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

You can run this formula in reverse whenever you are subtracting two perfect squares . For instance, if we see x 2 25 size 12{x rSup { size 8{2} } - "25"} {} , we recognize that both x 2 size 12{x rSup { size 8{2} } } {} and 25 are perfect squares. We can therefore factor it as x + 5 x 5 size 12{ left (x+5 right ) left (x - 5 right )} {} . Other examples include:

  • x 2 64 size 12{x rSup { size 8{2} } - "64"} {} = x + 8 x 8 size 12{ {}= left (x+8 right ) left (x - 8 right )} {}
  • 16 y 2 49 size 12{"16"y rSup { size 8{2} } - "49"} {} = 4y + 7 4y 7 size 12{ {}= left (4y+7 right ) left (4y - 7 right )} {}
  • 2x 2 18 size 12{2x rSup { size 8{2} } - "18"} {} = 2 x 2 9 size 12{ {}=2 left (x rSup { size 8{2} } - 9 right )} {} = 2 x + 3 x 3 size 12{ {}=2 left (x+3 right ) left (x - 3 right )} {}
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And so on. Note that, in the last example, we begin by pulling out a 2, and we are then left with two perfect squares. This is an example of the rule that you should always begin by pulling out common factors before you try anything else!

It is also important to note that you cannot factor the sum of two squares. x 2 + 4 size 12{x rSup { size 8{2} } +4} {} is a perfectly good function, but it cannot be factored.

Brute force, old-fashioned, bare-knuckle, no-holds-barred factoring

In this case, the multiplication that we are reversing is just FOIL. For instance, consider:

x + 3 x + 7 = x 2 + 3x + 7x + 21 = x 2 + 10 x + 21 size 12{ left (x+3 right ) left (x+7 right )=x rSup { size 8{2} } +3x+7x+"21"=x rSup { size 8{2} } +"10"x+"21"} {}

What happened? The 3 and 7 added to yield the middle term (10), and multiplied to yield the final term 21 size 12{ left ("21" right )} {} . We can generalize this as: x + a x + b = x 2 + a + b x + ab size 12{ left (x+a right ) left (x+b right )=x rSup { size 8{2} } + left (a+b right )x+ ital "ab"} {} .

The point is, if you are given a problem such as x 2 + 10 x + 21 size 12{x rSup { size 8{2} } +"10"x+"21"} {} to factor, you look for two numbers that add up to 10, and multiply to 21. And how do you find them? There are a lot of pairs of numbers that add up to 10, but relatively few that multiply to 21. So you start by looking for factors of 21.

Below is a series of examples. Each example showcases a different aspect of the factoring process, so I would encourage you not to skip over any of them: try each problem yourself, then take a look at what I did.

If you are uncomfortable with factoring, the best practice you can get is to multiply things out . In each case, look at the final answer I arrive at, and multiply it with FOIL. See that you get the problem I started with. Then look back at the steps I took and see how they led me to that answer. The steps will make a lot more sense if you have done the multiplication already.

Factor x 2 + 11 x + 18 size 12{x rSup { size 8{2} } +"11"x+"18"} {}

x + __ x + __ size 12{ left (x+"__" right ) left (x+"__" right )} {}

What multiplies to 18? 1 18 size 12{1 cdot "18"} {} , or 2 9 size 12{2 cdot 9} {} , or 3 6 size 12{3 cdot 6} {} .

Which of those adds to 11? 2 + 9 size 12{2+9} {} .

x + 2 x + 9 size 12{ left (x+2 right ) left (x+9 right )} {}

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Start by listing all factors of the third term. Then see which ones add to give you the middle term you want.

Factor x 2 13 x + 12 size 12{x rSup { size 8{2} } - "13"x+"12"} {}

x + __ x + __ size 12{ left (x+"__" right ) left (x+"__" right )} {}

What multiplies to 12? 1 12 size 12{1 cdot "12"} {} , or 2 6 size 12{2 cdot 6} {} , or 3 4 size 12{3 cdot 4} {}

Which of those adds to 13? 1 + 12 size 12{1+"12"} {}

x 1 x 12 size 12{ left (x - 1 right ) left (x - "12" right )} {}

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If the middle term is negative, it doesn’t change much: it just makes both numbers negative. If this had been x 2 + 13 x + 12 size 12{x rSup { size 8{2} } +"13"x+"12"} {} , the process would have been the same, and the answer would have been x + 1 x + 12 size 12{ left (x+1 right ) left (x+"12" right )} {} .

Factor x 2 + 12 x + 24 size 12{x rSup { size 8{2} } +"12"x+"24"} {}

x + __ x + __ size 12{ left (x+"___" right ) left (x+"___" right )} {}

What multiplies to 24? 1 24 size 12{1 cdot "24"} {} , or 2 12 size 12{2 cdot "12"} {} , or 3 8 size 12{3 cdot 8} {} , or 4 6 size 12{4 cdot 6} {}

Which of those adds to 12? None of them.

