VIII Grade Terms and Sequences

Hello friends, earlier we have studied about Proportional and non-proportional linear relationships. Now today we all are going to learn about the next topic that is terms in arithmetic sequences. It is one of the important areas of study which not just play an important role in eighth grade but also in our higher grades. Before proceeding further let's talk about the basic concept behind Sequence. In simplest mathematics manner it is a systematic list of objects. In mathematics the sequence list can be numbers, algebraic expressions, fractions etc. If the objects in the list are numbers then it is called as numeric sequence. Moving forward we will discuss numeric sequence topic in detail. Hence the sequence with number we mean the numeric sequence. Now each of the number used to form a sequence is called a term. In a sequence there can be finite number of terms or it can have infinite number of terms as well. Finite number of sequences can be stated as finite sequence and a sequence with infinite number of terms can be stated as infinite sequence. If we try to add a series of sequences then we get a series. So the total of the terms used in the sequences is called a series.


Let's take an example to understand the basic concept behind numeric sequences. A series : 1,3,5,7,9 is a finite number sequence with 5 terms. The another expression that is 3 + 6 + 9 + 12 + 15 + 18 is also a finite numeric sequence with six terms.
Similarly we can easily identify the infinite sequences like 3,1,6,4,5,4,...... etc.


Now moving further we are going to see the different types of sequences. There are various kinds of sequences in mathematics, but in eighth standard we all are going to discuss about three types of sequences.

1. Arithmetic Sequence :
   

In a simplest manner we can define it as the sequence of expressions or numbers in which the difference between a term and its previous term is a constant
For example : 2,4,6,8,10 ….... , 5,7,9,......................, etc.


In the above two sequences, each term is different from the previous term by a constant number and this statement is always true for any of the two consecutive terms. This kind of difference is known as the common difference. In simple words any of the term used in an arithmetic sequence is obtained by adding a constant number to the previously used term.
The most common form of an arithmetic sequence with the first term is “b” and the common difference d is given by
b, b + d, b + 2d,…………………………….b + (n – 1) d,………..


It is noticeable that first term is “b”, second term in the sequence is b + d and the third term in the above mentioned sequence is b + 2d etc.... So in this manner we proceed further and the nth term will be
b + (n – 1)d. In short we can say that an arithmetic sequence can be defined by the first term and the common difference.

2. Geometric Sequence
   

The next type of sequence is Geometric sequence which is basically a sequence in which the ratio of any term is a constant. Let's take an example to understand it better:
,100,200,400,800……………………….., 7,21,63,189……………………
In both of the above examples each term used in the sequence maintains a definite or we can say a constant ratio with the previous term. This can be stated in a more defined manner as if we take any term in the sequence and try to divide it by the previously formulated term we get the same number. This type of numbers in a sequence is called that common ratio of the geometric sequence.
In normal way we can understand it in a manner that any term of a geometric sequence is obtained by multiplying a constant number to the previous term
The common form of an arithmetic sequence with first term “a” and common ratio that is “r” is given by
a, ar,ar2,ar3,ar4………..ar(n-1)………….


Here it is also noticeable that the first term used is “a”, the second term in use are ar, and the third term in the sequence is ar2. So the nth term used in the sequence will be ar(n-1)
In short we can say that a geometric sequence is defined by the first term and the common ratio

3. Harmonic sequence :
   

The next kind of sequence used is harmonic sequence as it has its reciprocals in arithmetic sequence. This shows that if we are going to take any of the arithmetic sequence, the reciprocals of the terms used in the sequence are said to be in a harmonic sequence.
Let's take an example to understand it better : ½,1/3, ¼, 1/5, 1/6 …...... 1/5, 1/10, 1/15.... etc.
The most common form of the harmonic sequence can be expressed from the general form of the arithmetic sequence used by taking the reciprocal of the terms.
Let's take some example to illustrate the above mentioned terms and methodology:
Example 1: Here the problem is that the first term of an arithmetic sequence is 4 and the fifth term of the sequence is 20. Find the sequence
Solution: Let the first term b = 4
By using above mentioned sequence or general form we can find the fifth term from the formula for nth term b+ (n-1) d
Now the fifth term is b+(4-1)d= 20
b+3d= 20
4+4d=20
4d = 20-4=16
d= 16/4=4
So b=4 and d=4 the sequence is
4,8,12,16,20.........
Example 2: The next question is that the first term used in a geometric sequence is 6 and the fourth term used is 48. We need to find out the basic sequence form :
Solution : Let the first term is “a” = 6
Now the fourth term that can be calculated from the formula for nth term ar(n-1)
The fourth term which is to be used is ar(4-1) = ar3= 48
ar3= 48
6r3=48
r3= 48/6=8 so we can get r=2
So a=6 and r=2 the sequence we get after formulating this problem is :
6,12,18,24,..........


