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This module discusses the quadratic formula.

In "Solving Quadratic Equations by Completing the Square" I talked about the common mathematical trick of solving a problem once, using letters instead of numbers , and then solving specific problems by plugging numbers into a general solution.

In the text, you go through this process for quadratic equations in general. The definition of a quadratic equation is any equation that can be written in the form:

ax 2 + bx + c = 0 size 12{ ital "ax" rSup { size 8{2} } + ital "bx"+c=0} {}

where a 0 size 12{a<>0} {} . By completing the square on this generic equation, you arrive at the quadratic formula:

x = b ± b 2 4 ac 2a size 12{x= { { - b +- sqrt {b rSup { size 8{2} } - 4 ital "ac"} } over {2a} } } {}

This formula can then be used to solve any quadratic equation, without having to complete the square each time. To see how this formula works, let us return to the previous problem:

9x 2 54 x + 80 = 0 size 12{9x rSup { size 8{2} } - "54"x+"80"=0} {}

In this case, a = 9 size 12{a=9} {} , b = 54 size 12{b= - "54"} {} , and c = 80 size 12{c="80"} {} . So the quadratic formula tells us that the answers are:

x = ( 54 ) ± ( 54 ) 2 4 ( 9 ) ( 80 ) 2 ( 9 ) size 12{x= { { - \( - "54" \) +- sqrt { \( - "54" \) rSup { size 8{2} } - 4 \( 9 \) \( "80" \) } } over {2 \( 9 \) } } } {}

We’ll use a calculator here rather than squaring 54 by hand....

x = 54 ± 2916 2880 18 = 54 ± 36 18 = 54 ± 6 18 = 9 ± 1 3 size 12{x= { {"54" +- sqrt {"2916" - "2880"} } over {"18"} } = { {"54" +- sqrt {"36"} } over {"18"} } = { {"54" +- 6} over {"18"} } = { {9 +- 1} over {3} } } {}

So we find that the two answers are 10 3 size 12{ { {"10"} over {3} } } {} and 8 3 size 12{ { {8} over {3} } } {} , which are the same answers we got by completing the square.

Using the quadratic formula is usually faster than completing the square, though still slower than factoring. So, in general, try to factor first: if you cannot factor, use the quadratic formula.

So why do we learn completing the square? Two reasons. First, completing the square is how you derive the quadratic formula. Second, completing the square is vital to graphing quadratic functions, as you will see a little further on in the chapter.

Questions & Answers

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
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
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
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
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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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