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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.This module contains an exercise supplement for the chapter "Quadratic Equations".

Exercise supplement

Solving quadratic equations ( [link] ) - solving quadratic equations by factoring ( [link] )

For the following problems, solve the equations.

( x 2 ) ( x 5 ) = 0

x = 2 , 5

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( b + 1 ) ( b 6 ) = 0

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( a + 10 ) ( a 5 ) = 0

a = 10 , 5

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( y 3 ) ( y 4 ) = 0

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( m 8 ) ( m + 1 ) = 0

m = 8 , 1

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( 4 y + 1 ) ( 2 y + 3 ) = 0

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( x + 2 ) ( 3 x 1 ) = 0

x = 2 , 1 3

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( 5 a 2 ) ( 3 a 10 ) = 0

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x ( 2 x + 3 ) = 0

x = 0 , 3 2

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( y + 3 ) 2 = 0

y = 3

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16 y 2 49 = 0

y = ± 7 4

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a 2 + 6 a + 8 = 0

a = 4 , 2

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s 2 9 s + 8 = 0

s = 1 , 8

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11 y 2 = 6 y 2

y = 1 6 , 2

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3 ( y 2 8 ) = 7 y

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a ( 4 b + 7 ) = 0

a = 0 ; b = 7 4

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5 a 2 125 = 0

a = ± 5

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m 2 6 m + 5 = 0

m = 5 , 1

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x 2 + 3 x = 28

x = 7 , 4

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2 y 2 6 y = 8

y = 4 , 1

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2 r 2 = 5 3 r

r = 5 2 , 1

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Solving quadratic equations using the method of extraction of roots ( [link] )

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

s 2 10 = 0

s = ± 10

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3 y 2 = 75

y = ± 5

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Solve y 2 = 4 a 2 for y .

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Solve m 2 = 16 n 2 p 4 for m .

m = ± 4 n p 2

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Solve x 2 = 25 y 4 z 10 w 8 for x .

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Solve x 2 y 2 = 0 for y .

y = ± x

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Solve a 4 b 8 x 6 y 12 z 2 = 0 for a 2 .

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( x 2 ) 2 = 9

x = 5 , 1

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( a + 10 ) 2 = 1

a = 11 , 9

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( r 8 ) 2 = 10

r = 8 ± 10

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( a 2 ) 2 = 2

No real number solution.

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Solve ( x 2 b ) 2 = b 2 for x

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Solve ( y + 6 ) 2 = a for y .

y = 6 ± a

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Solve ( 2 a 5 ) 2 = c for a .

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Solve ( 3 m 11 ) 2 = 2 a 2 for m .

m = 11 ± a 2 3

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Solving quadratic equations using the method of completing the square ( [link] ) - solving quadratic equations using the quadratic formula ( [link] )

For the following problems, solve the equations by completing the square or by using the quadratic formula.

s 2 + 2 s 24 = 0

s = 4 , 6

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b 2 + b 8 = 0

b = 1 ± 33 2

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5 a 2 + 2 a 6 = 0

a = 1 ± 31 5

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y 2 = 2 y + 1

y = 1 ± 2

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r 2 + 2 r = 9

r = 1 ± 10

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10 x 3 + 2 x 2 22 x = 0

x = 0 , 1 ± 221 10

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6 r 3 + 6 r 2 3 r = 0

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15 x 2 + 2 x 3 = 12 x 4

x = 0 , 1 ± 181 12

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6 x 3 6 x = 6 x 2

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( x + 3 ) ( x 4 ) = 3

x = 1 ± 61 2

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( y 1 ) ( y 2 ) = 6

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( a + 3 ) ( a + 4 ) = 10

No real number solution.

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( 2 m + 1 ) ( 3 m 1 ) = 2

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( 5 r + 6 ) ( r 1 ) = 2

r = 1 ± 161 10

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4 x 2 + 2 x 3 = 3 x 2 + x + 1

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5 a 2 + 5 a + 4 = 3 a 2 + 2 a + 5

a = 3 ± 17 4

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( r 8 ) 2 = 70

r = 8 ± 70

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Applications ( [link] )

For the following problems, find the solution.

The revenue R , in dollars, collected by a certain manufacturer of inner tubes is related to the number x of inner tubes sold by R = 1400 16 x + 3 x 2 . How many inner tubes must be sold to produce a profit of $1361?

No solution.

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A study of the air quality in a particular city by an environmental group suggests that t years from now the level of carbon monoxide, in parts per million, in the air will be A = 0.8 t 2 + 0.5 t + 3.3.
(a) What is the level, in parts per million, of carbon monoxide in the air now?
(b) How many years from now will the carbon monoxide level be at 6 parts per million?

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A contractor is to pour a concrete walkway around a community garden that is 15 feet wide and 50 feet long. The area of the walkway and garden is to be 924 square feet and of uniform width. How wide should the contractor make it?

x 1.29 feet

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A ball thrown vertically into the air has the equation of motion h = 144 + 48 t 16 t 2
(a) How high is the ball at t = 0 ?
(b) How high is the ball at t = 1 ?
(c) When does the ball hit the ground?

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The length of a rectangle is 5 feet longer than three times its width. Find the dimensions if the area is to be 138 square feet.

w = 6

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The area of a triangle is 28 square centimeters. The base is 3 cm longer than the height. Find both the length of the base and the height.

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The product of two consecutive integers is 210. Find them.

x = 15 , 14 , or 14 , 15

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The product of two consecutive negative integers is 272. Find them.

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A box with no top and a square base is to be made by cutting out 3-inch squares from each corner and folding up the sides of a piece of cardboard. The volume of the box is to be 25 cubic inches. What size should the piece of cardboard be?

x = 18 + 5 3 3

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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
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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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