Learn Factor polynomials through some of the guidelines

Factoring is considered to be one of the essential processes to simplify many algebraic expressions, and it is a handy tool for solving any higher degree equations. In fact, factoring process is so important that little of algebra beyond this point get accomplished without understanding this phenomenon. In earlier chapters, you might have discovered the distinctions between the terms and factors which have been very much stressed. Factor polynomials are necessary and you might have wondered that terms refer to addition or subtraction and factors are multiplied or divided. Three of the most important form is as follows:

•        Terms that occur in indicated sum or differences

•        Factors that occur at an indicated product

•        An expression that is in factored form when an entire expression is in indicated product

Sometimes you don’t get to know if the form is factored or not in factored form, in such troubles, recall these examples: 6x(x + 3y), (2x + 3y)(4x – 2y), and (4x + 1)(x2 + 6x – 8) are some examples of factored form while 2x + 3y + z, 2(x + y) +z, and (x + y)(2x – y) + 5 are some example of non factored form. Note through these examples that you must always regard the entire expression. Some factors made up of terms that can contain factors but factored forms are mainly those that must confront with the above examples. Factoring is a process that helps in changing expression from terms like sum or differences to a product of factors.

Remove common factors:

Objectives are to determine the factors that are common to all terms of expression and factor common factors. To learn about this method, go through this example-

Example: factor 2x^2 + 6xy + 8xy^2

Solution: first, look for the factors of each term: 2x^2 comprises with the factors 1, 2, x, x^2, 2x, and 2x^2, 6xy with 1, 2, 3, 6, x, 2x, 3x, 6x, y, 6xy, 6y, and 8xy^2 with factors 1, 2, 4, 8, x, 2x, 4x, and so on. Here, you can see that 2x is common in all the factors. Now, take out this common from the expression and proceed by placing 2x to the set of parentheses.

      2x (             )

Within the term, place the parentheses by dividing the whole expression by the 2x, then it will give expression to be: 2x (x + 3y + 4y^2)

Note: this is based on a distributive property. So it is the reverse of the processes you have learned until now. You may think factored form to be 2(x^2 + 3xy + 4xy^2) which is wrong as it is not fully factored. You should be sure to factor expressions completely which is why it is best to choose the largest common factor. Many students don’t get greatest common factor because they are not careful. A factor is that part where you need to practice a lot. Find out examples and questions from exercise to be perfect in this region. Factor polynomials are considered to be one of the best ways to get perfect in solving any equation based expressions.

Learn the ways to solve linear equations

What is a linear equation? How to solve it? Here, in this article students will learn about linear equations and ways to solve it. The linear equation is a simple plain equation that contains simple variables with no square roots or fraction. It contains variables like x rather than complicated variables like x² solving them is simple and you can solve it in the same way you would solve the simple addition and subtraction.  In these equations instead of direct value, a variable is given. You have to find its value and then you will get the result.

However, a linear equation can vary from simple to more complicated expressions that take time to solve. More advanced methods are thus used for solving equations. To solve more complicated problems, students can use equation solver as well. Learn how to solve the simpler problems before we begin solving more complex problems.

Let’s learn through examples:

Example 1- solve equation x-9=3

Solution:

Here, the variable is x whose value we have to find. To solve for x, we need to eliminate 9 from x to find its value. To undo it, let’s add 9 to one side. Since the equation should be balanced on both sides, so add 9 to another side as well to make the balance. The equation will look something like this:

(X-9)+9=3+9

X-9+9= 12

Plus 9 and minus 9 will cancel each other, and thus we will get the value of x. So, x=12. This is called as addition-subtraction property.

Note- always remember that whatever we add or subtract at one side the same goes on another side to balance the linear equations. If you don’t add/subtract the same number to both the sides, then you will not get an answer.

You can also check whether the answer is correct or not. To check put the value of x in the equation.

X=12, on putting the value the equation will become

L.H.S:  12-9=3, R.H.S: 3, thus L.H.S=R.H.S

Value at both the sides is same; it means the answer is correct. If you get different values that means your answer is incorrect.

Some linear equations contain variables on both the sides. Let us learn how to solve such equations by taking next examples.

Example 2- 2x-4=x-6

Solution:

There are two ways to solve the above equation.

