**Factorization** is an essential topic of Algebra and
business mathematics, and it is therefore referred to as Algebra factorization
too. Factorization is executed by various methods. Today we learn the following
factorization with examples.

• Factorization by a common term

• Factorization by making pairs

• Factorization by any suitable formula

• Factorization by breaking the middle term

• Factorization by completing the square method

• Factorization by Quadratic formula.

## What is factorization by a common term?

**Factorization by the common term** is performed by taking
out any common variable or number in an equation. The common term is an
important part of the factorization by a common term. In this factorization, there
must be a variable like x, y, z, or any number that can be common.

## Examples of factorization by common term:

**• 6𝑥 ^{2}**

**-24y+48**

**→**6(𝑥^{2}-4y+8) __Ans__

**• 4𝑥 ^{2}y+5𝑥y^{2}-3𝑥**

**y**

**→**𝑥y(4𝑥+5y-3) __Ans__

## What is Factorization by making pairs?

In Algebra** factorization by making pairs** is to make pairs
or divide the terms into two pairs. In factorization by making pairs, there are
usually four terms we make the pair accordingly to get the like and common
terms.

## Examples of factorization by making pairs:

**• 16𝑥 ^{2}+32𝑥+3𝑥**

**+6**

**→**(16𝑥^{2}+32𝑥)+(3𝑥+6)

**→**16𝑥(𝑥+2)+3(𝑥+2)

**→**(16𝑥+3)(𝑥+2) __Ans__

**• 11y ^{2}+33y+7y+21**

**→**(11y^{2}+33y)+(7y+21)

**→**11y(y+3)+7(y+3) __Ans__

## What is factorization by any suitable formula?

In mathematics, **factorization by any suitable formula** is
solved by using an appropriate formula. Factorization by any suitable formula
can also be executed differently, but applying the formula will be time-saving
for the students.

## Examples of factorization by any suitable formula:

**• 169𝑥 ^{2}-156𝑥**

**+36**

**∴**** **formula= (𝑥-y)^{2}= 𝑥^{2}-2𝑥y+y^{2}

**→**(13𝑥)^{2}-2(13𝑥)(6)+(6)^{2}

**→**(13𝑥-6)^{2 }__Ans__

**• 324𝑥 ^{2}**

**-49**

**∴**** **formula= (𝑥)^{2}-(y)^{2 }= (𝑥-y)(𝑥+y)

**→**(18𝑥)^{2}-(7)^{2 }=
(18𝑥-7)(18𝑥+7)

**→**(18𝑥-7)(18𝑥+7) __Ans__

## What is the factorization by breaking the middle term?

**Factorization by breaking the middle term **is quite similar
to factorization by making pair. In factorization by breaking the middle term,
we break the middle term into two parts.

## How to break the middle in factorization?

It is very simple and easy to break the middle term in
factorization. Multiply the first term with the last term of the given equation.
Then, make the factors of that product that give the middle term number if
we add both factors. Let's clear with an example if the given equation is *7*y^{2}*+16y+4. *Now multiply 7 by 4, and the
answer will be 28. After that, get the factors of 28 i.e., 2,4,7,14, 28.

If we multiply 2 by 14 we will get 28 and in the same way, if we add 2 and 14 we will get 2+14= 16. Thus, 2y+14y will replace 16y, and we
will write the above equation in this way *7**y*^{2}*+2y+14y+4.*

## Examples of factorization by breaking the middle term:

**• 8𝑥 ^{2}+8𝑥**

**+2**

**→ **8𝑥^{2}+4𝑥+4𝑥+2

**→** (8𝑥^{2}+4𝑥)+(4𝑥+2)

**→ **4𝑥(2𝑥+2)+2(2𝑥+2)

**→ **(4𝑥+2)(2𝑥+2) __Ans__

**• 10y ^{2}+57𝑥y+11𝑥^{2}**

**→** 10y^{2}+2𝑥y+55𝑥y+11𝑥^{2}

**→** (10y^{2}+2𝑥y)+(55𝑥y+11𝑥^{2})

**→** 2y(5y+𝑥)+11𝑥(5y+𝑥)

**→** (2y+11𝑥)(5y+𝑥) __Ans__

## What is the factorization by a completing square method?

**Factorization by a completing square method** is quite
different from other factorizations. There are various steps to completing
square factorization.

## Simple steps of a completing square factorization:

1. co-efficient of 𝑥^{2} should be 1.

it should be like this 𝑥^{2} if it is not then we will divide the coefficient of 𝑥^{2} with an entire equation. For instance, if it is 5^{2 }then will divide 5 by the given equation.

2. Constant term will be moved to = side

3. coefficient of 𝑥 will be multiple with 0.5 or ½ and then
it will be squared the final answer will be added to both sides.

## Examples of factorization by a completing square method?

**• 10𝑥 ^{2}+80𝑥-120=0**

→Divide 10

→10/10𝑥^{2}+80/10𝑥-120/10=0

→𝑥^{2}+8𝑥-12=0

→ move -12

→𝑥^{2}+8𝑥=12

→Take the co-efficient of 𝑥 i.e., 8, and multiply it with 1/2

→[8x1/2]2

→[4]2

→16

→add 16 on both sides

→𝑥^{2}+8𝑥+16=12+16

→𝑥^{2}+8𝑥+16=28

→(𝑥+4)^{2}=28

→Take square root on
b/s

→√(𝑥+4)^{2}=√28

→𝑥+4= ±2√7

→𝑥= ±2√7-4

→𝑥= ±2(√7-2) __Ans__

**• 𝑥 ^{2}+24𝑥-25=0**

→ move -25

→𝑥^{2}+24𝑥=25

→Take the co-efficient of 𝑥 i.e., 24, and multiply it with 1/2

→[24x1/2]2

→[12]2

→144

→add 144 on both sides

→𝑥^{2}+24𝑥+144=25+144

→𝑥^{2}+24𝑥+144=169

→(𝑥+12)^{2}=169

→Take square root on b/s

→√(𝑥+12)^{2}=√169

→𝑥+12= ±13

→𝑥= ±13-12

→𝑥= +13-12 | -13-12

→𝑥= +1 | -25

→S.S= {1,-25} __Ans__

**• 𝑥**

^{2}+2𝑥-48=0

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