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This module is from Elementary Algebra</link> by Denny Burzynski and Wade Ellis, Jr. Methods of solving quadratic equations as well as the logic underlying each method are discussed. Factoring, extraction of roots, completing the square, and the quadratic formula are carefully developed. The zero-factor property of real numbers is reintroduced. The chapter also includes graphs of quadratic equations based on the standard parabola, y = x^2, and applied problems from the areas of manufacturing, population, physics, geometry, mathematics (numbers and volumes), and astronomy, which are solved using the five-step method.Objectives of this module: be able to solve quadratic equations using the method of extraction of roots, be able to determine the nature of the solutions to a quadratic equation.

Overview

  • The Method Of Extraction Of Roots
  • The Nature Of Solutions

The method of extraction of roots

Extraction of roots

Quadratic equations of the form x 2 K = 0 can be solved by the method of extraction of roots by rewriting it in the form x 2 = K .

To solve x 2 = K , we are required to find some number, x , that when squared produces K . This number, x , must be a square root of K . If K is greater than zero, we know that it possesses two square roots, K and K . We also know that

( K ) 2 = ( K ) ( K ) = K and ( K ) 2 = ( K ) ( K ) = K

We now have two replacements for x that produce true statements when substituted into the equation. Thus, x = K and x = K are both solutions to x 2 = K . We use the notation x = ± K to denote both the principal and the secondary square roots.

The nature of solutions

Solutions of x 2 = K

For quadratic equations of the form x 2 = K ,

  1. If K is greater than or equal to zero, the solutions are ± K .
  2. If K is negative, no real number solutions exist.
  3. If K is zero, the only solution is 0.

Sample set a

Solve each of the following quadratic equations using the method of extraction of roots.

x 2 49 = 0. Rewrite . x 2 = 49 x = ± 49 x = ± 7 C h e c k : ( 7 ) 2 = 49 Is this correct? ( 7 ) 2 = 49 Is this correct 49 = 49 Yes, this is correct . 49 = 49 Yes, this is correct .

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25 a 2 = 36 a 2 = 36 25 a = ± 36 25 a = ± 6 5
C h e c k : 25 ( 6 5 ) 2 = 36 Is this correct? 25 ( 6 5 ) 2 = 36 Is this correct? 25 ( 36 25 ) 2 = 36 Is this correct? 25 ( 36 25 ) = 36 Is this correct? 36 = 36 Yes, this is correct . 36 = 36 Yes, this is correct .

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4 m 2 32 = 0 4 m 2 = 32 m 2 = 32 4 m 2 = 8 m = ± 8 m = ± 2 2
C h e c k : 4 ( 2 2 ) 2 = 32 Is this correct? 4 ( 2 2 ) 2 = 32 Is this correct? 4 [ 2 2 ( 2 ) 2 ] = 32 Is this correct? 4 [ ( 2 ) 2 ( 2 ) 2 ] = 32 Is this correct? 4 [ 4 · 2 ] = 32 Is this correct? 4 [ 4 · 2 ] = 32 Is this correct? 4 · 8 = 32 Is this correct? 4 · 8 = 32 Is this correct? 32 = 32 Yes, this is correct . 32 = 32 Yes, this is correct .

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Solve 5 x 2 15 y 2 z 7 = 0 for x .
5 x 2 = 15 y 2 z 7 Divide both sides by 5 . x 2 = 3 y 2 z 7 x = ± 3 y 2 z 7 x = ± y z 3 3 z

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Use a calculator. Calculator problem.  Solve 14 a 2 235 = 0. Round to the nearest hundredth.
14 a 2 235 = 0. Rewrite . 14 a 2 = 235 Divide both sides by 14 . a 2 = 235 14
On the Calculator
Type 235 Press ÷ Type 14 Press = Press Display reads: 4.0970373
Rounding to the nearest hundredth produces 4.10. We must be sure to insert the ± symbol. a ± 4.10

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k 2 = 64 k = ± 64
The radicand is negative so no real number solutions exist.

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Practice set a

Solve each of the following quadratic equations using the method of extraction of roots.

x 2 144 = 0

x = ± 12

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9 y 2 121 = 0

y = ± 11 3

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Solve 4 n 2 = 24 m 2 p 8 for n .

n = ± m p 4 6

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Solve 5 p 2 q 2 = 45 p 2 for q .

q = ± 3

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Use a calculator. Solve 16 m 2 2206 = 0. Round to the nearest hundredth.

m = ± 11.74

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Sample set b

Solve each of the following quadratic equations using the method of extraction of roots.

( x + 2 ) 2 = 81 x + 2 = ± 81 x + 2 = ± 9 Subtract  2  from both sides . x = 2 ± 9 x = 2 + 9 and x = 2 9 x = 7 x = 11

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( a + 3 ) 2 = 5 a + 3 = ± 5 Subtract 3 from both sides . a = 3 ± 5

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Practice set b

Solve each of the following quadratic equations using the method of extraction of roots.

( a + 6 ) 2 = 64

a = 2 , 14

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( m 4 ) 2 = 15

m = 4 ± 15

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( y 7 ) 2 = 49

y = 0 , 14

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( k 1 ) 2 = 12

k = 1 ± 2 3

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( x 11 ) 2 = 0

x = 11

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Exercises

For the following problems, solve each of the quadratic equations using the method of extraction of roots.

a 2 8 = 0

a = ± 2 2

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x 2 10 = 0

x = ± 10

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3 x 2 27 = 0

x = ± 3

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For the following problems, solve for the indicated variable.

x 2 = 9 b 2 , for x

x = ± 3 b

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k 2 = m 2 n 2 , for k

k = ± m n

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k 2 = p 2 q 2 r 2 , for k

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2 y 2 = 2 a 2 n 2 , for y

y = ± a n

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9 y 2 = 27 x 2 z 4 , for y

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x 2 z 2 = 0 , for x

x = ± z

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5 a 2 10 b 2 = 0 , for a

a = b 2 , b 2

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For the following problems, solve each of the quadratic equations using the method of extraction of roots.

( x 2 ) 2 = 9

x = 5 , 1

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( a 5 ) 2 = 36

x = 11 , 1

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( a + 9 ) 2 = 1

a = 8 , 10

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( x + 4 ) 2 = 5

a = 4 ± 5

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( x + 1 ) 2 = a , for x

x = 1 ± a

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( y + 2 ) 2 = a 2 , for y

y = 2 ± a

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( x + 10 ) 2 = c 2 , for x

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( x a ) 2 = b 2 , for x

x = a ± b

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( x + c ) 2 = a 2 , for x

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Use a calculator.  calculator problems

For the following problems, round each result to the nearest hundredth.

8 a 2 168 = 0

a = ± 4.58

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0.03 y 2 = 1.6

y = ± 7.30

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1.001 x 2 0.999 = 0

x = ± 1.00

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Exercises for review

( [link] ) Graph the linear inequality 3 ( x + 2 ) < 2 ( 3 x + 4 ) .

A horizontal line with arrows on both ends.

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( [link] ) Solve the fractional equation x 1 x + 4 = x + 3 x 1 .

x = 11 9

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( [link] ) Find the product: 32 x 3 y 5 2 x 3 y 3 .

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( [link] ) Solve x 2 4 x = 0.

x = 0 , 4

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( [link] ) Solve y 2 8 y = 12.

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Questions & Answers

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
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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
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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