Radical Equations in Grade XII

Hello friends, Previously we have discussed about probability worksheets and in this session we will learn about the radical equations and inequalities problems in grade XII of tamilnadu stateboard syllabus. Radical equations are the type of equations having square root symbol and some of the variables with them. So such equations are the equation of roots. We can perform addition, subtraction, multiplication, and division on such type of equations. To deal with the radical equations we have to perform squaring of the radical expressions and after that we have to solve it in the proper manner as we solve other types of algebraic expressions. For example (√54 + 2x) is a radical equation.
In the radical equations there are the concepts of the nth roots of the expressions in them. The variables are enclosed with root symbol of nth degree where this n could be 1, 2, 3, …, n. The variable which comes in the radical part is called as radicand of the expression. For example in the expression √(5x + 8), (5x + 8) is the radicand part. Radical inequalities are also the part of radical equations. In the similar way radical inequalities are those inequalities which also involve the radical symbol or they are in the square root symbol. Some of the radical equations can be given as:
√(2x + 6), √(7xy) +5, √(x² y²) + x*y, √(7xy > 21) = (x y + 21), √(4x + 4) + 6 <= 12, etc…
Now talking about inequalities, in the grade XII, the inequality level is too complicated. The inequalities are the type of algebraic equations which has some inequality sign with it, and radical inequality is the expressions that have radical symbol with it. In inequality, some facts are used which state that
If x < y that have the meaning that number x is less than that of number y.
If x > y that have the meaning that number a is greater than that of number y.
if a<=x>=b, then x is greater or equals to a and b is less or equals to x.
If x ≠ y have the meaning that x is not equal to the number y. We can perform several operations in the inequalities like addition, subtraction, division, and multiplication as like other algebraic equations.
When we solve the inequality problems in general way, we use some of the standards like after multiplying or dividing them we flip the sign of the inequality. We use the number lines to solve the inequality problems. Here we also learn about the radical inequalities. The examples of the inequalities can be given as:
- (x / 2) < 8, 4 x + 8 < 13, √(4x + 4) + 6 <= 12,  √(x + 4) + 12 = 6, etc.
To solve such type of inequality problems we have to use the pattern of solving for normal radical and inequality.

So this is all about radical equation. I have given a brief Idea of how to deal with radical equation in XII grade and if you want to study about other topics like quartiles and Triangle congruence relationships visit various websites on the internet.

Learn factoring of Two Degree Polynomials

Hello friends today we are going to discuss about factoring of two degree and three degree polynomials which you need to study in grade XII of gujarat secondary education board. In earlier class you must have gone through about the basics of polynomials and For more practice you can use properties worksheets . We are here to study about 2^nd degree and 3^rd degree polynomial. Now let us discuss about the 2^nd degree polynomial.
2^nd degree polynomial:
                                    The degree of polynomial represent the highest degree on non zero term which contain variable.
2^nd degree polynomial:
                                    It means its variable has highest power of 2.
                                   Example -2X²+2y²+9=0
We can do fallowing operations on 2^nd degree polynomial.
Addition
Subtraction
Multiplication To know more about Two Degree Polynomials click here,
Now let us discuss Addition of polynomial with the help of fallowing examples
Example: 2X²+2Y²+2Z² + 3X²-Y²-Z²
                       For solving this type of problem we just need to use the property of associative and distributive law which you have studied in junior classes.
(2X²+2Y²+2Z²)+( 3X²-Y²-Z²)
2X² +3X² +2Y²-Y² +2Z²-Z²
 (2X² +3X²)+ (2Y²-Y²)+(2Z²-Z²)
 5X²+Y²+Z²
  Same things we can apply for subtraction but for more clearance we see one example
Example: Subtract 2X²+4Y²+6Z² from 4X²-6Y²-4Z²
(4X²-6Y²-4Z²)-( 2X²+4Y²+6Z² )
4X²-2X²-6Y^2-4Y²-4Z^2-6Z²
2X²-10Y²-10Z²
 After addition and subtraction we move to multiplication by this process we can change the degree of polynomial. How to multiply we see by example given blow
2X²+2Y²+2Z² multiply by 3X²-Y²-Z²
(2X²+2Y²+2Z²)* (3X²-Y²-Z²)
(2X^2*3X²)+( 2Y² *-Y²)+( 2Z² *-Z²)
6X⁴-2Y⁴-2Z⁴
   As you have seen in the above example that degree of polynomial can be changed by the multiplication in the above problem firstly the degree of polynomial was 2 now it is changed to 4
   Division can be also done by the polynomial that you will be going to study in further class. Now I think we have well understood about 2^nd degree polynomial now let’s move to 3^rd degree of polynomial as it is more important for you .Now you came to know that if polynomial is of  3^rd degree then its highest power will be 3.Like 2^nd degree we can implement same operation on.
Factoring 3^rd degree polynomials:
All algebraic expressions one or more than variables that are called polynomial .it terms are not in negative exponent.
Polynomial in algebraic expression P(x) =a0+a1x+a2x2……..an-1 xn-1+anxn,   where a0, a1,…an.and are real number and n is non negative integer .and in polynomial x real degree of n.
A polynomial of degree is in three term like ax3+bx2+cx+d, a? 0 that is also called cubic polynomial.
Some step for third degree polynomials