It can’t be factored. It is “prime.”

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Some things can’t be factored. Many students spend a long time fighting with such problems, but it really doesn’t have to take long. Try all the possibilities, and if none of them works, it can’t be factored.

Factor x 2 + 2x 15 size 12{x rSup { size 8{2} } +2x - "15"} {}

x + __ x + __ size 12{ left (x+"___" right ) left (x+"___" right )} {}

What multiplies to 15? 1 15 size 12{1 cdot "15"} {} , or 3 5 size 12{3 cdot 5} {}

Which of those subtracts to 2? 5–3

x + 5 x 3 size 12{ left (x+5 right ) left (x - 3 right )} {}

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If the last term is negative, that changes things! In order to multiply to –15, the two numbers will have to have different signs—one negative, one positive—which means they will subtract to give the middle term. Note that if the middle term were negative, that wouldn’t change the process: the final answer would be reversed, x + 5 x 3 size 12{ left (x+5 right ) left (x - 3 right )} {} . This fits the rule that we saw earlier—changing the sign of the middle term changes the answer a bit, but not the process.

Factor 2x 2 + 24 x + 72 size 12{2x rSup { size 8{2} } +"24"x+"72"} {}

2 x 2 + 12 x + 36 size 12{2 left (x rSup { size 8{2} } +"12"x+"36" right )} {}

2 x + 6 2 size 12{2 left (x+6 right ) rSup { size 8{2} } } {}

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Never forget, always start by looking for common factors to pull out. Then look to see if it fits one of our formulae. Only after trying all that do you begin the FOIL approach.

Factor 3x 2 + 14 x + 16 size 12{3x rSup { size 8{2} } +"14"x+"16"} {}

3x + __ x + __ size 12{ left (3x+"___" right ) left (x+"___" right )} {}

What multiplies to 16? 1 16 size 12{1 cdot "16"} {} , or 2 8 size 12{2 cdot 8} {} , or 4 4 size 12{4 cdot 4} {}

Which of those adds to 14 after tripling one number ? 8 + 3 2 size 12{8+3 cdot 2} {}

3x + 8 x + 2 size 12{ left (3x+8 right ) left (x+2 right )} {}

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If the x 2 size 12{x rSup { size 8{2} } } {} has a coefficient, and if you can’t pull it out, the problem is trickier. In this case, we know that the factored form will look like 3x + __ x + __ size 12{ left (3x+"__" right ) left (x+"__" right )} {} so we can see that, when we multiply it back, one of those numbers—the one on the right—will be tripled, before they add up to the middle term! So you have to check the number pairs to see if any work that way.

Checking your answers

There are two different ways to check your answer after factoring: multiplying back, and trying numbers.

  1. Problem : Factor 40 x 3 250 x size 12{"40"x rSup { size 8{3} } - "250"x} {}
    • 10 x 4x 25 size 12{"10"x left (4x - "25" right )} {} First, pull out the common factor
    • 10 x 2x + 5 2x 5 size 12{"10"x left (2x+5 right ) left (2x - 5 right )} {} Difference between two squares
  2. So, does 40 x 3 250 x = 10 x 2x + 5 2x 5 size 12{"40"x rSup { size 8{3} } - "250"x="10"x left (2x+5 right ) left (2x - 5 right )} {} ? First let’s check by multiplying back.
    • 10 x 2x + 5 2x 5 size 12{"10"x left (2x+5 right ) left (2x - 5 right )} {}
    • = 20 x 2 + 50 x 2x 5 size 12{ {}= left ("20"x rSup { size 8{2} } +"50"x right ) left (2x - 5 right )} {} Distributive property
    • = 40 x 3 100 x 2 + 100 x 2 250 x size 12{ {}="40"x rSup { size 8{3} } - "100"x rSup { size 8{2} } +"100"x rSup { size 8{2} } - "250"x} {} FOIL
    • = 40 x 3 250 x  ✓ size 12{ {}="40"x rSup { size 8{3} } - "250"x} {}
  3. Check by trying a number. This should work for any number. I’ll use x = 7 and a calculator.
    • 40 x 3 250 x = ? 10 x 2x + 5 2x 5 size 12{"40"x rSup { size 8{3} } - "250"x { {}={}} cSup { size 8{?} } "10"x left (2x+5 right ) left (2x - 5 right )} {}
    • 40 7 3 250 7 = ? 10 7 2 7 + 5 2 7 5 size 12{"40" left (7 right ) rSup { size 8{3} } - "250" left (7 right ) { {}={}} cSup { size 8{?} } "10" left (7 right ) left (2 cdot 7+5 right ) left (2 cdot 7 - 5 right )} {}
    • 11970 = 11970  ✓ size 12{"11970"="11970"} {}

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I stress these methods of checking answers, not just because checking answers is a generally good idea, but because they reinforce key concepts. The first method reinforces the idea that factoring is multiplication done backward . The second method reinforces the idea of algebraic generalizations.

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..
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
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
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.
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
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
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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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