Linear equation: Basic guide to solving each equation

This chapter is most important part of mathematics where you get to solve numbers of equations. Maybe you are the one who is afraid of this subject, but once you get the basic, you will find it amazing than any other chapters. Through this article, you can solve any Linear equation. You will get certain techniques which will improve you in solving multiple problems. These techniques involve rewriting of problems mainly in the form of symbols and simple terms. For example:

“Find that number which can give you 11 when added with 7.”

It may be written in several ways, for example, 7+? = 11, or maybe 7+n = 11, or 7+x = 11

And so on, you can take any term based on your choice and selection, but if you are solving equations through your book, then you will find more accurately x in place of any symbols like “?” or any other symbol. The above example, i.e., 7+x = 11 is called to be the first-degree problem. You can recognize first-degree problems when equation which contains variable with a maximum exponent of 1. The term placed at the left is to be maintained in such a way that equal sign can be placed on the right side. Thus, in the equation, you can see left contains “7+x”, while right contains “11”, where equality is created with the variable x.

Solving equations

When you solve any first degree problems, you get to know that equation may be false or true, and it only depends on what choice you are making for the variable x. Like the equation: 7+x = 11 will be false if any other number except four is substituted with this variable. For this equation, four is called the solution of the problem and the variable. You can get solution only if you are substituting the number from left to right in place of the variable.

Example: Determine if 0 is the solution of the equation

14x – 3 = 4x – 3

In this equation, you can substitute 0 with the x to check if the equation is correct or not.

Substituting it, you get 14(0) – 3 = 4(0) – 3 which is -3 = -3 or 3 = 3, so the equation is correct

Some other example: 2x – 3 = 8x + 6, check if equation is correct for -3

To check it, again substitute x with -3, then you will get: 2(-3) – 3 = 8(-3) + 6, which is -6 – 3 = -24 + 6, that is, -9 = -18, or 9 = 18 (here, you can see that left hand site does not equate with right hand side) hence, equation is not correct for -3.

Now, if you want to check what solution is correct for the variable to equate both sides of linear equation, then you can do some steps. Firstly, you should be sure to transfer variables and numbers on the other sides, like 2x – 3 = 8x + 6, to 2x – 8x = 6+3, that is, -6x = 9, where you get x = -9/3 as the solution. (You see through this example that when numbers and variables get transferred, change their symbols).

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How to solve factoring polynomials? Learn three ways to do

Factoring is essential to simplify the algebraic expressions. This topic is very useful for the students as without understanding the basic of factorial polynomials one would not be able to understand the later topics. Many students often confused between the factors and term, they think both are same, but no they are not. In any equation, we can add or subtract a term whereas factors need to be multiplied. You can understand them through their definitions.

Definition of factor and term:

A term can be added or subtract whereas when we multiply a term, then it is called as the factor. If the entire term in an equation is in product form, then it is a factored form of an equation. A term may or may not contain factors whereas a factor is made of a term. We can change the term into factor form by changing the expression into product form. However, remember that terms or value doesn’t get changed only its form gets manipulated.

For example- 3x(x+y) is an expression in factored form and3x+4y+z is an expression in non-factored form.

How to remove common factors?

Here you will learn how to get common factors and greatest common factor. To learn, let us take an example.

 Example – Factor 2×2+4xy+8xy2

Solution- first we will take out the factors for each term

The factors of 2×2 are 1, 2, x, x2, 2x and 2x^2

The factors of 4xy are 1,2,4,x,y,2x,4x,y,2y,4y,2xy and so on

The factors of 8xy2 are1,2,4,8,x,y,2x,4x,8x,xy,xy^2 , 8xy^2 and so on

Take the greatest common factor from these. Out of these, the H.C.F is 2x. Now divide the whole term by this factor. You will get-

2×2+4xy+8xy2= 2x(x+2y+4y2)

Check the correctness by solving the equation on another side, and if the L.H.S is equal to R.H.S, then it is true.

How to find greatest common factor using grouping method?

This method is not very popular, but it is quite useful when used.

Example- 2x^2-x+8x-4

Group the first and second terms, and you will get the two groups as shown here,

= (2x^2-x) + (8x-4)

=X (2x-1) +4(2x-1)

The common factor is 2x-1, the final factor form is 2x^2-x+8x-4= (2x-1) (x+4). Note that this method doesn’t work every time, and hence you cannot use it to solve every problem.

How to factor quadratic polynomials?

The quadratic equation is nothing but a second-degree polynomial, so the highest degree of expression will be 2. When finding factoring polynomials, you need to find a term which when multiply gives the third term of expression and when added gives the second term. If this condition is not satisfied then obtained factors are incorrect. It is easy to find the factors for some equations whereas some complex quadratic expressions do not give factors. In such situation, we use the quadratic formula.