1.      In numerical ,plug x=1,-1,-2,2 etc in P(x)

2.      If p (1) =0 then x-1 is factor of p(x) polynomial .

3.      If p(x) is a cubic ,then it divide it by x-1

4.      Now factorize quadratic polynomial.

Now lets the example of three degree

Ex: x3+7x2 +12x

Sol: let P(x) =x3+7x2 +12x

Let’s solve in simple form.

P(x) =x(x2+7x+12)

P(x) =x(x2+4x+3x+12)

P(x) =x(x(x+4) +3(x+4))

P(x) =x(x+4) (x+3)

This value is factor of this polynomial mean x=0, x=-4, x=-3
And this value is satisfying this polynomial...
: Let’s check my answer.
First you plug the value in given polynomial.
P(x) =x3+7x2+12x
Put x= -4
P (-4) = (-4)3+ 7(-4)2+ 12(-4)
P (-4) = -64+ 122 - 48
P (-4) =0
Here -4 values satisfy this polynomial. It means -4 is factor of this polemical.
In this way you can check your all value by this process.

In this section only these parts are important and You can also refer grade XI  blog for further reading on Polynomials in Grade XI.Read more maths topics of different grades such as Angles of triangles and polygons in the next session here.

Linear Equations and Inequalities

Hello friends today we are going to discuss about Linear equations, inequalities which you need to study in grade XII of every education board .As you all know about  solving linear equations and you all studied them in grade X and XI. Linear equation is a equation which have 1,2 ,3 or more than that unknown variables with some known variable.

Linear equation in one variable: This type of equation has only one solution or you can say unique solution.
Example: 3X+5=14
3X=9 or X=3
These equations are very simple to solve you just need to put x one side and other values on other side and u can easily get the desired result.
Linear equation with two variable: any equation is in the form of aX+bY+c=0 is called linear equation with two variable .where a,b,c are real numbers and X,Y are variables .For solving this type of  equations we need to have 2 equations because we are having two unknown variable in the form of  X,Y.
Example: 2X+3Y=15&3X+3Y=18
Now solving this type of equation we need to we need to add or subtract both the equation if sign are same then subtract them and if not same than just add them.
Now in above case we need to subtract on subtraction we will get X=3 and then put this value in above equation to find the value of y .than you will find the value of y is 3.
Note-in above example sign and number before variable are same so the problem solved easily but
if the number before the variable are not same than u need to multiply them by a constant. This constant you have to decide by your own.
Now we will discuss about the inequalities
Inequalities are basically linear equation type problem just the change is that you can change the sign according to inequalities but you can’t change sign in linear equation that is the basic difference between them. For solving inequalities you need to fallow fallowing steps
*Add or subtract both the number same side.
*On switching change the sign of inequalities.
*Multiply and Divide by same positive number
*if you Divide by negative number or multiply by positive number then you need to change the sign of inequalities. It will be more clear when we go through the examples.Know more about inequality here,

