#algebraic expression

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Do you know what a quadratic equation is? You must have learned it in school. It is simply defined as a polynomial equation that has only one variable with the highest exponent of 2. It is called polynomial because it has only one variable. What does it look like? =0 is the general form of the equation where “a” can never be equal to zero. The constant terms in the expression are “a, b and c.” here the single variable is “x.”

How to solve such equations?

You can solve it using any of the three ways.

1.    Factor the quadratic equation

2.    Use quadratic formula

3.    Complete the square

If you don’t know about these methods, then learn it here with simple examples. Or you can understand it better using algebra software. First, let us understand how to use the software to solve any problem.

·         Open the software

·         Write the equation

·         Click on “solve.”

·         Get the solution

·         Click on each step to understand the explanation

Now, learn each way to solve quadratic equations.

1.    Factor method

First, move all the terms from one side to another and zero should be at another side. Remember that should be positive, so don’t move it to another side. When you bring the terms to another side, change their signs, i.e., “+” will become “-“and vice versa. Now, add or subtract the terms with the same power of variables. For example, add or subtract the term with together and do the same with x term and integers. Next step is factoring, how to factor? Use the first and last term’s factor.  Once you get factors, set it equal to zero. Doing this gives you two values of x which may be even positive, negative or fractional.

Your answer may be correct or incorrect. To check it, put the value you got on solving it in the original problem, and if the result comes equal to zero, it is correct.

Example- x²-2x-8 =0  

Solution:

Here the value of a=1, b= -2 and c=-8. Let’s solve it using factor method. As all the terms are on one side, so no needs to shift them. Now, split the middle term.

x²-4x+2x-8 =0  

If we look to above expression, you will see that on subtracting 4 and 2 we get the middle term, i.e. (-2) whereas on multiplying them we get the last term i.e.8. It means we have correctly split the term. Now, make the groups get two factors.

x(x-4x)+2(x-4) =0  

(x-4) (x+2) =0  

Now, we will equate each factor to zero to get the values.

(x-4)=0, (x+2) =0  

x=4, x=-2

So, this is how we get the result.

2.    Quadratic formula

Do you know what the formula for solving quadratic equations is? It is written below:

We will solve the same problem using this method.

Example  

Put the value a=1, b= (-2), c= (-8)

Now, solve for each sign separately to get two values.

When we take positive sign:

When we take negative sign:

The basis of the algebra is to use the letters in order to represent a relationship that exit between the numbers before you specify what such numbers are. The algebraic expression has variables, constants, and algebraic operations. Transcendental numbers are not considered algebraic. The rational expression can be written in rational fraction through the use of the properties of arithmetic operations like associative properties and commutative properties among others. Algebra uses own terminology in order to describe some parts of its expression. There are variables, constant, operator, term, coefficient and exponent.

An algebraic number is a complex number which is a root of the non-zero polynomial while one variable is a rational coefficient.  All rational number and integers are called algebraic. This is not the same for complex and real numbers since they have transcendental numbers.  Most of the time, complex and real numbers are in the transcendental categories.

A set of algebraic number can be enumerable or countable. A set of algebraic number comes with a Lebesgue measure zero used as the subset of the complex numbers.  However, it does not mean that all complex numbers are also algebraic. For every algebraic number, there is the unique monic polynomial for a certain degree which has a number for its root. A polynomial is known as a minimal polynomial. When a polynomial number comes with n degree: then its algebraic number is known as being of degree n. algebraic number is always computable, and it is arithmetical and definable. A set of real algebraic number can be linearly ordered, densely ordered and countable without first and last elements.

An algebraic number can be found in different fields. It can be quotient, product, difference and sum when the denominator is not zero. When a root of a certain polynomial equation that has a coefficient is an algebraic number, it is at its turn algebraic. This means that a field for any algebraic number should also be algebraically closed. This is the smallest closed field in algebra, and it contains rationals, and it is also known as algebraic closure for rationals.

A number that may be obtained using an integer by the use of the finite number of an integer like division, multiplications, subtraction, and addition or taking the root of nth roots, if the n is a positive integer, they are all known as algebraic. However, the inverse is not the same, and it is different.  Some algebraic numbers may not be found in this way.  

An algebraic number is a number which may be defined implicitly or explicitly in terms of the polynomials that begin from a rational number. This is called closed-form number, and it can be defined in different ways. The numbers which may be defined implicitly or explicitly in the terms of polynomials, logarithms or exponential; are known as elementary numbers.