What are algebraic equations? Learn how to solve them

Algebraic equation is also known as polynomial equations. A polynomial expression is made up of constants and variable, combined using mathematical operators. When a polynomial expression is equated to some constant value, it becomes polynomial equation. To understand this in simple terms, go through the following example.

x²-5x+6

Is a polynomial equation, here x is a variable, 5 and 6 are constant and mathematical operators used are addition and square, but

x²-5x+6=0 is a polynomial equation or algebraic equation because it is equated to a constant (in this case zero).

The general form of a polynomial equation is

As one can notice the power f x keep on decreasing. The above equation is called a polynomial degree of order n.

What is degree of polynomial?

Degree of a polynomial is the highest power of variable. Degree of a polynomial is always a non-negative integer.

Depending upon the degree of polynomial, polynomial may de defined as quadratic equation, Linear equation or constants.

To be very particular a polynomial equation of degree 2 is called quadratic equation, polynomial equation of degree 1 is called linear equation and polynomial equation with degree of polynomial zero are nothing but constants values.

In this module we will focus on quadratic equations

The general form of a polynomial equation is ax²+bx+c= 0

Here,

·         a is the coefficient of x²

·         b is the coefficient of x And

·         c is the constant term

To solve quadratic equations there are three methods available, they are:

·         factorization

·         completing the square method

·         using quadratic formula

Let us see each method is detail.

Factorization method

In this method the quadratic equation is decomposed into 2 factors. (What are factors? Factors are those values, which if substituted into the original equation, then it result out to zero. ) for example in our example, x²-5x+6 =0 the equation can be written in the form, (x-2) (x-3)=0 . If one opens the bracket and multiply the terms then we get the original equation x²-5x+6 . Therefore (x-2) (x-3) are known as factors of the equation x²-5x+6 =0

How to find factors of any quadratic equation?

To find the factors of any quadratic equation, follow these steps:

Step 1 Decompose the middle term, i.e., the terms containing the x and it coefficient into 2 parts, such that the product of those two terms should be equivalent to the product of x² term and the constant term. Also, the summation of those decomposed two terms should give back the original middle term.

Step 2 Take common out from the first two terms, and from second two terms in such a way, that you get common factor among both these terms.

Step 3 Again take the common factor out, and then you are left with 2 final factors of the original equation.

If the above mentioned steps are not clear to you, go through the following steps:

Let’s start with

x²-5x+6 = 0

Now let us write 5x as -2x -3x

x²-2x -3x+6 =0

This is so because, product of (-2x) and (-3x) is (+6x²), which is same as product of (x²) and (6), i.e., (6x²), the product of x² term and constant term.

Now, take x common out from x² and (-2x) and (-3) common out from (-3x) and (6)

The equation becomes x(x-2)-3(x-2)=0

Now take (x-2) common out

 (x-2) (x-3)= 0

These are the factors of the original equation x²-5x+6=0
 

The values of x in this case would be 2 and 3.

Quadratic Formula Discovery

The quadratic formula was utilized for a few thousand years. The quadratic formula had additionally changed on events. Around 2000 years back, the Chinese, Egyptians and Babylonians were at that point acquainted with the territory of a square level with a length of its each side. By utilizing roughage parcels, they made sense of that they could stack together more nine bundles if the length of roof space were wide very nearly three times. The range of the other complex shapes could likewise be figured.

Be that as it may, they knew how the sides of the shapes could be worked out, and they had confronted the somewhat enormous problem. They ought to have known how the lengths of the sides are computed. The shape must be leveled with aggregate range with the length of sides.

Quadratic formula in Egypt
Around 1500 years back, Egyptians had not utilized numbers like they are utilized today. Words were utilized for communicating scientific problems. However, the sacred writing evaded the problem of quadratic formula by comprehending the territories of each side and developed a diagram. They made something like an augmentation table. The calculation was made snappy and quick. The Egyptians required processing all sides and shapes unfailingly. They just needed to allude to the diagram.
These tables still exist today. They may be numerically wrong yet they without a doubt demonstrates the start of the quadratic formula.

Quadratic formula in Babylon
The Babylonians had embraced a differing path for taking care of problems. They utilized numbers rather than words, conversely with the Egyptians. The numbers utilized by the Babylonians were significantly more the same like the numbers utilized today in spite of the fact that they depended on a hexadecimal model. Expansion and increase were less demanding to do with this framework. Around 1000 BC, Babylonian specialists could check the realness of their qualities. By 400 BC, they found a strategy called 'finishing the square' to solve problems with ranges.

Euclid and Pythagoras 
The principal numerical endeavor to concoct a quadratic formula was performed in 500 BC by the Pythagoras. Euclid did same in Egypt. He utilized a straightforward geometric strategy and thought of a formula for unraveling the formula. The Pythagoras had watched that proportions did not make any sense between the territory of square and length of sides and there was no other proportion except balanced. Euclid had remarkably believed that there would be silly numbers simply like there are sane numbers. He later discharged a book called "Components" and clarified the science for illuminating quadratic formulas in it. However his formula was not utilizing the same formula which is known today, his equation couldn't ascertain a square root.

Quadratic formula in Europe 
An outstanding Muslim mathematician named Mohammad Al-Khwarizmi effectively settled the quadratic formula in around 820 AD. He had not utilized numbers or negative arrangements. As his statement spread, a Jewish mathematician named Abraham Hiyya conveyed this learning to Spain in 1100. From that point forward, mathematicians from Europe picked and began utilizing the formula.