How to find factor for equations with the higher degree?

To solve such problems, take the common factor from expression and use the above methods to get the factor.

Example – 3×5+3×4-9×3+18×2

Solution- the common factor is 3×2, so we can write the expression as 3×2(x^3+x^2-3x+6) to solve further use any of the methods above.

Why do you need the Quadratic Formula Calculator?

The quadratic formula calculator has unique ways in calculating the equation.  It can solve the second order for the polynomial equation like ax2+bx+c=0   by the use of quadratic formula. The calculator solution shows the work by the use of the quadratic formula, and it solves entered the equation for complex and real roots. The calculator can determine if the discriminate is equal, greater or less to zero.

quadratic equation calculator
quadratic equation calculator

You can also find the quadratic formula calculator which shows how the formula has been used.  You can still use the graphing calculator if you want to solve a quadratic equation and even for these of the imaginary solution.

Because many people now want to work faster regardless of the field they are in, they choose to use an online calculator instead of counting manual. The calculator can solve the entire complicated problems faster and using efficient manner. The calculator will give accurate results compared to counting the equation manually. The calculator knows the formula to use, and it makes counting process easier. You can avoid boredom when you are getting the solution to the certain equation.

The calculator that was used before were limited, but this is no long the case since with technology and economy, the calculator are advanced, and people can get exact results for the problem they have. There are some functions like divide, multiply, minus which had been developed. The quadratic formula calculator is a complicated calculator, and it has been programmed to work on high-level equations.  The newest calculator is known as the scientific calculator, and it is being used in the companies and universities in order to ease the entire calculation process.

The quadratic formula calculator has anything you want in the calculator like binary functional, trigonometry, factorials, square root and the base 2.

When you use the calculator, you will enjoy the following benefits

It is convenient for people who want to do the complicated calculation by the use of the online scientific calculator.  Now you can work wherever you are as far as you can connect to the internet. You are able to use the calculator whenever you feel like it.

Easy

The online calculator may be used easily. There is the manual help with the help function that will guide you if you are not sure about the buttons that you can use in clicking to get to the calculation.

The user friendly interface: the calculator is not complicated and its interface is user friendly. The buttons have been arranged well as it happens with other normal scientific calculators.

The calculator can be used to perform both easy and complex calculation. The example is that you can use it so that you can calculate income tax benefits, property taxes or house loans.  It can be used by students or business people.

If you are not using the quadratic formula calculator, then you can use the quadratic formula which helps to solve quadratic equation and it is among the top five formulas found in math. Even if you should not be memorizing the formulas, this one should be memorized so that you can use it whenever you need it.

Functionality and Description of Math Calculators

A calculator is said to be a small electronic device that performs both involved and mathematical arithmetic operations. It’s also referred to as a machine that makes math operations easy to solve. These tasks are addition, subtraction, multiplication, and division. The advanced math calculators can do square roots, draw functions and can help in calculus. A computer or a smartphone is also known to be a calculator. A calculation calculator contributes to tackling mathematical problems.

Math Calculator
Math Calculator

Types of Math Calculators

Math calculators are found all over the internet and outside in the real world varying regarding cost. It’s crucial to know which math calculator you need to perform a certain job. There are three most common calculators;

  1. Handheld calculators:

Handheld calculators are the most basic calculator found in the world today. It’s simple to handle, pocket-sized, powered by the solar and used for basic mathematics only. The calculators come in attractive colors, designs and are often stylish. Basic calculators contain fewer functions and display results in just a line. The calculators are known as four function calculators since they do perform only addition, subtraction, multiplication and division only. We do have other calculators that can do basic functions such as tax, calculations and discount amounts. Handheld calculators can also copy values between the sub-display and the primary display, or independent actions can take place simultaneously.

  1. Printing calculators:

These calculators were used a long time before computers were created to run all the total numbers over a particular period. They feed off a role of tape to provide a record of the calculations made. They can print in two colors mainly black and red to represent positive and negative values form of identification. They are fast and known to keep up with most experienced Integer cruncher. Printing calculators are large; they are quick, and their display is better than the handheld calculators.

  1. Scientific calculators

They are calculators that offer a broad range of conversions, analysis, statistics and data plotting. The calculators are the most advanced calculators and provide a variety of functions to solve mathematical problems. Most scientific calculators used the VPAM method, a kind of notation for various algebraic functions. The calculator contains quite some math functions that offer mathematicians the ability to tackle complex tasks like fraction calculations, permutations, and combinations. They have some of the features listed below;

  • Dot-matrix display with a high-resolution screen that allows graphs and numbers to be spotted clearly.
  • 2-digit and 10-digit mantissa exponential display
  • A sophisticated memory that is quick and easily recalls previously done calculations
  • Re-execution and editing of formula
  • Editing of data input and back viewing.