Example: Solve this inequality
                       5X+4>-10X+14
Step one  add or subtract same value
5X+4-4>-10X+19-4
5X>-10X+15
5X+10X>15
15X>15
X>1
In this way just fallowing simple steps we can solve inequalities problem easily.
Example: solve this inequality
4X<16
Divide by 4 each side you can easily solve this problem
X<4
Now the example given blow is taken from compound inequality .These are complicated problems .Let us see one  problem .
Example:3X+9>18 and 4X-7<-11
Now for that type of problem you just need to take both the problems one by one and them just combine them.
3X>18-9
3X>9
X>9
And
4X<-11+7
4X<4
X<1
Now answer of the problem will be the combination of both that will be X>9,X<1 by this you can say all the number above 9 and blow 1 are the solution of this equation.
We have discussed about linear equation and inequalities,Inequalities are little difficult to solve  so you need to practice a lot and to know answers of interesting questions like can a rational number be negative wait till next session and You can also refer grade XII  blog for further reading on Formal/informal proofs.Read more maths topics of different grades such as Correlation and causation in the next session here.

Limits at Infinity in Grade XII

Hello friends I think you have understood the last articles today we are going to study about limits and infinity which you need to study in calculus in grade XII.As you are now familiar with functions, range and domain they are very important to understand the concept of math because these are the basic requirements for studying limits. Now let’s see how to solve limits?
Limit of any function can be defined as the behavior of the function near a particular input. Let’s suppose a function f has the output f(x) to every input x and the function has limit L and an input p. If f(x) is close to L when f(x) is close to p. In simple words we can say as f(x) moves closer to L x moves closer to p. and if we talk about infinity we know that it is impossible to reach there but we still try to reach it. For more on limits visit this.
Limits at infinity is used to describe the behavior of the function with respect to its limit and also describes its behavior as the independent variable increase or decrease without any bound. In actual the value we get from the limit is not the exact value but the value we get is very close to real value or we can say tends to that value.
Limit has a wide application in the world of mathematics. (Also see Upper Limits) It can be applied to simplify the function and also used to find the value of the function. General notation of limit is given below

By the above statement we can say that as x approaches to infinity 1/x approaches to 0. Whenever you see limits you just think of approaching to some value. In mathematical language we can say we are not talking about x tending to infinity. We just know that as x gets bigger, value of the function approaches to 0
Let’s see one example of limit approaching to infinity
y=3x
As  x=1  y=3
x=2 y=6
x=3 y=9
x=4 y=12
and so on
x=100  y=3000
Now as x approaches to infinity y approaches to infinity as well. Now we will see one more example
2x² -5x
2x² will always tend towards infinity and -5x always tends towards minus infinity so if x will increase where will the function tends?
It will always depend on the value of if x² will grow more rapidly with respect to x as x increases then the function will surely tend towards the positive infinity.
Now let’s talk about the degree of the function. The degree of the function can be defined as the highest power of variable for example
5x²+6x+7
In this x has highest power 2 so degree of the function will be 2.
By looking at degree of the function we can tell that limit will be positive or negative.
If degree of the function is greater than 0 then limit will always be positive.
If degree of the function is less than 0 then the limit will be 0
This is all about limits at infinity. I hope you have understood all the things in limit at infinity

In upcoming posts we will discuss about Linear Equations and Inequalities and standard deviation of normally distributed random variable. Visit our website for information on 12th biology syllabus Maharashtra board

Patterns and Functions in Grade XII

Hello friends as I believe you have understood the previous topics. Today we are going to discuss about Patterns, relations, and functions which you have to face in grade XII. As you have heard these terms in your previous classes but today we will discuss them in detail. Firstly we will learn about pattern.

You all are familiar with the term algebra pattern, here we are to discuss about the number pattern especially in series. It will be well understood by an example.

