, Two events are called to occur independently, if they to not depend on one another. Suppose 2 coins are thrown, we find that on both the coins we have the possibility of getting either a head or a tail. The possible outcomes are ( HH, HT, TH, TT ) but all these 4 outcomes are not dependent on one another, so they are  called Independent events.

Now let's talk about dependent events. When we conduct any experiment such that the observation of 2nd depends upon the observation of first experiment, then the event is called dependent event. You will study this type of math questions in higher grades.
Conditional Probability of any event is the probability condition when we have two events A and B associated with the same random experiment. Then probability of occurrence of  of event A under the condition that event b had already taken place is called conditional event.
In Grade VIII we will learn about basic experiments of probability.To know more about it refer this,
Let us take an experiment of throwing a coin. The possible outcomes for this event will be Head and Tail. Now a question arises that

•     what is the probability of getting a head in this event?

        We observe that the event has one probability out of two for getting a head, so
        P (Getting head ) = 1/2

•  what is the probability of getting a tail in this event?

      We observe that the event has one probability out of two for getting a tail, so

      P (Getting tail ) = 1/2

   Similarly if we study the pack of cards, we know that there are 52 cards in a pack. out of which 13 are ♣ ,13 are ♥, 13 are ♠ , 13 are ♦
   So probability of getting a ♣, if one card is drawn = 13/52  = 1/4
        probability of getting a ♥, if one card is drawn = 13/52  = 1/4

       probability of getting a ♠, if one card is drawn = 13/52  = 1/4

       probability of getting a ♦ , if one card is drawn = 13/52  = 1/4

     

      Further we elaborate and see, what is the probability of getting a red card, if a card out of a pack of 52 cards is drawn?
             P ( getting a red card ) = 26/ 52
                                               = 1/2
    what is the probability of getting a black card, if a card out of a pack of 52 cards is drawn?

             P( getting a black card ) = 26/ 52

                                               = 1/2

   In this way we solve various other problems related to it. To get information about Percent and Rates in IX Grade  and Mean in Grade VIII you can refer Internet.

Eighth grade quadratic equations

Friends, today we all are going to learn the basic concept behind one of the most interesting and important topic of Grade VIII mathematics that is quadratic equations. Before proceeding further let's talk about Binomial formulas first. The Binomial formulas are:

(a + b)² = a² + b² + 2ab First Binomial formula

(a – b)² = a² + b² -2ab Second Binomial formula

(a² – b²) = (a + b)(a – b ) Third Binomial formula

Here a and b are the variables, or they can be even more general expressions. In the first and second binomial formulas, expression on the left are perfect squares while the expression on the left hand side of the third formula is the difference of two squares. One of the important thing to notice is that the first and second binomial formulas are equivalent. What student needs to do is to replace b with -b to get from one to the other.

We can use distributive law to verify the binomial formulas straight from left to right. For example:
(a + b)² = a(a + b) + b(a + b)
= a² + ab + ba + b²
= a² + 2ab + b²
Now I am going to discuss about quadratic equations. A polynomial equation of the second order is known as Quadratic equation. The general form of quadratic equation is:
ax² + bx + c = 0
where x is a variable and a, b and c are constants. Here a is quadratic coefficient, b is a linear coefficient and c is a constant term or we can say that it is a free term. To solve any quadratic problem or equation refers to find the value of variable. Let's take y that makes the equation true. To understand it better and in more deeply let's take an example:
(y – 1)² = 25
As we can say that it is not a quadratic equation, we can convert it into an equivalent equation that is in that form, by suitable operations on both the sides of the equation.

(y + 1)² = 25 expand
y² – 2y + 1 = 25
y² – 2y – 24 = 0 a = 1, b = -2, c = -24
The last solution or equation is in the standard form, where a, b and c having the given values.
But the second equation can be solved much more easily than the first one (ax² + bx + c = 0 ).
(y + 1)² = 25 root
y – 1 = ±5 +1
y = 1 ±5 consider both cases the answer
y = 6 or y = -4 (these are the two solutions of the equation.) Students needs to verify this by substituting these values in the exact equation. If y = 6 we get 5² = 25 and if y = 4
then we got (-5)² = 25.