Before the creation of math calculators, math was done using pencil, mind, and paper. The four function calculator has by time evolved to be a readily available calculator that meets our needs. They are those powerful calculators that feel the industry and academic needs which do different functions from a core and ordinary calculators. These advanced calculators can be programmed, and some have print out features.

Solving Quadratic Equations – Use Algebra Calculator

Algebra is one of the important parts of mathematics, and it is right to say it is an essential branch of mathematics. Algebra includes three common parts in their equation such as unknown parts which are known as variable, coefficient and number values. According to the degree available in the unknown part, the algebra is known as a different name such as a linear equation, polynomial equation, quadratic equation, cubic equation, exponential equation and much more. Thus, the algebra is a large part of mathematics. Here we are going to learn solving quadratic equations in different methods.

About quadratic equation

It is one of the algebraic equation which includes two factors or roots of unknown value. If learn simply, the standard format of quadratic equation is-

ax² + bx + c = 0

Here the standard format of a quadratic equation is visible where you can find variable (x), constant or number term (c) and coefficient of x²  and x i.e. a and b.

Now solving the quadratic equation is quite simple with the latest algebra calculators but slightly difficult if solved by formula methods. Different methods you can use to solve the quadratic equations in which formula method is explained below in the content.

Consider an example to learn how to solve quadratic equation by formula method

x² + 3x – 4 = 0

Formula to identify the unknown term is: -b ± √b²-4ac⁄2a

According to the standard format of quadratic equation a = 1, b = 3 and c = -4

Put these values on the formula to find the value of a variable. You will get,

x = -3 ± √9 – 4.1. (-4)⁄2.1

x = -3 ± √9+16⁄2

x = -3 ± √25⁄2

x = -3 ± 5⁄2

Now, you can consider mathematical signs separately to identify both values separately that can balance the whole equation. Follow steps given below-

x = -3 + 5⁄2

x = 2⁄2

x = 1

And,

x = – 3 – 5⁄2

x = – 8⁄2

x = – 4

Now, you have two separate values of variable, i.e. x=1 and-4

This is formula method to solve any quadratic equation. It is quite simple, but you have to perform a much mathematical operation like multiplication, division, subtraction, and addition too. Also, you use other methods for solving a quadratic equation. In the quadratic equation, there are two roots are available that can balance the whole equation. Consider the same example to learn the root method of solving quadratic equations.  x² + 3x – 4 = 0

Here a = 1, b = 3 and c = – 4

To find out the variable values, use particular formula as-

α + β = – b⁄a

α β = c⁄a

Put the values to get the values of α and β

α + β = – 3⁄1

α + β = – 3

And

α β = -4⁄1

α β = – 4

Square both the term of ∝+β to find out unknown values as

α² + β² + 2αβ = 9

Now,

(α – β) ² + 2αβ + 22 αβ = 9

(α – β) ² + 2αβ + 22αβ = 9

(α – β) ² + 4αβ = 9

Put the values of α β in the above equation

(α – β) ² + 4 * – 4 = 9

(α – β) ² – 16 = 9

(α – β) ² = 25

α – β = 5

 

Now add α – β and α + β

You will get

2α = 2

Then

α = 1

Put the value of ∝ in any above equation

You will get

β = -4

Hence, you have a choice to use different methods to solve a quadratic equation. On the other hand, you can choose the algebra calculator that can easily find out variable values and deliver results in fast. In the algebra calculator, you just need to enter the quadratic equitation on the required field in the calculator. Thus, enter one next tap to solve the equation. The calculator solves the equation with the genuine method, and there is no short answer you will get. Thus, solving quadratic equations is quite easy with the calculator.

How to find Factors of any Algebraic Equation with the help of Factor Calculator?

Factoring calculator is a simple technique to find out the factors of any number with a real-time value. The algebra is the important and crucial part of the mathematics where most of the students get scared to solve an algebraic equation with higher degree coefficient. Every student has to do various operations like addition, subtraction, multiplication, and division for any equation. Thus, you have an option like either uses Vedic mathematics method or choose factor calculator.

Let us consider an example, how to find out the factors as:

(x + 2) (x – 2)

x² + 5x – 6 = 0

x³ + 4x² + 12x + 12 = 0

These are some examples of an algebraic equation that required huge attention while finding their factors. It is an easy task to find out factors of any numbers but difficult when you are going to find factors of an algebraic equation.