The other topic which we are going to understand consist of following terms that are Variables, expressions, equations and inequalities. Now take individual topic at a time :
Inequality: An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. Slope formula plays an important role in graphing linear inequalities. So we need to know what slope formula means. Slope of a line describes the steepness, incline or grade of the straight line.

Let's take an example to understand the linear inequalities: How to solve a compound inequality and graph the solutions?
Example: -6 < 2x - 4 < 12

-6 < 2x - 4 < 12
add 4 to all 3 parts
-2 < 2x <16
divide 2 from all 3 parts
-1 < x < 8

To graph the following equation, you put an open circle or we can say mark it by a dot on the point (-1,0) and then you put an open circle on the point (8,0).Then draw a line between the 2.

We need to be careful while solving inequalities, as they are harder to solve than equations and require more attention. You can multiply an equation by a positive or negative number and get an equivalent equation. But while solving Inequalities remember this: when multiplying it by a negative number, you need to change sign of the inequality.

A linear inequality describes an area of the coordinate plane that has a boundary line. In simple way in linear inequalities everything is on one side of a line on a graph.

In mathematics a linear inequality is an inequality which involves a linear function. For solving inequalities we need to learn the symbols of inequalities like the symbol < means less than and the symbol > means greater than and the symbol ≦ or ≤ less than or equal to etc.


Equation: An equation is basically explained as an assertion that two algebraic expressions are equal. Let's take an example: 3a + 1 = 4
Here we can get

a = 1 is the solution. This is an example of a linear equation. An example of a quadratic equation is a² - a - 2 = 0. This equation has the two solutions

a = -1 or a = 2.

Algebra and Variable relationship: If we talk in simple words then we will see that algebra is simply the art of replacing variables in place of numbers. In solving algebraic problems simplifying algebraic terms is important. Simplifying here refers to breaking the large expressions into smaller ones so that it becomes easy to solve.

So this is all about in our today's class. In next class we will continue with this topic. As this topic is very lengthy considering the syllabus of eighth standard.

VIII Grade Algebra

Friends today we all are going to start our eighth grade mathematics and I am going to discuss about Algebra section. Before proceeding further let's talk about the basic concepts of Algebra. Algebra is a very vast area of study of mathematics which almost covers 90 percent of mathematics. In simple mathematical manner we can say that it is a branch of mathematics which deals with the study of the rules of operations and relations. Equation term can be explained as a mathematical expressions which shows the equality of two expressions. For example b + z = 6. In most general manner we can say that any combination of literal symbols and numbers that results from algebraic operations (addition, subtraction, multiplication, division, raising to a power, and extracting a root) are called as Algebraic equations. The two most important types of such Algebraic equations are linear equations which is written in the form y = mx + b, and quadratic equations that can be represented in the form y = ax² + bx + c. In general Algebraic equations are useful for modeling real life phenomena.

Following steps are used to simplify an algebraic equation. Firstly remove all the fractions in the equation Then remove the parentheses . Combine all the like terms so that we get all the variables and terms together. Move all the variable terms by adding or subtracting on both sides of the equal sign so the variable terms are all on one side of the equal sign. And finally if there is any multiplication sign then remove it by dividing.

Let's take an example to understand it better. The problem is to find the product of the following algebraic equations:
(b + 5)(b – 3) here multiply each multinomial term to the another multinomial term like
b x b – 3 x b + 5 x b – 3 x 5 = b² + 2b – 15

Now let's take the first sub topic of eighth grade Algebra problem that is Proportional and non proportional linear relationships. First question comes in our mind is what is a proportion ? In simplest of manner we can say that forming a relationship with other parts or quantities is called as proportion. Relationship between the variables is basically a way in which the variables change. The relationship can be linear, directly proportional and non linear.