The first method - First, keep the numbers with x on one side and numbers without variable on another side by changing their signs. The equation will look like this:

                                                                                  2x-x=-6+4

Solving it will give this answer X=-2. To check it put the value of x in the original equation.

2(-2)-4= (-2)-6

-8=-8

Thus, a value on L.H.S and R.H.S are same. Hence the solution is correct.

The second method- you can solve the above problem using another method as well. Add 4 to both the sides to make the equation balance.

2x-4+4=x-6+4

2x=x-2

2x-x=-2

x=-2

The value of x is same as that we got from the first method. You can use the substitution method to check the correctness of the solution.

To solve more complex problems, you can take help of the equation solver as well. 

Different Solutions you can get with Fraction Calculator

With the fraction calculator, you can add, divide, multiply or subtract the fraction. You can reduce the fraction to reach to the lowest terms, to simplify them and to compare them with other fractions. You are able to convert the fractions into the decimal or percentage and to work with the mixed numbers and the improper fraction. You can solve to get x in the fraction if you are working with equations.

Fraction operation with manipulation

Fraction calculator: offers operation at the proper or improper fraction. They have the formulas used for dividing the fraction, multiplying, subtracting and adding

Subtracting and adding the fraction: you can subtract or add 10 fractions at once, and you can see the work in finding the right answers.

Mixed numbers: the calculator can be used to work with mixed numbers, fractions, and integers, the operation of the whole numbers, mixed numbers, integers with an improper or proper fraction. 

Mixed fraction are done in the same way as mixed numbers

Simplifying the fractions; you can convert the improper fraction into the mixed numbers. You may simplify the improper or proper fraction, and you can get the answer in the mixed numbers or in fraction.

Simplifying of the complex fraction calculator; you may simplify the fraction given numerators with the denominators to get two mixed numbers with a mixed fraction. You can also get a regular fraction with integers and fractions. 

Complex fraction calculator: you can divide, multiply, subtract and add the mixed numbers, integers and fraction with the calculator.

Decimal to the fraction: you can convert the decimal to be a fraction

Fraction to the decimal: you can convert the fraction to be a decimal

Percent into fraction: you can convert the percent to become a fraction

Least common denominator or LCD:  you may find the LCD of the mixed numbers, integers and fraction numbers. You can get the fraction equivalent with the LCD.

Greatest common factor or GCF: this is to find the GCF of the set numbers which shows the work by the use of prime factorization, factoring and Euclid’s algorithm.

Ratio calculator: you can use it to solve the problem of the proportion problems. 

Ratio to a fraction: you can convert ratio into fraction if you enter part to part and part to the whole ratio in order to find the equivalent of the fraction.

Ratio simplifier: you can simplify or reduce the ratio into the form of A into B with the work steps.

Equivalent fraction: you can generate the set of the fraction with the equivalent to the given fraction, integer, and mixed numbers.

The improper fraction with the mixed numbers: you can convert an improper fraction into the mixed numbers, and then you can see the work involved into the conversion. You can simplify the fraction, and you can reduce them to reach to the lowest term.

 Improper fraction to the mixed numbers: you can convert improper fraction in mixed numbers, and you can see which work is involved in the conversion. You may need simply the fraction in order to reduce it to the lowest terms

Mixed numbers with improper fraction: you can convert the mixed numbers to the improper fraction, and you may see which work is involved in the conversion.

Why Is Algebra So Important?

Algebra, often called as the gatekeeper subject, is not worthless as many people might think. It is utilized by experts going from circuit repairmen to engineers to PC researchers. It is no exactly a common right, says Robert Moses, author of the Algebra Project, which advocates for math proficiency in state-funded schools.

Algebra is basically like a game in which we follow certain rules, and we have to use logic. For example, if we take equation solver calculator, it will do the task within a second but if we want to learn how is it done, that would a more appreciable. Quick Math brings various online calculators that help the students to complete their homework nice and quick. But the good thing is that you can also learn the techniques and methods in easiest of ways.

Essential Algebra math is the first in a progression of larger amount math classes understudies need to succeed in school and life. Since numerous understudies neglect to build up a strong math establishment, a disturbing number of them graduate from secondary school not ready for school or work. Numerous end up taking medicinal math in school, which makes getting a degree a more drawn out, costlier procedure than it is for their more arranged schoolmates. Furthermore, it means they're more averse to finish a school level math course. For center scholars and their folks, the message is clear: It's less demanding to take in the math now than to relearn it later.

