<< Chapter < Page Chapter >> Page >
This module teaches the method of solving quadratic equations by factoring.

Consider the equation 4x 2 + 14 x 60 = 0 size 12{4x rSup { size 8{2} } +"14"x - "60"=0} {} . This is not an algebraic generalization, but an equation to be solved for x : that is, it asks the question “What x value, or values, will make this equation true?” We will be solving such equations in three different ways. The fastest and easiest is by factoring.

Using the techniques discussed above, we can rewrite this problem as follows. (Try it for yourself!)

4x 2 + 14 x 60 = 0 size 12{4x rSup { size 8{2} } +"14"x - "60"=0} {} Original form

2 2x 5 x + 6 = 0 size 12{2 left (2x - 5 right ) left (x+6=0 right )} {} Factored form

The second form may look more complicated than what we started with. But consider what this equation says. There are three numbers: 2, 2x 5 size 12{2x - 5} {} , and x + 6 size 12{x+6} {} . The equation says that when you multiply these three numbers, you get 0. Ask yourself this crucial question: How can you multiply numbers and get the answer 0 ?

The only way it can happen is if one of the numbers is 0. Take a moment to convince yourself of this: if several numbers multiply to give 0, one of those numbers must be 0.

So we have three possibilities.

2 = 0 size 12{2=0} {} 2x 5 = 0 size 12{2x - 5=0} {} x + 6 = 0 size 12{x+6=0} {}
( it just isn't ) size 12{ \( "it just isn't" \) } {} x = 2 1 2 size 12{x=2 { { size 8{1} } over { size 8{2} } } } {} x = 6 size 12{x= - 6} {}

The moral of the story is: when a quadratic equation is factored, it can be solved easily. In this case, the equation 4x 2 + 14 x 60 = 0 size 12{4x rSup { size 8{2} } +"14"x - "60"=0} {} has two valid solutions, x = 2 1 2 size 12{x=2 { { size 8{1} } over { size 8{2} } } } {} and x = 6 size 12{x= - 6} {} .

Consider this example:

x 2 9x + 20 = 6 size 12{x rSup { size 8{2} } - 9x+"20"=6} {}

A common mistake is to solve it like this.

x 2 9x + 20 = 6 size 12{x rSup { size 8{2} } - 9x+"20"=6} {} , solved incorrectly

  • ( x 4 ) ( x 5 ) = 6 size 12{ \( x - 4 \) \( x - 5 \) =6} {}
  • x 4 = 6 size 12{ left (x - 4 right )=6} {}
    • x = 10  ✗ size 12{x="10"} {}
  • x 5 = 6 size 12{ left (x - 5 right )=6} {}
    • x = 11  ✗ size 12{x="11"} {}

All looks good, doesn’t it? The factoring was correct. But if you try x = 10 size 12{x="10"} {} or x = 11 size 12{x="11"} {} in the original equation, you will find that neither one works. What went wrong?

The factoring was correct, but the next step was wrong. Just because ( x 4 ) ( x 5 ) = 6 size 12{ \( x - 4 \) \( x - 5 \) =6} {} does not mean that either x 4 size 12{ left (x - 4 right )} {} or x 5 size 12{ left (x - 5 right )} {} has to be 6. There are lots of ways for two numbers to multiply to give 6. This trick only works for 0!

x 2 9x + 20 = 6 size 12{x rSup { size 8{2} } - 9x+"20"=6} {} , solved correctly

  • x 2 9x + 14 = 0 size 12{x rSup { size 8{2} } - 9x+"14"=0} {}
  • x 7 x 2 = 0 size 12{ left (x - 7 right ) left (x - 2 right )=0} {}
  • x 7 = 0 size 12{ left (x - 7 right )=0} {}
    • x = 7  ✓ size 12{x=7} {}
  • x 2 = 0 size 12{ left (x - 2 right )=0} {}
    • x = 2  ✓ size 12{x=2} {}

You may want to confirm for yourself that these are the correct solutions.

Moral: When solving quadratic equations, always begin by moving everything to one side of the equation , leaving only a 0 on the other side. This is true regardless of which of the three methods you use.

x 2 + 14 x + 49 = 0 size 12{x rSup { size 8{2} } +"14"x+"49"=0} {}

  • x + 7 2 = 0 size 12{ left (x+7 right ) rSup { size 8{2} } =0} {}
  • x = 7 size 12{x= - 7} {}

Moral : If the left side factors as a perfect square, the quadratic equation has only one solution.

Not all quadratic functions can be factored. This does not mean they have no solutions! If the function cannot be factored, we must use other means to find the solutions.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Math 1508 (lecture) readings in precalculus' conversation and receive update notifications?

Ask