<< Chapter < Page Chapter >> Page >
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

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Advanced algebra ii: conceptual explanations. OpenStax CNX. May 04, 2010 Download for free at http://cnx.org/content/col10624/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Advanced algebra ii: conceptual explanations' conversation and receive update notifications?

Ask