How to factor a quadratic

In order to factor a quadratic, you have follow these steps:

• Arrange it in descending order (that is ax^2 + bx + c)
• Split the middle term into two parts.
• Now you have a four termed quadratic
• Group it into two groups each containing two terms
• Factor out the common factor from each group
• Factor out the common factor from both groups
• Write the answer as a product of two expressions

Each of these steps is explained below:

Arrange it in descending order: 
This means you have to rewrite the quadratic so that the first term is the one with 'x^2' in it, the second term is the one with only an 'x' in it, and the third term is the number. For example, arranging

3x + 2x^2 + 1 = 0

we get

2x^2 + 3x + 1 = 0

Split the middle term into two parts: 
The middle term is the one with an 'x' in it. To split the middle term into two parts, first find the products of the first and last term of the equation. Now you have to split the middle term into two parts such that the product of the two parts is equal to the product of the first and the last terms of the quadratic. For example, in the equation

2x^2 + 3x + 1 = 0

First term = 2x^2
Middle term = 3x
Last term = 1

Product of first and last terms = 2x^2 + 1 = 2x^2

Therefore we have to split 3x into two parts such that their product is 2x^2. Therefore we get 2x + x, because 2x * x = 2x^2.

Now you have a four termed quadratic 
Rewrite the quadratic with the two parts of the middle term instead of the middle term itself. For example in the above quadratic 2x^2 + 3x + 1 ,

Middle term = 3x
Two parts of middle term = 2x + x 

Therefore rewriting the quadratic, we get

 2x^2 + 2x + x + 1

Group it into two groups each containing two terms
The first two terms of the above quadratic will be considered one group, and the last two terms will be considered the other group. For example, we get

(2x^2 + 2x) + (x + 1)

Factor out the common factor from each group
From each group, we factor out the common factor and if there is no common factor from a group, then we take 1 or -1 as its common factor accordingly. From the above example, we get

2x(x + 1) + 1(x + 1)

Factor out the common factor from both groups
The common factor in both the groups above is (x + 1). So we will factor it out as follows"

(x + 1)(2x + 1)

Write the answer as a product of two expressions
The answer is a product of two expressions. From the above example, we get:

2x^2 + 2x + x + 1 = (x + 1)(2x + 1)

This is the method of how to factor a quadratic.