<< 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

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
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
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
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