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

Consider the equation:

x + 3 2 = 16 size 12{ left (x+3 right ) rSup { size 8{2} } ="16"} {}

We can solve this by analogy to the way that we approached absolute value problems. something squared is 16. So what could the something be? It could be 4. It could also be 4 size 12{ - 4} {} . So the solution is:

  • x + 3 = 4 size 12{x+3=4} {}
    • x = 1 size 12{x=1} {}
  • x + 3 = 4 size 12{x+3= - 4} {}
    • x = 7 size 12{x= - 7} {}

These are the two solutions.

This simple problem leads to a completely general way of solving quadratic equations—because any quadratic equation can be put in a form like the above equation. The key is completing the square which, in turn, is based on our original two formulae:

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} } } {}

As an example, consider the equation x 2 + 10 x + 21 = 0 size 12{x rSup { size 8{2} } +"10"x+"21"=0} {} . In order to make it fit one of the patterns above, we must replace the 21 with the correct number: a number such that x 2 + 10 x + __ size 12{x rSup { size 8{2} } +"10"x+"__"} {} is a perfect square. What number goes there? If you are familiar with the pattern, you know the answer right away. 10 is 5 doubled , so the number there must be 5 squared , or 25.

But how do we turn a 21 into a 25? We add 4, of course. And if we add 4 to one side of the equation, we have to add 4 to the other side of the equation. So the entire problem is worked out as follows:

Solving by Completing the Square (quick-and-dirty version)
Solve x 2 + 10 x + 21 = 0 size 12{x rSup { size 8{2} } +"10"x+"21"=0} {} The problem.
x 2 + 10 x + 25 = 4 size 12{x rSup { size 8{2} } +"10"x+"25"=4} {} Add 4 to both sides, so that the left side becomes a perfect square.
x + 5 2 = 4 size 12{ left (x+5 right ) rSup { size 8{2} } =4} {} Rewrite the perfect square.
x + 5 = 2 x = 3 size 12{ matrix { x+5=2 {} ##x= - 3 } } {} x + 5 = 2 x = 7 size 12{ matrix { x+5= - 2 {} ##x= - 7 } } {} If something-squared is 4, the something can be 2, or 2 size 12{ - 2} {} . Solve both possibilities to find the two answers.

Got questions? Get instant answers now!

Thus, we have our two solutions.

Completing the square is more time-consuming than factoring: so whenever a quadratic equation can be factored, factoring is the preferred method. (In this case, we would have factored the original equation as x + 3 x + 7 size 12{ left (x+3 right ) left (x+7 right )} {} and gotten straight to the answer.) However, completing the square can be used on any quadratic equation. In the example below, completing the square is used to find two answers that would not have been found by factoring.

Solving by Completing the Square (showing all the steps more carefully)
Solve 9x 2 54 x + 80 = 0 size 12{9x rSup { size 8{2} } - "54"x+"80"=0} {} The problem.
9x 2 54 x = 80 size 12{9x rSup { size 8{2} } - "54"x= - "80"} {} Put all the x size 12{x} {} terms on one side, and the number on the other
x 2 6x = 80 9 size 12{x rSup { size 8{2} } - 6x= - { {"80"} over {9} } } {} Divide both sides by the coefficient of x 2 size 12{x rSup { size 8{2} } } {} (*see below)
x 2 6x + 9 ̲ = 80 9 + 9 ̲ size 12{x rSup { size 8{2} } - 6x {underline {+9}} = - { {"80"} over {9} } + {underline {9}} } {} Add the same number (*see below) to both sides, so that the left side becomes a perfect square.
x 3 2 = 80 9 + 81 9 = 1 9 size 12{ left (x - 3 right ) rSup { size 8{2} } = - { {"80"} over {9} } + { {"81"} over {9} } = { {1} over {9} } } {} Rewrite the perfect square.
x 3 = 1 3 x = 3 1 3 size 12{ matrix { x - 3= { {1} over {3} } {} ##x=3 { { size 8{1} } over { size 8{3} } } } } {} x 3 = 1 3 x = 2 2 3 size 12{ matrix { x - 3= - { {1} over {3} } {} ##x=2 { { size 8{2} } over { size 8{3} } } } } {} If something-squared is 1 9 size 12{ { {1} over {9} } } {} , the something can be 1 3 size 12{ { {1} over {3} } } {} , or 1 3 size 12{ - { {1} over {3} } } {} . Solve both possibilities to find the two answers.

Got questions? Get instant answers now!

Two steps in particular should be pointed out here.

In the third step, we divide both sides by 9. When completing the square, you do not want to have any coefficient in front of the term; if there is a number there, you divide it out. Fractions, on the other hand (such as the 80 9 size 12{ - { {"80"} over {9} } } {} in this case) do not present a problem. This is in marked contrast to factoring, where a coefficient in front of the x 2 size 12{x rSup { size 8{2} } } {} can be left alone, but fractions make things nearly impossible.

The step after that is where we actually complete the square. x 2 + 6x + __ size 12{x rSup { size 8{2} } +6x+"__"} {} will be our perfect square. How do we find what number we want? Start with the coefficient of x 2 size 12{x rSup { size 8{2} } } {} (in this case, 6). Take half of it, and square the result . Half of 6 is 3, squared is 9. So we want a 9 there to create x 2 + 6x + 9 size 12{x rSup { size 8{2} } +6x+9} {} which can be simplified to x + 3 2 size 12{ left (x+3 right ) rSup { size 8{2} } } {} .

An image showing how to select the correct number and position it within the equation correctly when completing the square.

If the coefficient of x size 12{x} {} is an odd number , the problem becomes a little uglier, but the principle is the same. For instance, faced with:

x 2 + 5x + __ size 12{x rSup { size 8{2} } +5x+"__"} {}

You would begin by taking half of 5 (which is 5 2 size 12{ { {5} over {2} } } {} ) and then squaring it:

x 2 + 5x + 25 4 = x + 5 2 2 size 12{x rSup { size 8{2} } +5x+ { {"25"} over {4} } = left (x+ { {5} over {2} } right ) rSup { size 8{2} } } {}

Another “completing the square” example, in which you cannot get rid of the square root at all, is presented in the worksheet “The Generic Quadratic Equation.”

One final note on completing the square: there are three different possible outcomes.

  • If you end up with something like x 3 2 = 16 size 12{ left (x - 3 right ) rSup { size 8{2} } ="16"} {} you will find two solutions, since x 3 size 12{x - 3} {} can be either 4, or 4 size 12{ - 4} {} . You will always have two solutions if the right side of the equation, after completing the square, is positive.
  • If you end up with x 3 2 = 0 size 12{ left (x - 3 right ) rSup { size 8{2} } =0} {} then there is only one solution: x must be 3 in this example. If the right side of the equation is 0 after completing the square, there is only one solution.
  • Finally, what about a negative number on the right, such as x 3 2 = 16 size 12{ left (x - 3 right ) rSup { size 8{2} } = - "16"} {} ? Nothing squared can give a negative answer, so there is no solution.

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, 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