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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 by factoring.

Overview

  • Factoring Method
  • Solving Mentally After Factoring

Factoring method

To solve quadratic equations by factoring, we must make use of the zero-factor property.

  1. Set the equation equal to zero, that is, get all the nonzero terms on one side of the equal sign and 0 on the other.

    a x 2 + b x + c = 0
  2. Factor the quadratic expression.

    ( ) ( ) = 0
  3. By the zero-factor property, at least one of the factors must be zero, so, set each of the factors equal to 0 and solve for the variable.

Sample set a

Solve the following quadratic equations. (We will show the check for problem 1.)

x 2 7 x + 12 = 0. The equation is already  set equal to 0 . Factor . ( x 3 ) ( x 4 ) = 0 Set each factor equal to 0 . x 3 = 0 or x 4 = 0 x = 3 or x = 4
C h e c k : If x = 3 , x 2 7 x + 12 = 0 3 2 7 · 3 + 12 = 0 Is this correct? 9 21 + 12 = 0 Is this correct? 0 = 0 Yes, this is correct .

C h e c k : If x = 4 , x 2 7 x + 12 = 0 4 2 7 · 4 + 12 = 0 Is this correct? 16 28 + 12 = 0 Is this correct? 0 = 0 Yes, this is correct .
Thus, the solutions to this equation are x = 3 , 4.

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x 2 = 25. Set the equation equal to  0. x 2 25 = 0 Factor . ( x + 5 ) ( x 5 ) = 0 Set each factor equal to  0. x + 5 = 0 or x 5 = 0 x = 5 or x = 5
Thus, the solutions to this equation are x = 5 , 5.

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x 2 = 2 x . Set the equation equal to  0. x 2 2 x = 0 Factor . x ( x 2 ) Set each factor equal to  0. x = 0 or x 2 = 0 x = 2
Thus, the solutions to this equation are x = 0 , 2.

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2 x 2 + 7 x 15 = 0. Factor . ( 2 x 3 ) ( x + 5 ) = 0 Set each factor equal to  0. 2 x 3 = 0 or x + 5 = 0 2 x = 3 or x = 5 x = 3 2
Thus, the solutions to this equation are x = 3 2 , 5.

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63 x 2 = 13 x + 6
63 x 2 13 x 6 = 0 ( 9 x + 2 ) ( 7 x 3 ) = 0 9 x + 2 = 0 or 7 x 3 = 0 9 x = 2 or 7 x = 3 x = 2 9 or x = 3 7
Thus, the solutions to this equation are x = 2 9 , 3 7 .

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Practice set a

Solve the following equations, if possible.

( x 7 ) ( x + 4 ) = 0

x = 7 , 4

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( 2 x + 5 ) ( 5 x 7 ) = 0

x = 5 2 , 7 5

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x 2 + 2 x 24 = 0

x = 4 , 6

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6 x 2 + 13 x 5 = 0

x = 1 3 , 5 2

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5 y 2 + 2 y = 3

y = 3 5 , 1

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m ( 2 m 11 ) = 0

m = 0 , 11 2

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6 p 2 = ( 5 p + 1 )

p = 1 3 , 1 2

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r 2 49 = 0

r = 7 , 7

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Solving mentally after factoring

Let’s consider problems 4 and 5 of Sample Set A in more detail. Let’s look particularly at the factorizations ( 2 x 3 ) ( x + 5 ) = 0 and ( 9 x + 2 ) ( 7 x 3 ) = 0. The next step is to set each factor equal to zero and solve. We can solve mentally if we understand how to solve linear equations: we transpose the constant from the variable term and then divide by the coefficient of the variable.

Sample set b

Solve the following equation mentally.