Find the value of X in given series

20, 42 ,66, 92, X, 150, 182

For this type of series you just need to check the pattern and if you found the pattern then it is very easy to solve. Now for above problem

42 is greater than 20 by 22

66 is greater than 42 by 24

92 is greater than 66 by 26

Now you can check that the difference between the number is also in a pattern 22,24,26 and so on so next number in difference series will be 28 and X will be 92+28=120, and you solved the problem so you just need to find pattern then it is very easy. Now move to another example and find the advance pattern.

A group of lecturer were at a conference meeting. Every lecturer exchanges his class record with each additional lecturer who were there. If there were 11 lecturers how many class record were exchanged?

To solve this type of problem we just need to focus on number of class note exchanged so class notes will be exchanged=10+9+8+7+6+5+4+3+2+1=55

Number of class notes exchanged=55*2=110

This problem is little hard to understand but I don’t think it’s tough you just need to practice. Now let's move to function and relation.

If we denote relation by R. If R is the relation from set (C,D) and (C,D) belongs to R then we can say C is related to D under the relation of R. There are many types of relations which are as follows

identity Relations

The relation IB’on a set B is called identity relation on B, if IB = (x, y) | x = y

So it will be in identity only when x=y

Universal Relation

If a relation R is A*A than R in universal relation with A

Void Relation

As we know that null set is the sub set of every set so we can say that null set has a void relation with every set.

Reflexive relation

A relation R on a set B is called reflexive if (n, n) belongs to n in R for all n belongs to B

Symmetric relation

if any set (x,y) belong to R than (y,x ) will also belong to R

Transitive relation

If in a group two parameters are common and they are in a relation R then the other 2 parameters are in transtive relation

Now we will talk about functions

There are mainly 3 types of function

1. Injective Function

If we find the value of function at two parameters and value comes the same then it is called injective function

2. Surjective function

A function is said to be surjective only if its range lies within the domain

3. Bijective Function

A function is said to bijective only if it has the properties of injective and surjective. Also see Average Value of a Function.

This is all about patterns, relations, and functions and I hope that this article will help you in understanding this topic.

In upcoming posts we will discuss about Limits at Infinity in Grade XII and Line of best fit - least squares regression. Visit our website for information on 12th biology Maharashtra board syllabus

Exponents in on Grade XII

In previous article, one of the important topics of mathematics that is Quadratic functions was explored and in today's article we are going to move further in grade XII to learn Algebra online syllabus with Exponents and exponential functions. This topic has very short introduction of itself but various situations and applications of it, tends it to cover a wide portion of Algebra unit.
Let us first start with the general description of exponent after that we will move to the main topic of Exponential functions with their growth and decay property. Before coming into 12 standard students learn almost all the aspects of logarithm and in respect of those exponents as well. Whenever an exponent is described it is clear that presence of Euler constant 'e' is surely there in the function.  The main reason to use exponent in any function is that the corresponding constant change in any variable in parallel with other independent variable is measured with the help of graph. For more on exponents visit here.
If any function is described as f(x) = y^p then 'y' is called base and 'p' is its relative exponent. There are some pre-defined properties for exponential function as follows:
Suppose two integer variables are m and n, then
   • yⁿ = y × y × · · · × y with n factors
   • y^1/n means the nth root of a. That is, a1/n is that positive number which satisfies
                                 (y^1/n ) × (y^1/n )) × · · · × (y^1/n ) = y
       Where there are n factors on the left hand side.
   • y^m/n = (y^1/n ) × (y^1/n ) × · · · × (y1/n ) (y^1/n )
   • y⁻ⁿ = 1/ yⁿ
    
The above situations are used when x is an integer but if ‘x’ is a real number then:
Suppose ‘p’ is a rational number
                                
