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.

Equation Solver – A Guide to Solve Equations One by One

Equations seem to be a tough mathematics concept for many students, but it is not. Once you understand the right and proper concept to solve the equation, it becomes an interesting topic for you. Today, we are going to see certain techniques via which one can easily solve a linear equation. So, let’s start it: 

Solving equations

Equations consist of variables and constant terms. We have to find the value of given variable for example x. Let’s understand it with an example: 3 + x = 7

 In the given equation, we need to solve for X.  To solving this simple equation, we need to arrange the “Like terms” on both the sides, i.e., L.H.S and R.H.S.  After the arrangement of like terms, we get 4 as an answer:

3 + x = 7

x = 7 – 3

x = 4

There is only one variable, i.e., X in the equation and we have got two numbers. So, we have to arrange the numbers on one side. While changing the sides of constants and variable terms, make sure to change their signs as well. In the above example, x remains on L.H.S. However, we have shifted 3 on the R.H.S. After coming on the R.H.S, its sign becomes negative to positive and same goes with the variables also.

1.  Solving equation (using subtraction & addition properties)

 We have equation 3x + 3 = x +15, let’s start solving it:

3x + 3 = x +15

3x - x = 15 - 3

(In this step, we have arranged “like terms,” and thus terms with variables shifted on L.H.S and constant terms on the R.H.S.)

2x = 12

(After solving them, we have got 2x and 12.)

x = 6

After further multiplications, we have got answer 6 which is the right value of X.  Is it simple right? So, no let’s move on to the multiplication method.

2. Solving equation (Using multiplication property)

We solve equation 

plus going to discuss some important point’s one need to consider while using multiplication property.

I am going to explain it in a very simple way. In this method, we use multiply both the sides by using a nonzero quantity to get the answer. Let’s see how to do it? 

(In this step, we have canceled 4 as the numerator gets canceled by the denominator because of being a multiple of 4.  However, on the R.H.S we need to multiply 4 from 3 which results in 12.  Yes, you can direct multiple 4 from 3 but do this only when you understand the concept properly.

x = 12

When we solve for x, we got to answer 12.  The method is very easy, however; make sure to practice it more often for perfection.

3. Solving equation (using division property)

Let’s take an example 4x = 12. Now, in this one, we are going to divide the terms by a nonzero quantity to get the answer.

(In the above step, we have divided both the sides by 4 and thus got 3 as the value of x. talking about the L.S.H then 4 in denominator and numerator got canceled. On R.H.S, number 12 is divided by 4, and we got to answer 3. 

The above three concepts are easy to practice, so go for the concepts and improve your linear equation solving skills. 

Multiplying Calculator

Algebra solver is now available online to those students who find math difficult. If you are that student who uses a lot of time in doing your algebra homework, don’t worry anymore because this math solver can work out all the fractions and show the solutions. For many years students have been having difficulty in doing their fractions, but since this tool was introduced they find it more easily doing their calculations.

Fractions can be way difficult to absorb for some students when they first start with the topic. Especially, when dealing with equivalent fractions; it seems that they do not get the head or tail of anything. The concept in reality is very simple; but if the logic still eludes you, you can try using an equivalent fractions calculator, till you get the hang of it.

Your math teacher tells you the equivalent fractions are two or more fractions that have the same value, but have different forms. In simple words, these fractions can be defined as fractions having the same overall value, or the same simplest ratio. These fractions indicate the same part of a whole. The easiest way to explain equivalent fractions is using the example of a round apple pie. Whether you divide a pie into two pieces and take one, or out of four pieces take 2, or further on out of 8 pieces take 4; you still get one-half of the pie to eat. So we can conclude that 1/2 is same as 2/4 or 4/8; they all are equivalent fractions.

The technique to simplify a fraction is very easy; both numerator and denominator must be divided by the same number to reduce the fraction. So remember that the value of a fraction does not change if you multiply or divide the whole fraction by the same number. Remember that the number you use for division must divide without leaving a remainder. Both the numerator and denominator of a fraction must always be whole numbers; so you can know when you have arrived at the simplest answer & cannot reduce the fraction any further. And remember, only multiplication and subtraction is valid when reducing fractions; do not try to perform addition or subtraction while reducing your fractions.

The easiest way when working with conference is to use fraction calculator to check your answers. This will boost your confidence & in the long run help you with your math homework. But remember, that using the calculator will only validate your answer; you still have to learn how to arrive at that answer in the first place.

The prized scientific calculator of your elder sister can act as your equivalent fractions calculator; you can easily check your answers using it. On the other hand, there are many math websites today offering their own version of equivalent fractions calculator, you can easily search & bookmark your favorite one to verify your answers. You can also ask your nerd brother to write a short program for an equivalent fractions calculator; whether he uses C++ or Assembly, the code to develop these calculators is very simple.

Types of Equations

Algebra, as a topic in mathematics. Is being tacked right from primary school up to the highest level of learning. If you ask a junior mathematician what they understand by the term algebraic equation. A good number of them will give out an example that looks like this; 2x + 3x + 4x=. Yes, this is a form of an algebraic equation. I cannot refute since when you work on it, still you will come up with an answer. Many of the mathematicians have tackled some equations, but in the real sense good percentages of them do not understand or are well conversant with the types of equations. Funnily, some can never know the types of equations, but they can freely work on problems from varied types of equations.

Equation

Variables and constants when put together form an algebraic equation and it must have a symbol of equality to qualify as an algebraic equation.  Therefore an expression having the equal sign (=) is referred to as an equation. For example; 4a + 3b +7c = -7. With this rough background information. Let us discuss the types of equations that we are likely to encounter in any test or article:

1)    Linear Equations:

In this type of equation, the terms can be a constant or the product of a constant and one variable. A straight line kind of graph is as a result of two variables. Equation having two variables usually have a formula as; y = mx + c,

M is referred to as slope while c is the point at which it intercepts the y-axis.