The main year of variable based math is an essential for all more elevated amount math: geometry, algebra math II, trigonometry, and analytics. As per a study by the instructive not-for-profit ACT, understudies who take variable based math I, geometry, algebra math II, and one extra abnormal state math course are considerably more liable to do well in school math.

Algebra math is not only for the school destined. Indeed, even secondary school graduates set out straight toward the work power require the same math aptitudes as first-year school recruits, the ACT found. This study took a gander at occupations that don't require a professional education yet pay compensation sufficiently high to bolster a group of four. Specialists found that math and perusing abilities required to fill in as a circuit tester, handyman, or upholsterer were practically identical to those expected to succeed in school.

Algebra math is, to put it plainly, the passage to accomplishment in the 21st century. Besides, understudies make the move from solid number juggling to the typical dialect of variable based math; they create unique thinking abilities important to exceed expectations in math and science.

Understudies normally take algebra math in eighth or ninth grade. The advantage of contemplating it in eighth grade is that if your youngster takes the PSAT as a secondary school sophomore, she will have finished geometry. When she's prepared to take the SAT or ACT as a lesser, she will have finished algebra math II, which is secured in both of these school affirmations tests. Visit quickmath.com to get access to online equation solver and calculators. 

3 Easy Methods for Solving Quadratic Equations

Quadratic formula and the equations are an essential part of algebra. The equations can be solved by using four different methods. Yes there are quadratic equation calculators are available, but we don’t have access to such calculators all the time.  So, let’s discuss these methods one by one with an example.

Factoring method

The method requires a lot of practice as in it students need time to understand this method properly. In it, we solve expression 3x² - 11x - 4 = 0

Step 1: In the first step, we need to multiply 3 and 4 and then find out the LCM of its result. After multiplication, we get 12 and LCM 2, 2, 3 and 1. Now, there is a trick to solve the equation as you need to multiple these obtained numbers in such a way that it results in 11 when subtracted and 12 when multiplied.

3x² - 11x - 4 = 0

3x² - 12x + x - 4 = 0

In the above step, we have 12x and x in the middle of the equation. When we solve (-12x) and (-x), then we get (-11x). Just like that, we will get 12x² after multiplying 3x² and (4).

Step 2:  In the second step, we take 3x common and from the second half part 1 as a common number. Here what it looks like?

3x(x - 4) + 1(x - 4) = 0

(3x + 1) (x - 4) = 0

Step 3: In the last step, we need we will equate (3x + 1) and (x - 4) to 0. The two answers for variable X are (- 1⁄3) and 4.

Completing the square

The method is quite tricky however easy at the same time. We will solve equation

2x² - x – 1 = 0

By this method:

Step 1: At first we will divide the equation by 2 from both the sides and then the equation looks like this:

2x² - x – 1 = 0

Step 2: In next step, we need to add 1/2   from both the sides and this equation will be like this:  

x² - x⁄2 - 1⁄2 = 0

Step 3: In the above step, two numbers get canceled, i.e.,  after which the equation will reduce to:

x² - 1x⁄2 = 1⁄2

Step 4: In the fourth step, we will multiply the two numbers of the equation, i.e.

(1⁄2 * 1⁄2)² which results in in1/16.

x² - 1x⁄2 + 1⁄16 = 1⁄2 + 1⁄16

Step 5: Now, we simplify the RHS as it is an expansion of formula while you have to factorize the LHS.

(x - 1⁄4)² = 9⁄16

Step 6: Now in the last step, you need to square root both the sides to get the value of x. After that we will get:

x = 1⁄2 and 1.

Quadratic Formula

The easiest among all methods as just put the equation values in the formula to get answers. Here is the quadratic formula:

x = -b ± √b²-4ac⁄2a

We will solve expression 2x² - x -1 = 0, after putting values in the formula we will solve it in the following way:

x = - (-1) ± √ (-1)²-4(2) (-1) ⁄2 (-1)

Here the value of a = 2, b is (-1), and c is also (-1), make sure to put right sings along with the number. Sings play an important role as one single wrong sign will lead to wrong answers.

x = 1 ± √9⁄4

After which we will get two values of:

x = 1 ± 3⁄4

And we will get 2 values, i.e., x = 1 and x = (-1⁄2)

Along with this, you can even use quadratic formula calculator available on the internet. It is just like a calculator where you need to put numbers and get direct answers.