                               p=m/n;          here m, n are integers
                               
Now assume ‘x’ as a real number that cannot be written in a form of rational number. Two common examples of this type or irrational numbers are x = √2 and x = π.
For other possible real numbers that can be easily converted into fraction of rational form use decimal to fraction conversion for example:
                                     x = 3.914712334317 . . .
Then, if we are working to 3 decimal points we would write
                                            x ≈ 3.915
and this number can certainly be expressed as a rational number:                                                
                                       x ≈ 3.915 = 3915/1000                                                  
So, in this case                                                          
                                 y^x = y^3.914712... ≈ y^3.915 = y ^3915/1000
                 
and the final term: y ^3915/1000 can be determined in the usual way by using your calculator.
This is all about Exponents; now let us move towards the next part of this session that includes the description of Exponential functions.
Exponential Functions: - suppose a function is as f(x) = y^x
Now for this function the value of ‘x’ clearly varies according to a fixed base value and its appropriate exponent value.
So the functions (y^x) (as different values are chosen for a) are called exponential functions. Let us explain in more practical manner:
                 If y>z> 0 then    y^x > z^x       only if x>0    
And if x<0 then         y^x < z^x                   
Euler constant ’e’ is the most important and widely used exponential function whose particular base value is 2.7182818 . . . ,
                                          e = 2.7182818 . . .
The number ‘e’ has its own importance in mathematics like number π and, like π it is also an irrational. So ‘e’ also cannot be written as the quotient of two integers. The value of ‘e’ is stored in most of the science calculators to sort out exponential function.
There are various ways of calculating the value of ‘e’. For example, ‘e’ represents the end point of sequence of numbers as follows:
(2/1)¹ . (3/2)² . (4/3)³ . ......  (18/17)¹⁷.....  ..  (65/64)⁶⁴.....
Decimal form of these ( up to 6 decimal points) is as:          2.000000,   2.250000,    2.370370,    ...,   2.637929,     . . . , 2.697345,    ...
This slowly converging unit can represent as single unit as ‘e’s expression as:
e= Lim _{m-> infinity} [( m + 1) /m]^m
There is one more way to calculate the value of ‘e’ and that is infinite series:
e = 1 + 1/1! + ½! + 1/3! + ¼! .........+ 1/m! + .....
Here students know that
                         m! = m × (m − 1) × (m − 2) × . . . (3) × (2) × (1)
According to the standard analysis it can be shown that first eleven terms of this series represents the value of ‘e’ with almost negligible error of 3 × 10⁻⁸.
In general all functions of the form of a^x are called exponential functions but still to be more specific we refer to e^x as
The exponential function.
 Exponential growth:-
If s > 1 then it can be easily represented no matter how large K is:                                  
                                      s^x/x^k   → ∞ as x → ∞
It means if K is a fixed value and can be chosen as large as wanted, still if the value of ‘x’ increases then s^x will also increase and being ahead of it always. It will happen until the value of‘s’ is greater than 1. This phenomenon of s^x growth as x increases is called exponential growth.
 Exponential decay
As explained in previous point that the behavior of ‘e^x’ as x → ∞ is known as exponential growth. Just in similar passion,
 if we explain or characterize the expansion of e^–x then that procedure will called as Exponential decay.
This is all about Exponential functions and their relative properties. As the definition of exponential function defines that in it we observe the relative change in any variable in respect of any other variable. This variation may occur in two different styles as Direct and Indirect.
Let us first talk about the direct variations in exponential function. If two variables are said to be direct variant of each other then it means that if one of them get doubles then other one also changes in same ratio, similarly if the second one is reduced in some amount then first one is also decreased in same ratio. For example: assume a function as y = x + 1
If value of ‘x’ grows up then at the same time value of ‘y’ also gets increased.
Now let’s see “what is Inverse variation?”, this type of variation causes opposite change in any value whenever the related value has increased or decreased. For example, if x and y are two values that are related to each other with Inverse relation then if value if x increases as it doubles then at the same time the value of y gets decreased by half of its value. This type of change gets complicated when more than two independent variables are included in the function. Let us take example of Newton’s kepler theorem:
Formula is as F = Gm/r²
So it represents that whenever mass of the body gets doubled then the strength of force also gets doubled and when distance is increased then the strength of the force gets fall in its fourth.
The thing is that both variations: direct and Inverse, are occurring together so determination of the value gets complicated.
This is all about Exponents, Exponential function with their growth and decay functionality, and types of variations in exponents; all of these grade XII Algebra topics comes as a single chapter in your syllabus because there queries solution requires implementation of their common properties, which are explored in detail in this article.
In the successive article of this one, we will discuss the next included algebra topic in grade XII syllabus and that is co-relation of data.
After reading this article still there is any issue or doubt remains in your mind or you want to solve related math queries of it, and then access the online math tutoring websites where students like you can ask their queries and surely will have immediate assistance by online math tutor. These math education websites provides proper interactive session to students so that they can ask their problems on the regular basis to be clear in every math concept. Students can use some available options like online chat, relevant study material, video seminars, unsolved and solved math worksheets, online math calculators for quick solution of the problem, and many more. All these options increase the ease level of the students’ study that causes their better involvement in studies.
Grade XII math syllabus is comparatively vast than any other standard and the main difficulty in it is that all the topics of it usually have the math application problems whose introductory part was studied by you guys in your previous classes. So always learn every math topic with your pure attention to execute it with ease in the final examination because math is a kind of practical subject that can enrich your aggregate percentage when potentially studied otherwise it also can ruin your overall performance in other subjects.