Before proceeding further let's talk about Rational expression. Rational expression is an expression which can be written as a fraction a/b. Here a is the numerator and b is a denominator. The most important thing to understand is that denominator can never be zero.
Let's take an example:
The numbers 5/3 and -6/11 are rational numbers.
The number 5 is also an expression : 5/1 = 5 and any number in decimal form also an expression for example: 3.33 is a rational number : 3.33 = 333/100.
Some theorems which tell about Rational Expressions:
First one is that any integer is a rational. For example a number n = n/1.
Second: the representation of rational number as a fraction is not unique. Like 3/4 = 6/8 = -9/-12.
Third : Every nonzero rational number or a rational that do not contain a 0 has a representation in lowest term.
Closure Property of Rational Number shown as:
Adding and Subtracting of a fraction is done by using this property: x/y + a/b = xb + ay / yb.
For Multiplying a fraction this property comes in an account: a/b x c/d = ac/bd
and for dividing a fraction we use this property: a/b divide c/d = ad/bc.

For simplifying rational expression, we must need to have  good factoring skills. It requires two steps in solving a rational expressions.
factor the numerator and denominator is the first step and the second step is divide all common factors that the numerator and denominator have.
Dividing a rational number is the most difficult part as it requires key skills. Now we are going to learn how can we divide a rational expression:
12/5 divide by9/5 then we need to take a reciprocal of 9/5 . The reciprocal of 9/5 is 5/9.
Multiply 12/5 with the reciprocal. 12/5 x 5/9 = 12/9 = 4/3.


Linear relationship can be stated that the two variables always form a straight line when graphed. The most common example to understand this is the distance time graphs of the stationary object and any of the object which is moving with the constant velocity. Directly proportional can be defined as the rate of increase in one variable is similar to the rate of increase in the other variable. The example to understand: force is directly proportional to its mass. Non Linear relationship means that the rate of increase in one variable is distinct from the rate of increase in the other variable.

Let's discuss about Non Proportional linear relationship. This can be explained in the general form of linear expression that is y = mx + b, Here b is not equal to zero, m is the slope of the line or we can say that it represents the constant rate of change, b = Y intercept form. The graph of a non proportional linear relationship is a straight line which can never pass through the origin.

Let's take an example to understand the Non proportional linear relationships (y = mx + b, b not equal to 0).
The taxi provider company charges a flat fee of Rs. 50 plus Rs. 30 per mile to ride in a taxi.
Assumed that the flat fee is acquired as soon as the person enters the taxi.
What we need to do is to find out the cost of the taxi ride, multiply the total number of miles traveled by Rs. 30 and then add it with Rs. 50 (Rs. 50 is the flat fee) to the product.
If we are going to simplify the above equation then we need to follow this : If y shows the total cost of a taxi ride of x miles, then the relationship can be represented as an equation in the form of y = mx + b, here m represents the cost per mile (Rs. 30/mile) and b represents the flat fee (Rs. 50).
Total Cost = Cost per mile * Number of Miles + Flat Fee
Y = 30 * X + 50, or we can also make it y = 30x + 50


Another example to show Non-proportional Linear relationship with negative slope. The problem is that a ten inch candle burns at a constant rate of one inch per hour.
What we need to do: To find out the exact height of the candle you need to multiply the number of hours that the candle burns by 1 inch per hour, and than subtracting the product from the candle's initial height that is ten inches.
Solution : Similarly if y shows the height of the candle after x hours of burning, then the relationship can be explained in the form of an equation that is Y = mx + b, here m represents the rate at which the candle burns which is I inch per hour and b shows the initial height of the candle which is 10 inches.
Height of Candle = Rate at which it Burns * Number of Hours Burned + Initial Height
y = -1 * x + 10, or it can make it y = 10 -1x


We can explain proportional relationship by the following methodology. Let's take a general method to understand it. In any of the relationship let's take between y and z is proportional, it means that as y changes , x also changes by the same percentage. It shows that if y grows by 20 percent of y, z also grows by 20 percent of z. In an algebraic form we have already discussed it that is y = mx, where m is a constant.


Directly proportional means : A very common delusion is that two variables are directly proportional if one increases as the other increases or vice versa. Another thing is that two variables are explained to be directly proportional if and only if their ratio is a constant for all the values of each variable. Thus the most important outcome is that one variable is divided by the other , the answer is always a constant.