( 2 x 3 ) ( x + 5 ) = 0
2 x 3 = 0 Mentally add 3 to both sides . The constant changes sign . 2 x = 3 Divide by 2, the coefficient of  x . The 2 divides the constant 3 into  3 2 .  The coefficient becomes the denominator . x = 3 2 x + 5 = 0 Mentally subtract 5 from both sides . The constant changes sign . x = 5 Divide by the coefficient of   x , 1 .The coefficient becomes the denominator . x = 5 1 = 5 x = 5
Now, we can immediately write the solution to the equation after factoring by looking at each factor, changing the sign of the constant, then dividing by the coefficient.

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Practice set b

Solve ( 9 x + 2 ) ( 7 x 3 ) = 0 using this mental method.

x = 2 9 , 3 7

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Exercises

For the following problems, solve the equations, if possible.

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

x = 1 , 3

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( x 5 ) ( x 1 ) = 0

x = 1 , 5

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( x 6 ) ( x 3 ) = 0

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( x 4 ) ( x + 2 ) = 0

x = 2 , 4

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( x + 6 ) ( x 1 ) = 0

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( 2 x + 1 ) ( x 7 ) = 0

x = 1 2 , 7

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

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

x = 3 4 , 2 3

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( 5 x 1 ) ( 4 x + 7 ) = 0

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( 6 x + 5 ) ( 9 x 4 ) = 0

x = 5 6 , 4 9

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( 3 a + 1 ) ( 3 a 1 ) = 0

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x ( x + 4 ) = 0

x = 4 , 0

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y ( 3 y 4 ) = 0

y = 0 , 4 3

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x ( 2 x + 1 ) ( 2 x + 8 ) = 0

x = 4 , 1 2 , 0

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y ( 5 y + 2 ) ( 2 y 1 ) = 0

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( b + 7 ) 2 = 0

b = 7

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x ( x 4 ) 2 = 0

x = 0 , 4

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y ( y 7 ) 2 = 0

y = 0 , 7

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x 2 4 = 0

x = 2 , 2

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x 2 + 36 = 0

no solution

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a 2 100 = 0

a = 10 , 10

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b 2 49 = 0

b = 7 , 7

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3 a 2 75 = 0

a = 5 , 5

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y 3 y = 0

y = 0 , 1 , 1

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b 2 = 4

b = 2 , 2

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a 2 = 36

a = 6 , 6

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2 x 2 = 4

x = 2 , 2

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7 b 2 = 63

b = 3 , 3

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3 b 2 = 48

b = 4 , 4

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y 2 + 10 y + 25 = 0

y = 5

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x 2 2 x 1 = 0

no solution

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a 2 + 4 a + 4 = 0

a = 2

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b 2 14 b = 49

b = 7

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2 m 3 + 4 m 2 + 2 m = 0

m = 0 , 1

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3 m n 2 36 m n + 36 m = 0

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a 2 + 2 a 3 = 0

a = 3 , 1

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x 2 + 9 x + 14 = 0

x = 7 , 2

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b 2 + 12 b + 27 = 0

b = 9 , 3

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x 2 13 x = 42

x = 6 , 7

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a 3 = 8 a 2 15 a

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6 a 2 + 13 a + 5 = 0

a = 5 3 , 1 2

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12 a 2 + 15 a + 3 = 0

a = 1 4 , 1

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12 a 2 + 24 a + 12 = 0

a = 1

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2 x 2 = x + 15

x = 5 2 , 3

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4 y 2 = 4 y 2

no solution

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Exercises for review

( [link] ) Simplify ( x 4 y 3 ) 2 ( x y 2 ) 4 .

x 12 y 14

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( [link] ) Write ( x 2 y 3 w 4 ) 2 so that only positive exponents appear.

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( [link] ) Find the sum: x x 2 x 2 + 1 x 2 3 x + 2 .

x 2 + 1 ( x + 1 ) ( x 1 ) ( x 2 )

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( [link] ) Simplify 1 a + 1 b 1 a 1 b .

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( [link] ) Solve ( x + 4 ) ( 3 x + 1 ) = 0.

x = 4 , 1 3

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
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
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
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
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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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