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.