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This module discusses how to solve quadratic equations by factoring.

When we multiply, we put things together: when we factor, we pull things apart. Factoring is a critical skill in simplifying functions and solving equations.

There are four basic types of factoring. In each case, I will start by showing a multiplication problem—then I will show how to use factoring to reverse the results of that multiplication.

“pulling out” common factors

This type of factoring is based on the distributive property , which (as you know) tells us that:

2x 4x 2 7x + 3 = 8x 3 14 x 2 + 6x size 12{2x left (4x rSup { size 8{2} } - 7x+3 right )=8x rSup { size 8{3} } - "14"x rSup { size 8{2} } +6x} {}

When we factor, we do that in reverse. So we would start with an expression such as 8x 3 14 x 2 + 6x size 12{8x rSup { size 8{3} } - "14"x rSup { size 8{2} } +6x} {} and say “Hey, every one of those terms is divisible by 2. Also, every one of those terms is divisible by x size 12{x} {} . So we “factor out,” or “pull out,” a 2x size 12{2x} {} .

8x 3 14 x 2 + 6x = 2x __ __ + __ size 12{8x rSup { size 8{3} } - "14"x rSup { size 8{2} } +6x=2x left ("__" - "__"+"__" right )} {}

For each term, we see what happens when we divide that term by 2x size 12{2x} {} . For instance, if we divide 8x 3 size 12{8x rSup { size 8{3} } } {} by 2x size 12{2x} {} the answer is 4x 2 size 12{4x rSup { size 8{2} } } {} . Doing this process for each term, we end up with:

8x 3 14 x 2 + 6x = 2x 4x 2 7x + 3 size 12{8x rSup { size 8{3} } - "14"x rSup { size 8{2} } +6x=2x left (4x rSup { size 8{2} } - 7x+3 right )} {}

As you can see, this is just what we started with, but in reverse. However, for many types of problems, this factored form is easier to work with.

As another example, consider 6x + 3 size 12{6x+3} {} . The common factor in this case is 3. When we factor a 3 out of the 6x size 12{6x} {} , we are left with 2x size 12{2x} {} . When we factor a 3 out of the 3, we are left with...what? Nothing? No, we are left with 1, since we are dividing by 3.

6x + 3 = 3 2x + 1 size 12{6x+3=3 left (2x+1 right )} {}

There are two key points to take away about this kind of factoring.

  1. This is the simplest kind of factoring. Whenever you are trying to factor a complicated expression, always begin by looking for common factors that you can pull out.
  2. A common factor must be common to all the terms. For instance, 8x 3 14 x 2 + 6x + 7 size 12{8x rSup { size 8{3} } - "14"x rSup { size 8{2} } +6x+7} {} has no common factor, since the last term is not divisible by either 2 or x size 12{x} {} .

Factoring perfect squares

The second type of factoring is based on the “squaring” formulae that we started with:

x + a 2 = x 2 + 2 ax + a 2 size 12{ left (x+a right ) rSup { size 8{2} } =x rSup { size 8{2} } +2 ital "ax"+a rSup { size 8{2} } } {}
x a 2 = x 2 2 ax + a 2 size 12{ left (x - a right ) rSup { size 8{2} } =x rSup { size 8{2} } - 2 ital "ax"+a rSup { size 8{2} } } {}

For instance, if we see x 2 + 6x + 9 size 12{x rSup { size 8{2} } +6x+9} {} , we may recognize the signature of the first formula: the middle term is three doubled , and the last term is three squared . So this is x + 3 2 size 12{ left (x+3 right ) rSup { size 8{2} } } {} . Once you get used to looking for this pattern, it is easy to spot.

x 2 + 10 x + 25 = x + 5 2 size 12{x rSup { size 8{2} } +"10"x+"25"= left (x+5 right ) rSup { size 8{2} } } {}
x 2 + 2x + 1 = x + 1 2 size 12{x rSup { size 8{2} } +2x+1= left (x+1 right ) rSup { size 8{2} } } {}

And so on. If the middle term is negative , then we have the second formula:

x 2 8x + 16 = x 4 2 size 12{x rSup { size 8{2} } - 8x+"16"= left (x - 4 right ) rSup { size 8{2} } } {}
x 2 14 x + 49 = x 7 2 size 12{x rSup { size 8{2} } - "14"x+"49"= left (x - 7 right ) rSup { size 8{2} } } {}

This type of factoring only works if you have exactly this case : the middle number is something doubled , and the last number is that same something squared . Furthermore, although the middle term can be either positive or negative (as we have seen), the last term cannot be negative.

All this may make it seem like such a special case that it is not even worth bothering about. But as you will see with “completing the square” later in this unit, this method is very general, because even if an expression does not look like a perfect square, you can usually make it look like one if you want to—and if you know how to spot the pattern.

The difference between two squares

The third type of factoring is based on the third of our basic formulae:

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
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