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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 )} {}

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 )} {}

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 )} {}

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.”

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 )} {}

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} } } {}

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 )} {}

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"} {}

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

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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