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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

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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