In upcoming posts we will discuss about Patterns and Functions in Grade XII and Standard distributions. Visit our website for information on 12th physics syllabus Maharashtra board

Quadratic Equations in grade XII

In previous article of Algebra I, we had already explained the fundamentals of Rational expressions and polynomials, today in this article we are going to continue with another topic of algebra 1 that is Quadratic equations and quadratic functions. The introductory part of quadratic equation was learned by students in their previous classes but still to make you guys remember it, let us start with the basics concept of Quadratic equations.
As we all know that every algebraic equation is a polynomial function and so on Quadratic equation is also, but the point is that every quadratic equation is a non-linear polynomial function of second order and it is in the form of monomial because quadratic equation has many derivatives but all of them are of a single variable, Suppose x is an unknown variable then its quadratic equation could be of the form:
Ax² + Bx + C = 0, here A, B, and C are integer constants. One compulsion to have this form as quadratic is that “A is not equal to zero”, if it does then the equation is said to be linear not quadratic.
For example:x² + 2x = 4 is said to be quadratic because it can be written in the form of Ax² + Bx + C = 0,
Let us take another example;
Suppose given equation is as 2x² – 1/x + 2 = 0
Here 1/x can be rewritten as x^-1, and quadratic can’t have x raised to the power other then 1 or 2. So the equation is not quadratic.
Now let us come to the main point and that is “ how to solve any quadratic equation?”, in previous article you guys have learned the factoring procedure and in between that session it was nicely explained that factoring can be used to solve any algebraic equation, so quadratic equation is also an algebraic form.  Let’s see how factoring is being applied to evaluate quadratic equation solutions.
There are 3 simple steps need to be executed to sort out the quadratic equation:
1.    First, write the given equation into standard form and for this you can use distributive property and additional property to combine like terms and put them into one side of ‘=’ sign.
2.    Now factor the obtained equation by using simple factoring technique.
3.    Now evaluate the possible values of variable by putting each factor equals to zero according to Zero product law.

Let us take an example and observe the actual execution of above defined steps:
(6x + 8)x = 14
Apply distributive property
6x² + 8x = 14
Imply addition property
6x² + 8x – 14 = 0 ( in standard form)
Now apply the factoring technique as:
6x² – 14x + 6x -14 = 0
Now take the common factor out
6x( x-1) – 14 ( x-1) = 0
(x - 1) (-6x + 14) = 0
Use the zero product law
 (x - 1) = 0 or (-6x + 14) = 0
x = 1 or x = -14/6 = -7/3