Now the next topic we all are going to understand in next section is Sequences: A sequence is basically a systematic listing of objects. In mathematical world the list formed from sequence can be numbers, algebraic expressions etc. If the objects in the list are numbers then it is a numeric sequence. There are many types of sequences in mathematics but the most common types of sequences are:
Arithmetic Sequence
Geometric Sequence
Harmonic Sequence
In next class we will discuss sequences in detail and also going to learn about the types of sequences in mathematics.

Eighth Grade syllabus

Hello friends, here I am going to discuss about the overall syllabus of eighth grade mathematics which we supposed to do. Students often think that the only purpose of figuring things out is to get the answer with the minimum amount of ado and effort. But there are certain things which student needs to follow to get the desired answer in the most optimum way. If student opt to choose shortest path or longest path to arrive an answer then there is a possibility of getting a wrong answer so students need to follow the optimum path to arrive an answer. A daily practice and hard work is necessary to arrive at an answer. The most important thing to remember is that there is no alternative of hard work.

Following are the topics which we all are going to study in eight standard.

1. Algebra : Analyzing and representing linear functions and solving linear equations and systems of linear equations.
2. Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle.
3. Data Analysis and Number and Operations and Algebra: Analyzing and summarizing data sets.

In more generalize manner, the topics to cover are

1. Integers
2. Perimeter
3. Area
4. Algebraic Expression
5. Equations
6. Perimeter
7. Fractions
8. Decimals

In Grade 8, student must focus on the following main categories:

(1) Understanding linear relationship along with proportional relationship and numerical relationship. Concentrating more on arithmetic sequences, variables sequences etc.....formulating and reasoning about expressions and equations, including molding a combination in bivariate data with a linear equation, and solving linear equations and systems of linear equations. Clinching the concept of a function and using functions to describe quantitative relationships.

(2) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem. Few of the other concepts are : Geometric concepts, triangle inequality theorem, tessellations along with graphing equations (Linear equations or non-linear equations).

(3) Analyzing and measuring 3 dimensional figures and applying Pythagorean theorem and calculating measurements of figures using formulas for measurements. In addition to these, we are also going to learn scale drawing and proportions to convert to equivalent measurements.
(4) In the number and operation category of eight syllabus, I am going to discuss about problem involving percents along with rationalization techniques used with rational numbers and scientific notations and unit rate. Understanding and formulating the properties of real numbers, estimation of solutions, numbers, sequences etc...
(5) Finally we all are going to see the most interesting topic of eighth standard that is Probability and statistics where we formulate the dependency of events and calculating conditional probability. Some of the statistics analysis like mean, mode, median along with sampling techniques.

Now I am going to discuss about the most important topics of mathematics that are Algebra, Geometry and measurements, and Data analysis, and number and operations and algebra. So start with the first section that is Algebra.
Algebra :

In the algebra section what students going to use is linear functions, and systems of linear equations to represent and analyze the information provided and to solve a variety of problems. Students just recognize a proportion (Y/X = K, or Y = KX) as a special case of a linear equation having a form of

y = mx + b, understanding the properties that the constant of proportionality (K) is the slope and the resulting graph is a line through the origin. Students need to understand that Slope of a line is a concept which tells us how a straight line angles away from the horizontal or, we can say that it describes the steepness, incline or grade of the straight line. What students need to understand is that the slope (m) of a line is a constant rate of change.  Student recognize that tabular and graphical representations are usually only partial representation (translate among verbal, tabular, graphical, and algebraic representations of functions), and they describe how such conditions of a function as slope and Y intercept appear in distinct representations. Proceeding further students need to solve systems of two linear equations in two variables and relate the systems to pairs of lines that are parallel, or are the same lines in the plane. The overall result of the above discussion is that students need to use linear equations, systems of linear equations, linear functions, and their understanding of the slope of a line to analyze situations and solve problems.