Note the symbol ± in the above sequence of the equation. The square root value of the number 25 is positive by conventions and equals +5. Meanwhile, our task at that stage is not just evaluating a square root value as such, but our main focus should be there to answer the question for what values of y does (y – 1)² equal 25?
There are two such values y – 1 = -5 and y – 1 = + 5, and student needs to consider both the possibilities.
Now to understand it more wisely, consider the more general equation
(y – r)² = s (***)
where r and s are considered known and as before y need to be determined. The above equation can be solved just like we solve one before this.
Here r and s are considered known and as earlier y needs to be determined.
We can solve this in the same way as we can solve the above solved problem.
(y – r)² = s root
y – r = ±root s +r
y = r ±root s is the required answer

The most important way to solve a quadratic equation is to convert them to the above mentioned form. This process is known as completing the square. It is mainly based on the first and second binomial formulas.
Let's take an example to understand the basic concept behind and how this works with our equation in standard form:
y² – 2y – 24 = 0
If in the equation, the constant term was 1 instead of -24 than it would be a perfect square. To make it perfect square what we just need to do is add 25 on both the sides and get the desired value
y² – 2y + 1 = 25
It can be rewritten as
(y – 1)² = 25
Now the one thing is
(y – r)² = y² – 2yr + r²
What students need to do is to simply look at the factor of y, than halve it and in next step square it and in final step add the appropriate constant that makes the constant equal to that desired value. Another simplest of the way to find out that constant value is to subtract whatever constant is there and in final stage add the desired value. To solve it in this manner, work the leading coefficient (multiplying y²) must equal to 1. If it doesn't happen than we need to divide the first by the leading coefficient on both sides.
Now I am going to discuss about quadratic formula. Now what we need to know is what actually a quadratic formula is:
Take the standard form of the quadratic equation:

The general form of quadratic equation is:

ax² + bx + c = 0

where x is a variable and a, b and c are constants.

We already study that we can solve this general equation by completing the square exactly like we would solve it if the coefficients assumed specific values.

Quadratic equations can be solved by using following methods : factoring , completing the squares, graphing, Newton's method, and with the help of Quadratic formula.

The Quadratic formula. Quadratic equation is ax² + bx + c = 0 and it has the solutions

here the expression under the square root sign is known as discriminant of the quadratic equation. Discriminant is denoted by the upper case Greek delta.

Delta = squared b – 4ac .

If the discriminant is zero then there is only one exact real root, also known as double root.

X = -b/2a.

The ‘±’ symbol indicate as ‘plus or minus’, which means that we need to work out the formula twice, once with a plus sign in that position, then again with a minus sign.

Clearly there are three cases while finding a discriminant:

D > 0 there are two real solutions

D = 0 there is one real solution

D < 0 there is a associated complex pair of solutions.

If you wants to apply the quadratic formula to a particular quadratic equation, what we need to do is just convert the equation to standard form and substitute the appropriate values of a, b and c in the formula. In between this is the more efficient and less error flat to make sure that student understand the binomial formulas and the basic concept behind completing the square and the most important is to solve a quadratic equation from scratch. In addition to these there are some of the formulas like the binomial formulas, the distributive law and different rules for manipulating powers, which needs to be remembered by the students.

Let's take an example to understand it better, we need to solve the following equation

x² - 4x - 5 = 0 , here no coefficient is written before x so we can use 1 as a coefficient of x. Now here a = 1, b = -4, and c = -5 now substitute this values in the above equation we get two values for the same that is x = 5 or x = -1.

The technique used for graphing quadratic equations is the same as for graphing linear equations. The most basic quadratic equation is y = x^2.. A quadratic graph is a parabola it can be generated by using quadratic math help calculator.

For solving quadratic equations we can also use math helper or solvers available over internet. For graphing quadratic equations we need to graph a proper parabola which is quite difficult , so we can use math helper which can provide us the useful x and y intercepts through which it becomes easy to graph a proper parabola of a particular quadratic equation.

Now in next class we all are moving forward and going to see the problems involving quadratic equations and try to solve them in a better and faster manner. In addition to these we are also going to understand the quadratic equation graphs.