Finding factors for linear equation, let us consider some examples as-

  1. 3x – 6 = 0
  2. 2x + 4x – 24 = 48
  3. 13x / 4 = 39

These are some linear equation whose factors you are finding. It is quite easy to find factors of linear equation. First transfer the number opposite to the equal sign. If there are more than two variables, then try to add them and transfer remaining part to the different side of the equation. You need to eliminate the coefficient term to find the factors of any linear equation. Read the content, to know the method that how to solve an algebraic equation solution-

For first equation:

  • 2x – 5

2x = +5

x=+5/2

  • 3x – 6 = 0

3x = 6

x = 6/3 = 2

For second equation

2x + 4x -24 = 48

6x – 24 = 48

6x = 72

x = 12

Now, the factor is 12

For third equation

13x / 4 = 39

x = 3*4

x = 12

It is very simple to find out factors for a linear equation. On the other hand, you feel difficulties while solving higher degree order of algebra equations. Consider a quadratic equation to find out factors by factoring calculator. Quadratic equation can be solved by the below standard formula as-

math

 

Put the values of a, b, and c in the formula to find out the factors of given algebraic or quadratic equation. For better understanding consider a quadratic equation as

x² + 5x + 6 = 0

Now, put the values of number, coefficient, and coefficient of now, you will get the figure as

math

Therefore, x = -2

By solving above equation, you can get the value for X. The major concern is that the student needs to perform various mathematical operations which are difficult to solve. Thus, you can find factoring calculator to find the factors of given equation. Here you do need to enter a question in the required field then you will get the right solution of any algebraic equation. Whether the equation is linear, quadratic or higher order degree, you can find its factors within seconds. You will get the explained solution or the short answer.

  • Using factoring calculator, you need to perform less mathematical operations.
  • Factoring calculator can easily solve your algebraic equation.
  • With the help of factoring calculator, you can directly put variables and get your desired answer.
  • Use digital algebraic calculator to find out factors

These are some benefits of using the factoring calculator to find out factors of any algebraic equation.

Solve for X – Easy Methods to get the Value of X

If you are looking for different ways via which you could solve for x, then you are welcome here. The content is meant for those students who face problems while solving radicals and exponents in algebra. After reading this content, you will get the right method to solve it. So, let’s start this learning session.

Basic linear equation

We are going to solve simple linear equation 5x – 8 = x + 4 and find out the value of X.

Step 1: At first, we will arrange 5x and x on the left-hand side and 4 and on the right-hand side.

5x – 8 = x + 4

Step 2: After that, we will solve the arrange number and here is the further procedure:

math
math

Important note: Before solving any equation, students should solve exponents if there is any to make the equation easy.

Solve it by isolating the terms (in case of exponents)

If you didn’t get it as what I meant by the word “isolating” then here is an explanation for you. It means that you should make one term disappear to get the value of X. The trick is quite popular plus makes the procedure easy. So, how to solve for x by this method?

Example: 2x² + 14 = 46

Step 1: In the above example, we are going to isolate term 14 and thus we ill subtract both the sides by 14.  Keep in mind that you should choose an opposite side for isolating a term, i.e., here 14 is positive and thus we have chosen (-14) in it.

2x² + 14 – 14 = 46 – 14

2x² = 32

Step 2: Now you see that number 14 has gone and now we can find the value of x easily. Here are its further steps:

math
math

Now, we know that 16 is the square of 4 and thus we will square root both the sides and get:

x = 4

Solving fractions by cross multiplication method 

Fractions are easy to solve, by using multiplication methods we can get the value of X. So, here is the example:

math
math

Step 1:  We will solve this by cross multiplication method where we will multiply 4) by 4 and 3 by 4.

(x – 4) (4) = (2) (4)

4x – 16 = 8

Step 2: Now we will combine the like terms like this:

4x = 8 + 16

4x = 24

X = 24 ⁄ 4

And finally, we get x = 6. The cross multiplication method is very easy but does not forget to reduce the problem into simplest form.

Solve for x using radical signs

For this one, you need to learn some basic concepts of radical sign or square roots. Now, let’s solve example

math
math

Step 1: In the first step, we will isolate the number 5 by adding 5 on both the sides like this:

math
math

Step 2: Now we are going to square both the sides:

math
math

Step 3: Again, we will subtract both the sides by 7 so that 2x remain on the LHS:

math
math

So, we have got the answer 9, and you can verify this answer by putting the value of x in the problem.

Hope so all the methods explained above let you solve for x easily and make your mathematics stronger.