Geometry and measurements:

In this section students deal with facts and figures. Geometry is an important area of mathematics which deals with the shape, size, relative position of figures, and the properties of space. It is all about shapes and their properties. Geometry is of two types : Plane geometry and solid geometry. Plane geometry deals with the shapes on a flat surface like lines, circles and triangles ... shapes that can be drawn on a piece of paper whereas Solid geometry is all about three dimensional objects like cubes, prisms and pyramids. Students need to use fundamental facts about distance and angles to describe and analyze figures and situations in two and three dimensional space and to solve problems, including those with multiple steps. What we are going to understand or prove is that particular configurations of lines give rise to similar triangles as the congruent angles created when a traversal cuts parallel lines. Students apply this statements for the similar triangles which helps them to solve a variety of problems. Some of the problems related to above statements are to find heights and distances. They must use the facts about the angles that are formed or formulated when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students also use Pythagorean theorem and explain why it is valid by using different methods. The Pythagorean Theorem states that in a right triangle the squares of the two short sides add to the square of the long side. If we call the lengths of the two short sides “a” and “b”, and the length of the long side “c” this leads to the familiar statement

c² = a² + b²

Students apply the Pythagorean theorem to calculate distances between points in the Cartesian coordinate plane to measure lengths and analyze polynomials and polyhedra.

Data analysis and number and operations and algebra :

Here we are going to discuss about probability, conditional probability, statistical analysis etc. Probability is a way of telling or expressing a knowledge that an event will occur or has occurred. The probability of an event occurring given that another event has already occurred is called a conditional probability. Statistics, on the other hand is the practice or science of collecting and analyzing numerical data in large quantities. It is basically a study of the collection, organization, analysis and interpretation of the data. So, the two things probability and statistics together play an important role in finding out measures of central value, measures of spread of different data and this helps in comparing of two data. Both probability and statistics are interrelated with each other and play an important role in analyzing the data. Students have to use descriptive statistics which comes loaded with concepts like mean, median, and range, to summarize and compare data sets. They also need to organize and display data to act and answer questions. Comparing the information comes out from the mean and the median and investigating the different effects that changes in data values have on these measures of center. Students need to understand the most important concept that a measure of center alone does not completely describe a data set because very different data set can share the same measure of center. Students have an option to select any of the two (mean or the median) as the appropriate measure of center for a given purpose.

Angles and triangles:

Here students need to use different ideas about distance and angles, and how they vary or behave under the following situations or conditions translations, rotations, reflections, and dilations, and ideas about congruence and similarity to describe and analyze two-dimensional figures and to solve problems. If we talk about concepts behind triangle then they need to show that the sum total of the angles in a triangle is the angle formed by a straight line. The various configurations of the lines play an important role to find out a similar triangles because of the angles created when a transversal cuts parallel lines. The another topic which students needs to work out is Volume by solving problems involving cones, cylinders and spheres. Now in next class we all going to practice each and every section of eighth grade mathematics syllabus. Starting from the Algebra section to the Probability section.

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.

How to tackle eighth standard Algebra

Friends we all know about Numbers and how to combine them using various operations like addition, subtraction, multiplication and division. This area of study is what we call as Arithmetic. The more advanced area of study of Algebra are distinct from arithmetic in which addition or basic operations to specific numbers involves entities, what we called as variables that have no particular value or we specify it as unknown value. Variables in common are generally denoted by upper or lower case letters. Some of the basic examples to show variables are “3a” , “x”, “ 7 + d” here a, x and d are variables having unknown values.

Let’s talk about algebraic expression in mathematics. An algebraic expressions is basically a collection of letters and numbers combined together to form an expression. These letters and numbers are combined together by the four basic arithmetic operations. Some of the examples of algebraic expressions are 7a, 7a + b, 7a – 4b, a / (a + b), a², (a + b)²
Here all the numbers used in algebraic expression are called constants and all the letters used in the above expressions like “a” and “b” are variables. If the expression comes with no variables then the algebraic expression is stated as arithmetic expressions. For example 4 + 7 / 6

Variables are generally used to explain the general situations or real time situations and they can also be used to solve problems that in anyway would be very difficult or even impossible to solve. Whenever we are going to solve algebraic expressions or any kind of word problems, we will see the use of these applications.

Now I am going to discuss about another topic which is very important that is an equation. In simplest mathematical manner we can say that an equation is an allegation in which two algebraic expressions used are equal. It is further stated in two different ways that are:
The first case: the given equation is true for all the values of the variables. In such kind of situation, equation is called an identity. The example to show an identity or we can say that values of variables which is true for the given equation. Most simple example is
x + y = y + x
It is also known as commutative law of addition. Another most common and well known identity is the first binomial formula or algebraic formulas like
(a + b)² = a² + 2ab + b²

The second case: In this situation the equation is true for some values of the variables. In such kind of problems or equations what we need to do is to identify those values of the variables for which the equation is true. This process is known as solving the equation. Moving forward we will study how to solve equations but for instance let’s take an example to understand the case in better manner
3a + 1 = 7
in this case obviously
a = 2 is the solution.