2)    Radical Equations:

This is a very common type of an equation. How can you identify a radical equation? Very simple, if you see a square root sign anywhere accompanying the equation then it’s a radical equation. Radicle equations on many occasions are being handled by mathematicians at the secondary level of learning.  Below is an example of a radicle equation;

 √x+14=28

Have you tackled such kind of equation?

3)    Quadratic equations:

When you see or hear the term Quadra, what comes into your mind? Many of us will think about four in the first case. Yes grammatically Quadra, means four. Mathematically we have the quadratic equation. This type of equation in which one variable has got the variable with an exponent of 2. It usually has a structure like this, ax² + bx + c = 0, a ≠ 0 for example;

4x² + 6x – 74 = 0 is an example of a quadratic equation.

4)    Rational Equations:

Such kind of equations deals with rational expressions. The term rational expression may sound unfamiliar, yet you know it right? Let me shade some light. A rational expression is a fraction in which both the numerator and the denominator are polynomials.

X2/2 = X+2/4

These types of equations may tend to be hectic to solve. Why? Because they consume a lot of time as some involve the collection of like terms together in order to move on. But with factor calculator, all is made easy.

The Results you Should Expect from Equation Solver

With mathematics, the equation is the statement of the equality, and it contains one or different variables.  Equation solver helps to maintain the value of these variables and to see how the equality can be true. The variables can also be called unknowns, and there are the values of the unknowns, and they satisfy the equality also named solution of an equation. You can find two different types of the equations; they are identity equation and conditional equation.  Identity equation will keep coming true for all the values of the variables. The conditional equation will be true for certain values.

Every side of the equation is named as the member of the equation. Every member should have one or even more terms.

The Ax² +Bx+ C= Y features two members, the Ax² + B x +C and Y, the left members feature three terms while the right members have only one term.  The variables in the members are Y and X while the parameters are the A, B or C.

The equation is said to be analogous to the scale at which the weight would have been placed.  If the equal weight for some time is placed on two pans, the two weight will make the scale to reach a balance.  If the scale should be kept in balance, then the equal grain amount should be removed from the two pans so that the scale can be kept in the balance. This is the same when it comes to solving the equation, since if there is a need to keep this equation in the balance, then the division, multiplication, subtraction, and addition done on one part, it should be repeated at the other side. This is to help in keeping the equality.

If it is geometry, then the equation is meant to describe the geometric figures. The equation is considered to be a parametric equation or implicit equation. They all have different solutions.  The objective will be different because, in the place of counting or giving an explicit solution, the equation is used instead for studying the property of figures.  This is what it is called algebraic geometry, and it is an important part of mathematics.

Algebra studies come with two different equations; they are linear equations and polynomial equations. Polynomial equation comes with P(x) =0 and the P here is the polynomial. The Linear equation comes with the form ax+b= O, and here a and b are the parameters.  The equation solver for any of the two families, it involves the use of the algorithmic or of geometric technique that originated from the mathematical analysis or linear algebra.  The algebra will also study the Diophantine equation, and the coefficient and the solution in the game are the integers. These techniques that are being used are not the same, and they do come from a number theory. The equation is difficult in the general, and one has to do the research in order to find the absence and the existence of the solution.  If they do exist, then the person has to count how many solutions he can get. 

Quadratic Formula : A Better way of Solving Quadratic Equations

Algebra is regarded as the most important part of mathematics. There are many interesting topics in algebra that you may get to learn. However quadratic equations are known to be some of the most common discussions among the learners. There can be many different ways to solve quadratic equations well. However, with the need of some easy methods, the emergence of the quadratic formula is considered to be great. There are many important things that you may need to know about this topic in order to be perfect in it. Here, we are going to discuss some easy ways of solving problems for the beginners.

Quadratic Formula

Significance

There are many different types of methods and formula that are used to solve equations whether linear or quadratic. However, there can be many complexities associated with the solutions of the quadratic equations however there have been many other formula that are designed specially to make this procedure easy. Some quadratic equations cannot be factored readily because of different reasons. Therefore you need to take up certain necessary steps so that the equation is ready to be factored. You can also choose to factor the trinomial and then use the square root property to solve the equations.  

However, all these factoring and finding the square root will make the solution lengthy and more confusing for the student. Therefore the best way to simplify the solution is to use the quadratic equation for getting the answer. We will further learn about the use of the formula and also what it exactly is. 

The formula is the simplified form of the complete result that you may get through the factoring of the trinomial. The formula is: 

For a trinomial that is in the form ax² + bx + c = 0. Thus with this, you can simplify the solution for your problems and get the right answers in the least time with least efforts.

Clear the concept

Now understand in deep, recall the steps that were involved in completing the square for the equation. We will do it by completing the square of the general equation i.e.

ax² + bx + c = 0 and see how the formula came into existence.

·         It starts from the equation x² + bx + c = 0

.

·         Now rewrite so that the variable is isolated to one side: .x² + bx 

·         Add (b/2)²

 to both the sides and complete the square.

·         Now solve for the value of x after rewriting the square of the binomial. 

 

Thus you just need to learn a single formula, and you can eliminate the so many long processes involved in the solution to get the right answers. This way you will always be able to solve the quadratic equation in the best way and also get the right answers at once. There are many more interesting things that you will get to know as you practice for the formula. Therefore from the above discussions, it can be said that the quadratic formula is a no different concept. It is just the summarized form of the complete steps involved in solving the quadratic equations. Therefore it is better to use this to solve the algebraic problems rather than using long steps and getting confused.