Learn and Use Quadratic Formula with an Example

Mathematics is quite tricky but an interesting subject.  It is full of equations and calculations that have much significant importance in different fields. Learning this subject is very useful to excel in various fields and also to clear many competitive exams.  So, today we will learn about a simple yet useful topic that is quadratic equations. To solve any quadratic equation you need to know its formula and steps to use the quadratic formula. In this article, you will get the introduction of quadratic equations along with its different forms and examples.

Quadratic Equation

Quadratic equation – its introduction

It is a second-degree equation that means it has at least one term with a square. The equation is always defined in its standard form. The standard form of this equation is ax² + box +c = 0 where a, b and c are the constants or coefficients. In this equation, there is a variable that is ‘X’ with an unknown value. Rest of the coefficients has known value. Using the quadratic formula, we usually find the value of this unknown variable. An equation is called quadratic if:

·         Its first variable that is ‘a’ is not equal to zero while the other variables can be zero.

·         The third variable is constant or an absolute term.

·         It must be a second-degree equation.

·         Value of unknown variable, i.e., x must satisfy the equation and hence called the root of an equation.

To understand what the quadratic equation is, here are examples of quadratic equation in its different form.

·         Standard form equation: 3x2+4x-12=0

·         Equation without the linear coefficient: x²-8=0

·         Equation without constant: 2x2+ 16x=0

·         Equation in factor form: (x+2) (x-4) =0

·         In another form: x(x-2) = 8, on multiplying and moving eight the equation comes in standard form and becomes x²-2x-8=0

These were some of the examples of quadratic equation in its different forms. Let’s see the procedure to solve these equations. Using a quadratic formula you can solve any quadratic equation, you can also use the other method where equate the factor to zero and get the value. But this method does not work for every problem as sometimes it can be quite messy because it does not get factored. The formula for the standard equation is:

                                                                     x= (-b±√ (b²-4ac))/2a

To use this formula, first, arrange the equation in the standard form.  Remember that the coefficient b will get squared completely means along with the sign. On solving the problem, you will get two roots, one with a positive value and other with the negative value. Take an example to understand the formula.

For example:

To solve x²+2x-4=0 , You can find the factor or simply put the values of a, b and c in the above formula to get the answer. Let’s solve using the quadratic formula.  The value of coefficients a, band c is 1, 2 and-4 respectively. Put them in the formula.

x= (-2±√ (2²-4.1.-4))/2.1

= (-2±√(4 +16))/2

= (-2±√20)/2

= (-2±2√5)/2   

The two roots of equation are -2+2√5/2 and -2-2√5/2

If you understand this example, then try solving other problem as well. Go with the simple problems and then move toward the tough questions. I hope you understand how to use the quadratic formula.

Factoring Calculator - How to Factor an Algebraic Equation?

Factoring calculator is an essential technique to simply most of the algebraic equation that helps to find out the real value of a variable. The algebra is one of the important parts of mathematics and most of the students scared to solve an algebraic equation and find out their factors. Actually, students need to use various mathematical operations such as addition, multiplication, division and subtractions two. On the other hand, keeping essential formulas to solve the given equation is also frustrating for students. Thus, you have the option to choose the factor calculator that can help you to solve any of the algebraic equation.

Consider some examples to find out their factors as

(x + 3) (x – 3)

x² + 2x – 3 = 0

x³ + 3x² + 12x + 12 = 0

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

To find factors for linear equation consider several examples as-

1.    2x + 5 = 0

2.    2x + 4x – 12 = 48

3.    13x / 2 = 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 variable element. If there are more than two variables, then try to add them and transfer remaining part in different side. You need to eliminate the coefficient term to find the factors of any linear equation. Read the content below in the article to know the solution-

For first equation: 2x + 5 = 0

2x = -5

X = -5/2

For second equation

2x + 4x – 12 = 48

6x – 12 = 48

6x = 60

x = 10

Now, the factor is 10

For third equation

13x / 2 = 39

x = 3 * 2

x = 6

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

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

x² + 5x +10 = 0

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

By solving above figure, you can get two values 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 on 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.

·         Benefits of factoring calculator to solve algebraic equation

·         No need to perform various mathematical operations

·         Use digital algebraic calculator to find out factors

·         Get answer in short and expanded form

·         The factoring calculator is best to get factors in few seconds

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