This is the specified way to sort out quadratic equation by using factorization. There is one more way to sort out these types of equations and that is Extracting square roots. After extracting square roots equation automatically comes into the correct form, for example:
( 2x-5 )2 + 5 = 3
Use additional; property to add -5 both sides of the ‘=’ sign
( 2x-5 )2 = -2
Now extract square roots
±√(2x-5)² = ±√(-2) (negative and positive sign both are included when square root of any value is taken)
(2x – 5) = ±i√ 2 ( simplify the radical and apply the definition of ‘I’)
Apply addition property of equality
2x = 5 + ±i√ 2
Now use division property of equality
x = (5 + ±i√ 2)/2
There are two more ways to evaluate any quadratic equation, one of them is “completing the square” and other is “ quadratic formula”. We will discuss both of them but let us start with Completing the square method.
Let’s take an example and in parallel of its solution we will explain the implementation of this method,
Given equation is as: 3x² + 4x – 7 = 0
First step: use addition equality property and isolate the variable terms like x² and x, at one side of the equation.
 3x² + 4x = 7
Second step: use division equality property and divide each term on both sides by integer coefficient of term ‘x²’.
3x²/ 3 + 4x/3 = 7/3
x² + (4/3)x = 7/3
Third step: now take coefficient of term x, square it and add in both sides of equation; here X’s coefficient is 4/3 so its ½ is 2/3, now square of (2/3) is added to both sides of the equation:
x² + (4/3)x + (2/3)² = 7/3 + (2/3)²
x²+ (4/3)x + (4/9) = 7/3 + (4/9)
Multiply 7/3 with 3/3 to have the same denominator on RHS
x² + (4/3)x + (4/9) = 21/9 + (4/9)
x² + (4/3)x + (4/9) = 25/9
Next step: factor the left side by using the standard formula:
( a² + 2ab + b²) = (a + b )²
So , x² + (4/3)x + (4/9) = 25/9
(x + 2/3 )² = 25/9
Next step: use; extract the square root technique for finding the relative roots of the equation
√(x + 2/3 )² =±√( 25/9)
x + 2/3 = ± 5/3
Again use addition property of equality
x = -2/3  ± 5/3
Now two equations can be formed as
x = -2/3  + 5/3  OR x = -2/3  - 5/3
By solving these equations
x = 1 or x = -7/3
This method is used to solve complex problems and for some easy problems prefer the previous method, which is much easier than this one.
Now it’s time to talk about the standard quadratic formula and its use to evaluate Quadratic function. If quadratic equation is in the form of
Ax² + Bx + C = 0 then the standard quadratic formula is represented as:
; here (b² – 4ac) is called determinant of the function, which decides the behaviors of the roots of quadratic equation.
While using this formula, the only task for students is to convert the given equation into standard form, for example:
Given equation is: (y + 3)² = ( y – 2)
Rewrite the above equation as its factors:
( y + 3) ( y + 3) = (y - 2)
Now use the standard distributive property to combine like terms:
y² + 6y + 9 = y – 2
Use addition equality property:
y² + 6y + 9 – y + 2 = 0
y² + 5y + 11 = 0
Now the equation is in standard form so we can implement the quadratic formula, first grab the values of a, b and c from the equation.
a = 1, b = 5 and c = 11.
Put this value in the formula


; by using the ‘i’ definition roots can be easily determined.
This is all about Quadratic equations and functions. In this article we have explored all the possible ways of solving any quadratic equation. These four ways of evaluating quadratic equation covers the topic of quadratic function in grade XII math. Students of this class most of the times fixed in troubles because there are several ways in which a similar problem can be asked and every time there is a bit of changes in the solution process, so it is hard to memorize all the ways and forms of quadratic equation. This task only can be accomplished by doing practice as much they can and to help students in it Online math tutor websites do their role by giving immediate assistance and various worksheets to solve.  If any kind of doubt stays in student's mind related to any topic then the whole fundamentals gets messed up, so it must be required that student asks his queries on regular basis and this will happen only when student does not hesitate in front of tutor which most often happen. To troubleshoot this problem, Online services uses online Chatting option via text or video, to build a better conversation between students and tutors, and because most of us are friendly with Internet platform, the access of this service gets easier.

In grade XII math content Algebra is covering most of the problem region so it gets more essential to learn its each and every problem with all the possible ways of query presentation. For having the detailed descriptions of other topics of Grade XII math syllabus, keep following the successive articles that will surely add a wide range of knowledge.

In upcoming posts we will discuss about Exponents in on Grade XII and Methods of data representation. Visit our website for information on Maharashtra state board