The given example is of a linear equation. We will further study about quadratic equations, but for know the example of a quadratic equation is a² – a – 2 = 0. If we are going to find out values for variables then, this equation has the two results that are:
a = -1 or a = 2

Now I am going in deep with the topic so the next topic is, How to evaluate an algebraic expression? To evaluate or to explore an algebraic expression refers to substitute or place specific values (desired values) for its variables. Let’s take a simple example of an algebraic expression to understand it better:
algebraic = 2a + 1

Now we are going to evaluate the given algebraic expressions. Now what we need to do is to try various values of variable which satisfy the given algebraic expression. Lets take a = 3, which provides us: algebraic = 2 x 3 + 1 = 7. we can say that the value of algebraic at a = 3 is 7. let’s take an example which is little tougher or  a bit complex then the above one
Now if we use above algebraic and put value of a = 2b + 1, where b is another variable which gives
Algebraic = 2 x (2b + 1) + 1 = 4b + 2 + 1 = 4b + 3
Now we can further solve this by substituting different values for variable b.

Before proceeding further towards inequalities let’s talk about equivalent expressions: Two expressions are equivalent if their values are equal for all possible evaluations of the two expressions. In other words presenting them with an equality sign between the expressions gives an identity.

Now I am going to talk about eighth grade linear equations topic:
In simple mathematical manner we can say that any equation that when graphed produces a straight line, then the equation is called as Linear Equation or we can say that any equation is a linear if it can be written in the linear form : ax = b. Here x is the variable, a and b are the constants. The common form of a linear equation in the two variables like x and y is
y = mx + b
where x and y are two variables and m and b are two constants.

The constant m determines the slope or gradient of that line and the constant b shows the point at which line crosses the Y-axis. Constant b is also known as Y-Intercept.
There are three possible solutions for the linear equations:
Unique Solutions: if a not equal to b then only possible solution: x = b/a
No Solution: If a = 0 and b is not equal to 0 then it has no solution.
Infinite Solutions or many solutions: if a = 0 and b = 0. In this solution 0x = 0 and there are infinite solutions for all the values of x.
Let’s take an example : 2a + 3 = a + 5
For solving this we need to subtract 3 from both the sides 2a = a + 2 that will result in a = 2.
The most important thing to understand is that, for solving a linear equation we need to recognize that the equation is linear or non linear and if linear then convert it to the simplest form. Lets take an example to elaborate it well:
(a – 2)(a – 3) = (a + 1)(a + 2)
what we need to do is to apply distributive property to both the sides of the equation
axa – ax3 – ax2 + 2x3 = axa + 2xa + 1xa + 1x2
on further simplifying this we get
a² – 5a + 6 = a² + 3a + 2
This is what we get, we can further simplify it by subtracting a² from both the sides.
Now after telling the basic concept behind linear equations, I am going to discuss about inequalities for class eighth. An inequality tells that two values are not equal. For example a ≠ b shows that a is not equal to b. The most important thing to understand is the use of inequality symbols. For solving inequalities we need to learn the symbols of inequalities like the symbol < means less than and the symbol > means greater than and the symbol ≤ less than or equal to etc.

In eighth standard we learn about linear inequality. So what linear inequality means is the first query comes in our mind. A linear inequality describes an area of the coordinate plane that has a boundary line. In simple way in linear inequalities everything on One side of a line on a graph. In mathematics a linear inequality is an inequality which involves a linear function.

The most important thing is to understand, how the inequality sign reverse when negative value comes. So to understand linear inequality, take an example: How to solve a compound inequality and graph the solutions?
Example: -6 < 2x - 4 < 12
-6 < 2x - 4 < 12
add 4 to all 3 parts
-2 < 2x <16
divide 2 from all 3 parts
-1 < x < 8
To graph the following equation, you put an open circle or we can say mark it by a dot on the point (-1,0) and then you put an open circle on the point (8,0).Then draw a line between the two.

We need to be careful while solving inequalities, as they are harder to solve than equations and require more attention. You can multiply an equation by a positive or negative number and get an equivalent equation. But while solving Inequalities remember this: when multiplying it by a negative number, you need to change sign of the inequality.